Involutions in Fischer’s sporadic groups
Paul Taylor
May 18, 2011
Abstract
For each of Fischer’s sporadic simple groups and their automorphism groups, we catalogue the suborbits of the conjugation action on
each of the involution conjugacy classes. Representative elements for
the suborbits are provided, where possible as words in the standard
generators and elsewhere they are given electronically as base images.
Additionally we use this data to determine the structure of the commuting involution graphs on the two largest involution classes in F i24 .
1
Introduction
The three sporadic groups F i22 , F i23 and F i240 arose from Fischer’s classification of the 3-transposition groups. The defining property of such a group
is that it is generated by a conjugacy class of involutions (usually referred to
as the ‘transpositions’ by analogy with the symmetric groups), products of
any two of which have order not exceeding three. Fischer’s classification of
these groups gave rise to three exceptional examples, now named F i22 , F i23
and F i24 , the first two of which were also new sporadic simple groups, as
was the derived subgroup F i240 of the third. The groups were constructed
by considering the automorphism group of the graph which has the class of
transpositions as its vertices; a pair of vertices joined by an edge if and only
if they commute. The involutions in these classes are therefore obviously
well understood. But the groups have other involution classes besides the
transpositions, so we might wish to investigate these as well.
An obvious question is then: what is the structure of the equivalent
graphs on the other classes of involutions? This type of graph is known
as a ‘commuting involution graph’ and has in recent years been studied
for a wide variety of groups (see, for example, [1], [2], [6]). In particular,
in [3] the graph structure was determined for all involution classes in the
Fischer groups save two largest classes of F i24 . We determine the structure
of these outstanding cases as part of the present work: that is, we discover
the diameters of the graphs as well as the sizes of the disks of vertices at a
given distance from a fixed vertex.
1
Study of the commuting involution graph leads naturally to consideration of the suborbit structure of the class, that is the orbits under the action
of a point stabilizer: CG (t) for t an element from the relevant class of involutions. This is because, once an involution t ∈ X is fixed, the set of vertices
at a given distance from it in the graph is a union of these suborbits. It
is the cataloguing of the CG (t)-orbits that is the main focus of the present
work.
Throughout this paper, G denotes one of the groups F i22 , F i23 , F i240
or one of their automorphism groups F i22 : 2 and F i24 (F i23 having no
outer automorphisms). Also, X is a G-conjugacy class of involutions. In the
online Atlas [11], generators for these groups are provided in GAP [7] and
Magma [4] formats. We employ the latter package for our computation.
Atlas [5] conventions are adopted for the names of conjugacy classes.
For each pair (G, X) under consideration we fix an element t ∈ X, and
then we wish to determine the size of, and find a representative element
x ∈ X for, each CG (t)-orbit on X. For all but the pairs (F i240 , 2B) and
(F i24 , 2D) we give these representatives in terms of words in the generators
a, b (or c, d for an automorphism group), these words giving us elements
g that conjugate t to the desired representatives. The online Atlas uses
standard generators [10] so these words can be used to derive explicit representatives in any representation of the group.
In the cases (F i240 , 2B), (F i24 , 2D) we give representatives as base images
for a specified base and strong generating set, using the smallest-degree
permutation representations of these groups available in [11]. We also in
these two cases uncover the structure of the commuting involution graph
C(G, X).
In the following section we summarise the main results. Section 3 describes how these were obtained, and Section 4 tabulates the full data on
the CG (t)-orbits.
This work also appears in the author’s PhD thesis [9].
2
Results
For each of the involution conjugacy classes in the Fischer sporadic groups
and their automorphism groups, Table 1 gives the number of suborbits and
the number of sets XC into which they fall—these sets are defined in the
following section. We also outline the structure of the commuting involution
graphs for (F i240 , 2B) and (F i24 , 2D). In Section 4 we provide tables listing
the full details of the CG (t)-orbits for all thirteen pairs (G, X) except for
the classes of transpositions: 2A in F i22 and F i23 and class 2C in F i24 .
2
G
F i22
F i22 : 2
F i23
F i240
F i24
X
2A
2B
2C
2D
2E
2F
2A
2B
2C
2A
2B
2C
2D
Suborbits
3
14
136
4
56
74
3
12
303
13
233
3
232
Nonempty XC
3
11
57
4
31
38
3
10
92
11
104
3
84
Table 1: Involution suborbits in the Fischer groups
2.1
Commuting Involution Graphs
Knowledge of the CG (t)-orbits allows us to easily compute the structure
of the commuting involution graphs for the two previously unknown cases,
which we summarise here in terms of their disk structures. For our fixed
t ∈ X and i ∈ N, the ith disk is defined as ∆i (t) = {x ∈ X | d(t, x) = i}
where d(−, −) is the usual distance metric on C(G, X). A simple argument
demonstrates that the graph is vertex-transitive so that the disk sizes are
the same whatever t ∈ X we choose, and that once t is fixed each of the
discs ∆i (t) is a disjoint union of CG (t)-orbits.
Theorem 1. Let C(G, X) be the commuting involution graph of G ∼
= F i240
on X = 2B. Then for t ∈ X,
• ∆0 (t) = {t} has size 1 and is composed of one CG (t)-orbit.
• ∆1 (t) has size 3,324,762 and is composed of six CG (t)-orbits.
• ∆2 (t) has size 3,755,093,739,776 and is composed of one hundred and
seventy-four CG (t)-orbits.
• ∆3 (t) has size 4,064,208,224,256 and is composed of fifty-two CG (t)orbits.
The CG (t)-orbits comprising the graph are as detailed in Table 10.
Theorem 2. Let C(G, X) be the commuting involution graph of G ∼
= F i24
on X = 2D. Then for t ∈ X,
• ∆0 (t) = {t} has size 1 and is composed of one CG (t)-orbit.
3
• ∆1 (t) has size 3,682,503 and is composed of six CG (t)-orbits.
• ∆2 (t) has size 822,139,288,316 and is composed of one hundred and
ninety-one CG (t)-orbits.
• ∆3 (t) has size 2,021,240,770,560 and is composed of thirty-six CG (t)orbits.
The CG (t)-orbits comprising the graph are as detailed in Table 11.
3
Computation
All computation is carried out in the smallest-degree permutation representation of the relevant group available in [11]. In these representations
the necessary computations (testing for CG (t)-conjugacy and computing
the order of CCG (t) (x) for a representative x ∈ X to find the orbit size)
are straightforward.
Our initial strategy is to build up a list of CG (t)-orbit representatives
by random search. We take a random element g ∈ G and form x = tg ∈ X.
This element is tested for CG (t)-conjugacy against all elements in the list
thus far. If it is found to be a representative for a new orbit, its size is
calculated.
However, given the size of the groups at hand and the number of suborbits to be located, this strategy would take a long time to find every suborbit.
The following tricks are employed to speed up the search.
3.1
Structure Constants
Definition 3. For G, X and t as above, and C a further G-conjugacy class,
we define
XC = {x ∈ X | tx ∈ C}.
It is easy to see that for any class C of G, the set XC is a (possibly
empty) union of CG (t)-orbits on X. Further, the values of |XC | are readily
calculated from the character table (for instance using the GAP function
ClassMultiplicationCoefficient). So we can partition X into the subsets XC and consider each one in turn, knowing from their sizes when we
have found all the CG (t) orbits that comprise such a set.
In practice it is often not a simple matter to quickly determine the class
of a given z = tx, so this information cannot be used directly. However it is
obviously simple to determine the order of z so we can at least check when
we have found all the suborbits with a given order of z. Then if a random
x ∈ X is formed later having z of that order, we can discard it without
having to carry out any CG (t)-conjugacy testing.
4
Once all the suborbits have been located, we determine the class of
z = tx for every representative x. In Section 4, the suborbits are listed
grouped according to which set XC they are contained in.
3.2
Powering Suborbits
We note the following observation.
Lemma 4. Let x ∈ XC , so that z = tx ∈ C and suppose n is a divisor of
the order of z. Let D be the conjugacy class containing z n . We define the
element x(n) = tz n , and note the following properties.
(i) x(n) ∈ XD ;
(ii) if x, y ∈ X are CG (t)-conjugate then so are x(n) and y (n) ; and
(iii) All CG (t)-conjugates of x(n) arise as y (n) for some CG (t)-conjugate y
of x.
Proof. We see that x(n) = t(tx)n = xtx . . . tx, so clearly x(n) ∈ X, and since
tx(n) = z n ∈ D we obtain (i). From the definition it is immediate that
if h ∈ CG (t) conjugates x to y then it also conjugates x(n) to y (n) , giving
(ii). Similarly if (x(n) )h = y for h ∈ CG (t) then (xh )(n) = y and so we get
(iii).
When we have found a representative x for a new suborbit, we form and
check x(n) for each n dividing the order of z. This allows us much more
easily to alight on suborbits which are small and so hard to find by random
searching, but which can be ‘powered down’ to in this manner from much
larger orbits. For example, when G ∼
= F i22 and X is the class 2C, there is
a suborbit in X2C of size 9, which we would be very unlikely to find in a
random search, however, a representative is easily found as x(4) for x in an
orbit of size 55296 contained in X8A .
3.3
Suborbit invariants
While CG (t)-conjugacy testing in the groups we consider is possible, it is
relatively computationally expensive. So to minimize its use we compute
for each x ∈ X considered two suborbit invariants: the cycle type of z = tx
and the orbit sizes of ht, xi. Elements having different invariants are not
CG (t)-conjugate so we avoid some extra computation.
3.4
Inverse suborbits
Our suborbit representatives are formed by taking random elements g ∈ G
−1
and setting x = tg . Now consider the element x0 = tg . We note the
following properties:
5
−1
Lemma 5. Let t ∈ X, g ∈ G. Set x = tg and x0 = tg . Then
(i) the CG (t)-orbits containing x and x0 are of the same size;
(ii) x and x0 lie in the same XC ; and
(iii) if y = th is CG (t)-conjugate to x then y 0 = th
x0 .
−1
is CG (t)-conjugate to
Proof. We have that CCG (t) (x) = CG (t) ∩ CG (tg ) and CCG (t) (x0 ) = CG (t) ∩
−1
−1
CG (tg ), so clearly CCG (t) (x)g = CCG (t) (x0 ). Then |CCG (t) (x)| = |CCG (t) (x0 )|
and so their orbits have the same size, giving (i). For (ii), we note that
−1
(tx)g t = tx0 . Now let c ∈ CG (t) so that xc = y. So tgc = th , that is,
−1
−1
−1
−1
−1
gch−1 ∈ CG (t). Then we see that x0(gch ) = tg (gch ) = tch = th = y 0 ,
giving (iii).
So when we find a representative x of a new CG (t)-orbit, we can form the
element x0 , and we need only to check it for CG (t)-conjugacy against x to see
if it represents a new suborbit (if it were in a previously discovered suborbit
then when that orbit were found we would have uncovered the orbit xCG (t)
by the same process). In some of the classes we consider a large number of
the CG (t)-orbits fall into pairs of ‘inverse’ orbits in this manner so use of
this technique can save much computational time.
4
Suborbit tables
In each of Tables 2–9, each row corresponds to a CG (t)-orbit (note that a
row may extend to more than one line if there is a long entry in the third
column; each orbit size listed in the second column marks the start of a new
row). The first column gives the class C of G for which the suborbit lies in
XC . The second column gives the size of the suborbit and the third gives
information on how to obtain a representative element for that suborbit.
This is either a word in the standard generators of G giving an element g
so that tg is a suborbit representative; or a list of symbols Ci0 or Di for D
a class of G: the former meaning that the suborbit is the inverse of the ith
listed suborbit in XC and the latter meaning the suborbit can be powered
to from the ith listed suborbit in XD .
For Tables 10 and 11, covering the two larger examples (F i240 , 2B) and
(F i24 , 2D), words for representative elements are not provided. Instead,
these tables simply list the sizes of the CG (t)-orbits, and which disk of the
commuting involution graph C(G, X) each suborbit lies in. Where two or
more suborbits in a particular XC have the same size and lie in the same
disk, their entries are collapsed into one row of the table. This is denoted
by an entry of the form C1,2 in the first column. Representative elements
for these groups may be downloaded from
6
http://www.maths.manchester.ac.uk/~ptaylor/Fi24/,
where they are stored as base images (relative to a base which is also
provided). We also provide there a partial list of CG (t)-orbit representatives
for the class 2D in the group G ∼
= 3.F i24 . This class is of interest because
G occurs as a maximal subgroup in the Monster sporadic group M, hence
these suborbits can be used to investigate the so-called ‘Monster graph’ [8].
7
Table 2: G ∼
= F i22 , X = 2B, t = (ababab3 )12 .
C
1A
2B
2C
3A
3C
4A
4D
4E
5A
6D
6I
Orbit size
1
270
360
1152
4320
1024
40960
34560
138240
69120
69120
442368
46080
368640
Representative
4A1
4E1 6D1
4E2 6I1
4D1
6D1
6I1
babab3 a
b2 a
aba
4E10
ba
b2
b
Table 3: G ∼
= F i22 , X = 2C, t = ((ababab3 )2 ab3 abab3 )6 .
C
1A
2A
Orbit size
1
48
2B
9
216
288
576
2C
432
432
432
432
3456
768
1536
8192
12288
24576
98304
1152
9216
13824
6912
6912
13824
13824
6912
6912
27648
3A
3B
3C
3D
4A
4B
4C
Representative
t
6A1 6A2 6B1 6E1 6E2 10A1 10A2 14A1
14A2
4A1 4C2 4E1 4E4
4A3 4B1 4C1 4E5 6D1 6D2
4B3 4E7 4E1 0 6D4 6I1 10B1
4A2 4B2 4B4 4C3 4E2 4E3 4E6 4E8 4E9
6C1 6D3 6D5 6I2 6I3 10B2
2C20 6F1
2C10 6F2
b5 ab4 ababababababab4
4D1 4D2 4D3
4D4 6F3 6F4 6G1 6H1 6J1 6J2 6K1
6A1 6D2 6D3 6D4 6F2 6F4 15A1
6A2 6D1 6D5 6F1 6F3 15A2 21A1
6B1 6C1 6G1 9A1 9B1 9B2
6E2 6I3 6J2
6E1 6H1 6I1 6I2 6J1
6K1 9C1
8A2 8B1
8A1 8A3 8B2 8B4 12A1 12A2 12C5
8A4 8B3 12C1 12C2 12C3 12C4
4B20 12D2
4B10 12D1
4B40 12D3 12J1 20A2
4B30 12D4 12J2 20A1
ababab4 ababab2
4C10
8C1 12H1 12H2
8
4D
13824
13824
13824
110592
4E
1152
1152
6912
6912
13824
13824
13824
13824
27648
27648
147456
221184
9216
9216
24576
73728
13824
13824
18432
18432
55296
73728
73728
13824
13824
55296
55296
221184
221184
73728
73728
110592
221184
221184
884736
294912
884736
55296
55296
110592
110592
18432
18432
110592
5A
6A
6B
6C
6D
6E
6F
6G
6H
6I
6J
6K
7A
8A
8B
8D3
8D2
8D5
8D1 8D4 8D6 12E1 12E2 12F1 12F2
12G1 12G2 12K1 12K2
b2 ab6 abab
4E10
4E40
b3 ab2 ab4 ab2 a
4E60 12I2
4E50 12I1
abab6 abab2
4E70
4E1 00 12I4
4E90 12I3
10A2 10B1 15A1
10A1 10B2 15A2
6A02 30A1
6A01 30A2
18C1 18C2
12A1 12A2 12H1 12H2 18D1
12C1 12C3 12D2 12I2
12C2 12C4
12D1 12I4
12D3 12I3
12C5 12D4 12I1
6E20
bab6
b5 aba
6F10
b5 ab
6F30
12E1 12E2 12F1 12F2
12G1 12G2
12J1
ab2 ab3 a
12J2
6J20
ababab4 a
12K1 12K2
14A1
14A2 21A1
b2 ab6
8A01
8A04 24B1
8A03 24B2
ab2 ab7
8B10
8B40 24A2
9
8C
8D
9A
9B
9C
10A
10B
12A
12C
12D
12E
12F
12G
12H
12I
12J
12K
13A
13B
14A
15A
110592
221184
221184
221184
221184
221184
221184
221184
294912
294912
884736
1769472
442368
442368
442368
442368
73728
73728
110592
110592
110592
110592
221184
110592
110592
221184
221184
442368
442368
442368
442368
442368
442368
221184
221184
110592
110592
221184
221184
442368
442368
884736
884736
1769472
1769472
884736
884736
884736
884736
8B30 24A1
ab2 ab3 ab2
8D50
ab3 ab2
8D60
8D20
abab2 a
b3 ab6 a
18D1
18C2
18C1
baba
10A02 30A2
10A01 30A1
20A2
20A1
ab3 ab2 ab
12A01
ab5 aba
12C10
24A2
24B2
24A1 24B1
12D20
ab3 a
12D40
b2 abab2 a
12E20
b
12F20
ba
bab5
12G01
ab4 a
bab2 abab
12I20
b5 abab2 a
12I40
b2 abab4
12J20
bab3 ab
ab
12K10
b5 a
ab3
14A02
b4 a
30A1
30A2
10
18C
18D
20A
21A
24A
24B
30A
884736
884736
884736
884736
884736
1769472
442368
442368
442368
442368
884736
884736
abab3 a
18C10
b4
bab
20A01
aba
24A02
b6
24B20
abab
bab2
30A01
Table 4: G ∼
= F i22 : 2, X = 2D, t = d9 .
C
1A
2B
3C
4A
Orbit size
1
1575
22400
37800
Representative
t
4A1
cdcdc
cd4 c
Table 5: G ∼
= F i22 : 2, X = 2E, t = (d6 cdc)9 .
C
1A
2B
Orbit size
1
27
540
1080
2C
3A
3B
3C
3240
2304
5120
5760
11520
2160
3240
17280
25920
51840
103680
8640
8640
51840
51840
27648
414720
138240
34560
34560
103680
4A
4C
4D
4E
5A
6C
6D
Representative
t
4A1 4A2 4E2
4C1 4E4 6D2 6I3 10B3
4A3 4C2 4E1 4E3 6C1 6D1 6D3 6I1 6I2
6I4 10B1 10B2
4D1 6H1
6D1 6D2 6D3 15A1
6C1 9A1 9B1
6I2
6H1 6I1 6I3 6I4
8B2 8B4
8A1
8A2 8B1 8B3 12A1 12A2 12B1
8C1
8C2 12G1
12F1
4E20
d2 cdcd5 c
4E40 12H1
4E30 12H2
10B1
10B2 10B3 15A1
12A1 12A2 12G1 18C1
12H1
12H2
12B1
11
6H
6I
7A
8A
8B
8C
9A
9B
10B
11A
12A
12B
12F
12G
12H
15A
16A
18C
103680
103680
103680
138240
138240
552960
103680
103680
34560
34560
103680
103680
414720
414720
552960
552960
414720
414720
829440
3317760
138240
138240
414720
829440
829440
414720
414720
1658880
1658880
1658880
1658880
12F1
6I20
cd
cd8 c
cdcd
d2 cd3
cd2 cdcd2
8A01
d2 cd4 cdcd3
8B10
8B40
d5 cd2
16A2
16A1
18C1
cd3 c
10B20
cd2 cd
dcdcdcd3 c
dcd2 cd
d2 c
12A01
cd3
d2 cd2 cdc
d2
dcdcd
12H10
d3
d2 cd
16A01
dcd
Table 6: G ∼
= F i22 : 2, X = 2F , t = (d3 c)15 .
C
1A
2B
Orbit size
1
63
315
945
2C
3A
3780
56
240
2520
4480
2240
20160
53760
3780
11340
3B
3C
3D
4A
Representative
t
4E1 6I1
4A1 4A2 4E2 4E4 6D1 6D2 6D5
4A3 4C1 4E3 6C1 6D3 6D4 6D6 6I2 6I3
6I4 10B1
4D1 6F1 6F2 6H1 6H2 6K1
6D1 6D4
6D2 6F1
6D3 6D5 6D6 6F2 15A1 21A1
6C1 9A1 9A2 9B1
6H1 6I1 6I3
6H2 6I2 6I4
6K1 9C1
8B2 12B1 12C1 12C2
8A2 12B2 12C3 12C4
12
4C
4D
4E
5A
6C
6D
6F
6H
6I
6K
7A
8A
8B
8C
9A
9B
9C
10B
11A
12A
12B
12C
12G
12H
15120
45360
60480
15120
15120
45360
45360
241920
120960
2520
15120
15120
15120
22680
90720
60480
60480
60480
60480
20160
60480
60480
181440
483840
1451520
181440
181440
181440
181440
362880
80640
725760
161280
967680
725760
2903040
120960
120960
60480
181440
362880
30240
30240
90720
90720
725760
181440
181440
181440
8A1 8B1 12A1 12A2 12B3
8C1 12G1
12J1 12J2
d2 cd3 cd3 cdcd3 cd
4E10
4E40 12H2 12H3
4E30 12H1 12H4
10B1 15A1
12A1 12A2 12G1 18C1 18C2
12C1 12H4
12B1 12C2 12C3
d2 cd4 cdcd3 cd2
6D30
12B2 12C4 12H1
12B3 12H2 12H3
6F20
d2 cd4 cd2 cd2 cd
6H20
d2 cd4 cdcd
dcd2 cd2 cd
dcd2 cdcd2 cdcd
6I20
d2 cd6 cd
12J1 12J2
21A1
8A02 24B2
8A01 24B1
8B20 24A2
8B10 24A1
d2 cdcd4 cd
18C2
18C1
dcdcdcd2 cdcdcd
dcdcdcdcdcd
dcd
d2 cdcd2
dcd5 cd4 cd
12A01
24A1
24B1
24A2 24B2
dcd7 cd3 cdcd
12C10
dcdcd4 cdcdcd
12C30
d2 cd5 cd2 cd
d2 cdcd2 cd3 cd
12H10
12H40
13
12J
15A
18C
21A
24A
24B
181440
483840
483840
1451520
725760
725760
2903040
725760
725760
725760
725760
dcd2 cdcd
dcd2 cd2 cdcd
12J10
d2 cd5 cd
18C20
dcd4 cd2 cd
dcd3 cd
24A02
dcd3 cdcd
24B20
d2 cdcd
Table 7: G ∼
= F i23 , X = 2B, t = a.
C
1A
2B
2C
3A
3C
4B
4C
5A
6C
6K
Orbit size
1
1386
12672
62370
5632
630784
7983360
1596672
1596672
14598144
709632
28385280
Representative
t
4C1 6C1
4C2 6K1
4B1
6C1
6K1
babab
babab2 abababab
4C10
b
b2 ab2 abab2 ababab2 abababab
bab
Table 8: G ∼
= F i23 , X = 2C, t = ((b2 a)3 ba)6 .
C
1A
2A
Orbit size
1
180
2B
540
3456
4320
2C
810
12960
12960
12960
103680
3A
1536
Representative
6A1 6A2 6A3 6B1 6E1 6E2 6E3 6E4 6J1
6J2 10A1 10A2 10A3 10A4 14A1 14A2
14A3 14A4 26A1 26A2 26B1 26B2
4A2 4C1 4C3 4C5 4C1 0 6C2 6K2 10B2
4A1 4A3 4C2 4C4 4C6 4C7 4C8 4C1 2
6C1 6C6 6D2 6K1 6K3 6K4 10B1 10B6
14B3 22A1
4A4 4C9 4C1 1 6C3 6C4 6C5 6D1 6K5
6K6 6K7 10B3 10B4 10B5 14B1 14B2
14B4
4B1 4B2 4B3 4D2
2C30 6I2 6M1 10C2
2C20 6I3 6M2 10C1
4B4 4B5 4D1 6I1 6I4
4B6 4B7 4D3 6F1 6G1 6G2 6H1 6I5 6I6
6I7 6L1 6L2 6L3 6L4 6M3 6M4 6M5
6M6 6N1 6N2 6O1 6O2 10C3 10C4
6A1 6C1 6C2 6C3 6I1 6I3 6I6 15A1
14
23040
3B
81920
3C
73728
122880
368640
3D
4A
983040
2949120
103680
103680
414720
414720
4B
103680
103680
103680
414720
1244160
1658880
1658880
4C
4D
103680
103680
207360
207360
414720
414720
1244160
1244160
1244160
1244160
1244160
1244160
1244160
1244160
9953280
5A
1327104
6635520
6635520
6A
69120
6A2 6A3 6C4 6C5 6C6 6I2 6I4 6I5 6I7
15A2 15A3 21A1 21A2 39A1 39B1
6B1 6D1 6D2 6F1 6H1 9A1 9B1 9B2
9B3 9C1 9C2 9D1 9D2
6E1 6K3 6K6 6L1 6M5
6E4 6G2 6K1 6K2 6K5 6L2 6L3 6M2
6M4 15B1
6E2 6E3 6G1 6K4 6K7 6L4 6M1 6M3
6M6 15B2
6J2 6N2 6O1 9E1
6J1 6N1 6O2 9E2
4A02 12A3 12J1 20A1
4A01 12A4 12J3 20A2
4A04 12A1 12A5 12D1 12J4 12J5 20A4
28A2
4A03 12A2 12A6 12D2 12J2 12J6 20A3
28A1
8A2 8B4
8A4 8B5
8A6 8B2
8A8 8B1 0 12C1 12G4
8A9 8B8 12C2 12G1 12G2 12G3
8A7 8A1 0 8B1 8B3 8B6 12B1 12B2
12E1 12E3 12G5 12H1 12H2 12M1
12M2
8A1 8A3 8A5 8B7 8B9 12B3 12B4 12C3
12E2 12E4 12H3 12H4 12M3 12M4
(abab2 )2 abab(ab2 )4 (ab)5 (ab2 )2 (ab)4 a
4C10
4C40 12I1
4C30 12I2
4C60 12N1
4C50 12N2
4C1 00 12I8 12N5 20B2
4C1 10 12I5 12I6
4C1 20 12I9 12N4 20B3
4C70 12I7 12N6 20B1
4C80 12I3 12I4
4C90 12I1 0 12N3 20B4
4D20
(ba)5 b2 ababab2 (ab)3 ab2 a
8C1 8C2 12F1 12F2 12K1 12K2 12K3
12K4 12L1 12L2 12L3 12L4 12O1 12O2
12O3 12O4
10A1 10B2 10B3 10C1 15A1
10A4 10B4 10B6 10C3 15A2 15B1 35A1
10A2 10A3 10B1 10B5 10C2 10C4 15A3
15B2
6A02 30B3
15
6B
6C
6D
6E
6F
6G
6H
6I
6J
6K
6L
6M
6N
6O
69120
276480
1474560
110592
138240
414720
414720
829440
1658880
2211840
2211840
1105920
1105920
2211840
2211840
6635520
3317760
3317760
13271040
414720
829440
829440
1244160
1658880
1658880
9953280
8847360
8847360
1105920
2211840
3317760
3317760
3317760
3317760
6635520
3317760
3317760
6635520
19906560
3317760
3317760
6635520
6635520
9953280
9953280
26542080
26542080
26542080
6A01 30B4
30B1 30B2 42A1 42A2
18A1 18A2 18B1 18B2 18B3 18B4
12A1 12I2 12I6
12A4 12I1 12I7
6C40 30A1
6C30 30A2
12A2 12A6 12I3 12I4 12I9 30A4
12A3 12A5 12I5 12I8 12I1 0 30A3
12D2 18E2 18E3 18E4
12D1 18E1 18F1 18F2
6E20
ab(ab2 )2 (ab)3 ab2 (ababab2 )2 a
bababab2 ab2 ab2 abababab2
6E30
12B1 12B2 12B3 12B4 12F1 12F2
6G02 30C1
6G01 30C2
12E1 12E2 12E3 12E4 12K1 12K2 12K3
12K4 18C1 18C2 18D1 18G1 18G2
12C1 12G3
6I30
ababab2 ab2 (ab)4 (ab2 )3 ab(ab2 )2 ababa
12C2 12G1 12G2 12G4
bab2 ababab2 ababababab2 ababababa
6I50
12C3 12G5
6J20 18H2
6J10 18H1
12J4 12N2
12J3 12N1 12N6
12J5 12N5
12J1 12N3
6K60
abababab2
12J2 12J6 12N4
6L02
abab2 ab2 abababab2 ababab2 ab2 a
12H1 12H2 12L2 12L4
12H3 12H4 12L1 12L3
bab2 abab2 ab2 ab2 abababab2 a
6M10
6M40
abab2 ababababababab2
6M60
ab2 ab2 ab2 abababababababab2
bababab2 ababab2 a
6N10
12M1 12M2 12O1 12O2
16
7A
8A
8B
8C
9A
9B
9C
9D
9E
10A
10B
10C
79626240
2654208
26542080
39813120
3317760
3317760
3317760
3317760
9953280
9953280
19906560
19906560
19906560
19906560
3317760
3317760
9953280
9953280
9953280
9953280
19906560
19906560
19906560
19906560
39813120
39813120
17694720
4423680
13271040
26542080
8847360
26542080
17694720
53084160
53084160
159252480
13271040
13271040
19906560
19906560
13271040
13271040
19906560
19906560
39813120
39813120
19906560
19906560
39813120
12M3 12M4 12O3 12O4
14A2 14B2
14A1 14A3 14B4 21A1
14A4 14B1 14B3 21A2 35A1
8A04
8A03
bababab2 abababab2 ababab2 abab2 a
babab2 ab2 ab2 ab2 abababababab2
8A06
babab2 ababababababab2 a
8A08 24C3
8A07 24C4
8A1 00 24C2
8A09 24C1
8B20
abababab2 ababababab2 ab2 ababab2
8B40
bab2 ab2 abababababababab2
bababab2 ababababab2
8B50
8B1 00 24A4
8B90 24A3
8B80 24A2
8B70 24A1
8C20 24B1 24B3
8C10 24B2 24B4
18D1 27A1
18B1 18E1 18E2
18B4 18E3
18B2 18B3 18E4
18A2 18C1 18F1
18A1 18C2 18F2
18G2
18G1
18H1
18H2
10A02 30B3
10A01 30B4
10A04 30B1
10A03 30B2
20A1 20B4
20A2 20B1
10B40 30A1
10B30 30A2
20A3 20B3 30A4
20A4 20B2 30A3
10C20
bababab2 abababab2 ab2 a
10C40 30C2
17
11A
12A
12B
12C
12D
12E
12F
12G
12H
12I
12J
12K
39813120
159252480
3317760
3317760
3317760
3317760
9953280
9953280
6635520
6635520
19906560
19906560
3317760
9953280
39813120
26542080
26542080
13271040
13271040
13271040
13271040
39813120
39813120
9953280
9953280
9953280
9953280
39813120
13271040
13271040
39813120
39813120
3317760
3317760
9953280
9953280
9953280
9953280
19906560
19906560
19906560
19906560
13271040
13271040
13271040
13271040
39813120
39813120
39813120
10C30 30C1
22A1
12A02
abab2 abababababab2 ababab2 a
12A04
bab2 abababab2 ababab2
12A06 60A2
12A05 60A1
12B20
bab2 ababababababababab2
12B40
abab2 abababababab2 abab2 a
24A1
24A3
24A2 24A4
12D20 36B1
12D10 36B2
12E30 36A2
12E40
12E10 36A1
abababab2 ab2 ababab2
24B2 24B3
24B1 24B4
12G03
24C2
babab2 ababab2 ababab2
24C4
24C1 24C3
12H20
bab2 ab2 ababab2 ab2 abababab2 a
12H40
bab2 ab2 ababababa
12I20
abababab2 abababab2 ab2 ababab2
abababab2 abababababab2 abab2
abababab2 ababab2 ab2 ab2
12I30
12I40
12I80
babababababababab2
12I1 00
babababababab2 ab2 ab2 a
12J30
12J40
bababab2 ababababab2 abab2 a
ababab2 ababab2 ab2 ababab2 a
abababab2 abab2
12J50
bababab2 ababab2 ababab2
18
12L
12M
12N
12O
13A
13B
14A
14B
15A
15B
17A
18A
18B
18C
18D
18E
39813120
39813120
39813120
39813120
39813120
39813120
39813120
26542080
26542080
79626240
79626240
13271040
13271040
39813120
39813120
39813120
39813120
79626240
79626240
79626240
79626240
53084160
159252480
53084160
159252480
26542080
26542080
79626240
79626240
39813120
39813120
79626240
79626240
13271040
39813120
79626240
53084160
159252480
318504960
26542080
26542080
26542080
26542080
79626240
79626240
79626240
79626240
159252480
26542080
12K30
bab2 ab2 ab2 ab2 abababababa
12K10
bab2 ab2 abababab2 abab2 a
bab2 ab2 ab2 abababab2 a
12L02
12L01
abab2 abababababab2 a
12M10
b
12M30
bababababababab2
12N10
bab2 ab2 abab2 ababab2 a
12N30
abab2 abababababab2
12N50
12O30
ababab2 ababab2
bab2 abababab2 ab2 ab2 a
12O20
26A2
26A1 39A1
26B2
26B1 39B1
bab2 ababab2 ab2 ababababa
14A01
14A04 42A2
14A03 42A1
bababab2 abababababababa
14B10
28A2
28A1
30A1 30B3
30A2 30A3 30B2
30A4 30B1 30B4
30C2
30C1
bababab2 ab2 abab2
18A02
babab2 ab2 ab2 abababab2
bababababab2
18B10
babab2 ababababab2
18B30
babababab2 ab2 a
18C10
36A1 36A2
36B1
19
18F
18G
18H
20A
20B
21A
22A
24A
24B
24C
26A
26B
27A
28A
30A
30B
30C
39813120
39813120
79626240
26542080
79626240
159252480
159252480
159252480
159252480
26542080
26542080
79626240
79626240
79626240
79626240
79626240
79626240
159252480
159252480
159252480
79626240
79626240
79626240
79626240
79626240
79626240
79626240
79626240
79626240
79626240
79626240
79626240
159252480
159252480
159252480
159252480
318504960
159252480
159252480
39813120
39813120
79626240
79626240
79626240
79626240
79626240
79626240
159252480
159252480
bababab2 abababababa
18E20
36B2
abababab2 ab2 abab2
abababababab2 ab2 ababab2
bab2 ababababa
18G01
bab2 ababab2
18H10
bab2 abababab2 ab2 abababab2
20A01
20A04 60A1
20A03 60A2
20B20
bababab2 ab2 a
bababab2 abababab2
20B30
42A2
42A1
ababab2 abababab2
24A04
24A03
babababab2 a
ababababab2 ab2 ab2
24B20
bab2 ababab2 abababab2 a
bababab2 a
24B30
24C20
bab2 abababababab2
babababababab2
24C30
bababab2
26A01
ababab2 ababababab2
26B10
bababab2 ab2
abab2 ababababab2
28A01
30A02
bababababab2 abab2 a
60A2
60A1
abab2 abababab2
30B10
babababab2 ab2 abab2
30B30
30C20
bab2 ababab2 ababab2
20
35A
36A
36B
39A
39B
42A
60A
318504960
159252480
159252480
159252480
159252480
318504960
318504960
159252480
159252480
159252480
159252480
bababababab2 a
36A02
bab2 ab2 abababa
36B20
abab2 ababab2
bab2 abababab2
babab2
bababababab2 ab2 a
42A01
babababab2
60A01
Table 9: G ∼
= F i240 , X = 2A, t = a.
C
1A
2A
2B
3A
3C
3E
4A
4B
5A
6A
6F
Orbit size
1
720
123552
1216215
56320
20500480
60825600
389188800
88957440
88957440
1423319040
19768320
2767564800
Representative
4B2 , 6F1
4B2 , 6A1
4A1 , 6A1
6A1
6F1
(ba)3 b2 (ab)3
(ba)4 b
(b2 a)2 ba(b2 a)2 b(ab)3
4B10
b2 ab
b2 (ab)5 (ab2 )2 ab
bab
21
Table 10: G ∼
= F i240 , t ∈ X = 2B
CG (t)-orbit
t
2A1
2A2
2B1
2B2
2B3
3A1
3B1
3C1
3C2
3D1
3E1
4A1
4A2
4A3
4A4
4B1,2
4B3,4
4C1,2
4C3,4,5
4C6
5A1
5A2
6A1
6A2
6A3
6B1
6C1
6C2
6D1
6D2
6E1
6F1
6F2,3,4
6F5
6G1
6H1
6I1,2,3
6I4
6J1
6K1
6K2
Orbit Size
1
24192
45360
3402
816480
2177280
258048
917504
1032192
10321920
165150720
278691840
1306368
4354560
104509440
139345920
52254720
209018880
78382080
313528320
1254113280
334430208
836075520
34836480
55738368
69672960
278691840
123863040
371589120
209018880
836075520
1114767360
139345920
278691840
1114767360
4459069440
4459069440
836075520
3344302080
13377208320
5016453120
20065812480
22
(factored)
1
27 · 33 · 7
24 · 34 · 5 · 7
2 · 35 · 7
25 · 36 · 5 · 7
28 · 35 · 5 · 7
212 · 32 · 7
217 · 7
214 · 32 · 7
215 · 32 · 5 · 7
219 · 32 · 5 · 7
215 · 35 · 5 · 7
28 · 36 · 7
29 · 35 · 5 · 7
212 · 36 · 5 · 7
214 · 35 · 5 · 7
211 · 36 · 5 · 7
213 · 36 · 5 · 7
210 · 37 · 5 · 7
212 · 37 · 5 · 7
214 · 37 · 5 · 7
216 · 36 · 7
215 · 36 · 5 · 7
212 · 35 · 5 · 7
215 · 35 · 7
213 · 35 · 5 · 7
215 · 35 · 5 · 7
217 · 33 · 5 · 7
217 · 34 · 5 · 7
213 · 36 · 5 · 7
215 · 36 · 5 · 7
217 · 35 · 5 · 7
214 · 35 · 5 · 7
215 · 35 · 5 · 7
217 · 35 · 5 · 7
219 · 35 · 5 · 7
219 · 35 · 5 · 7
215 · 36 · 5 · 7
217 · 36 · 5 · 7
219 · 36 · 5 · 7
216 · 37 · 5 · 7
218 · 37 · 5 · 7
Disk
∆0
∆1
∆1
∆1
∆1
∆1
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
Table 10 (cont.)
CG (t)-orbit
7A1
7A2
7B1,2
8A1,2
8A3,4
8A5,6
8A7,8
8B1
8B2
8C1,2,3,4
9A1
9B1
9B2
9C1
9D1
9E1,2
9F1
10A1,2
10A3
10A4
10A5
10B1,2,3
10B4,5
11A1
12A1
12A2
12B1,2
12B3,4
12C1
12C2
12D1,2,3,4
12D5,6
12E1,2
12E3,4
12F1,2
12G1,2
12H1,2
12I1
12I2
12J1
12J2
12K1,2
12K3,4
12L1,2,3,4
12L5,6
12M1,2
Orbit Size
63700992
6688604160
11466178560
278691840
836075520
2508226560
10032906240
10032906240
20065812480
10032906240
2972712960
445906944
2229534720
4459069440
5945425920
8918138880
26754416640
5016453120
6688604160
668860416
10032906240
5016453120
10032906240
80263249920
1672151040
3344302080
1114767360
3344302080
5016453120
10032906240
1672151040
5016453120
3344302080
6688604160
13377208320
13377208320
20065812480
20065812480
40131624960
20065812480
40131624960
10032906240
20065812480
6688604160
20065812480
40131624960
23
(factored)
218 · 35
218 · 36 · 5 · 7
220 · 37 · 5
215 · 35 · 5 · 7
215 · 36 · 5 · 7
215 · 37 · 5 · 7
217 · 37 · 5 · 7
217 · 37 · 5 · 7
218 · 37 · 5 · 7
217 · 37 · 5 · 7
220 · 34 · 5 · 7
218 · 35 · 7
218 · 35 · 5 · 7
219 · 35 · 5 · 7
221 · 34 · 5 · 7
220 · 35 · 5 · 7
220 · 36 · 5 · 7
216 · 37 · 5 · 7
218 · 36 · 5 · 7
217 · 36 · 7
217 · 37 · 5 · 7
216 · 37 · 5 · 7
217 · 37 · 5 · 7
220 · 37 · 5 · 7
216 · 36 · 5 · 7
217 · 36 · 5 · 7
217 · 35 · 5 · 7
217 · 36 · 5 · 7
216 · 37 · 5 · 7
217 · 37 · 5 · 7
216 · 36 · 5 · 7
216 · 37 · 5 · 7
217 · 36 · 5 · 7
218 · 36 · 5 · 7
219 · 36 · 5 · 7
219 · 36 · 5 · 7
218 · 37 · 5 · 7
218 · 37 · 5 · 7
219 · 37 · 5 · 7
218 · 37 · 5 · 7
219 · 37 · 5 · 7
217 · 37 · 5 · 7
218 · 37 · 5 · 7
218 · 36 · 5 · 7
218 · 37 · 5 · 7
219 · 37 · 5 · 7
Disk
∆3
∆3
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆3
∆3
∆2
∆3
∆2
∆3
∆2
∆2
∆2
∆2
∆2
∆2
∆3
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
Table 10 (cont.)
CG (t)-orbit
13A1
13A2
14A1,2
14A3
14A4
14B1,2
15A1,2
15B1
15C1,2
16A1,2,3,4
17A1
18A1
18B1,2
18C1
18D1,2
18D3,4
18E1,2
18F1
20A1,2
20A3,4
20B1,2,3,4,5,6
21A1
21B1
21C1,2
21D1,2
22A1
24A1,2
24B1,2
24C1,2
24D1,2
24E1,2,3,4
24F1,2,3,4
24G1,2,3,4
26A1,2
27A1
27B1
27B1
28A1,2
29A1
29B1
30A1,2
30A3,4
30B1,2
33A1
33B1
Orbit Size
8918138880
80263249920
6688604160
13377208320
40131624960
80263249920
6688604160
53508833280
26754416640
40131624960
160526499840
40131624960
13377208320
80263249920
13377208320
20065812480
80263249920
80263249920
13377208320
40131624960
20065812480
80263249920
160526499840
80263249920
80263249920
80263249920
40131624960
40131624960
80263249920
80263249920
40131624960
40131624960
40131624960
80263249920
160526499840
160526499840
160526499840
80263249920
160526499840
160526499840
20065812480
40131624960
80263249920
160526499840
160526499840
24
(factored)
220 · 35 · 5 · 7
220 · 37 · 5 · 7
218 · 36 · 5 · 7
219 · 36 · 5 · 7
219 · 37 · 5 · 7
220 · 37 · 5 · 7
218 · 36 · 5 · 7
221 · 36 · 5 · 7
220 · 36 · 5 · 7
219 · 37 · 5 · 7
221 · 37 · 5 · 7
219 · 37 · 5 · 7
219 · 36 · 5 · 7
220 · 37 · 5 · 7
219 · 36 · 5 · 7
218 · 37 · 5 · 7
220 · 37 · 5 · 7
220 · 37 · 5 · 7
219 · 36 · 5 · 7
219 · 37 · 5 · 7
218 · 37 · 5 · 7
220 · 37 · 5 · 7
221 · 37 · 5 · 7
220 · 37 · 5 · 7
220 · 37 · 5 · 7
220 · 37 · 5 · 7
219 · 37 · 5 · 7
219 · 37 · 5 · 7
220 · 37 · 5 · 7
220 · 37 · 5 · 7
219 · 37 · 5 · 7
219 · 37 · 5 · 7
219 · 37 · 5 · 7
220 · 37 · 5 · 7
221 · 37 · 5 · 7
221 · 37 · 5 · 7
221 · 37 · 5 · 7
220 · 37 · 5 · 7
221 · 37 · 5 · 7
221 · 37 · 5 · 7
218 · 37 · 5 · 7
219 · 37 · 5 · 7
220 · 37 · 5 · 7
221 · 37 · 5 · 7
221 · 37 · 5 · 7
Disk
∆3
∆3
∆3
∆3
∆3
∆2
∆3
∆3
∆2
∆2
∆3
∆2
∆2
∆2
∆3
∆3
∆2
∆3
∆2
∆2
∆2
∆3
∆3
∆2
∆2
∆3
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆3
∆3
∆3
∆3
∆3
∆3
∆3
∆3
∆3
∆2
∆3
∆3
Table 10 (cont.)
CG (t)-orbit
35A1
36A1,2
36B1,2
36C1,2
36D1,2
39A1
39B1
39C1
39D1
42A1
42B1,2
42C1,2
45A1
45B1
60A1,2
Orbit Size
160526499840
40131624960
40131624960
80263249920
80263249920
160526499840
160526499840
160526499840
160526499840
80263249920
80263249920
80263249920
160526499840
160526499840
80263249920
25
(factored)
221 · 37 · 5 · 7
219 · 37 · 5 · 7
219 · 37 · 5 · 7
220 · 37 · 5 · 7
220 · 37 · 5 · 7
221 · 37 · 5 · 7
221 · 37 · 5 · 7
221 · 37 · 5 · 7
221 · 37 · 5 · 7
220 · 37 · 5 · 7
220 · 37 · 5 · 7
220 · 37 · 5 · 7
221 · 37 · 5 · 7
221 · 37 · 5 · 7
220 · 37 · 5 · 7
Disk
∆3
∆2
∆2
∆2
∆3
∆3
∆3
∆3
∆3
∆3
∆2
∆2
∆3
∆3
∆3
Table 11: G ∼
= F i24 , t ∈ X = 2D
CG (t)-orbit
1A1
2A1
2A2
2A3
2B1
2B2
2B3
3A1
3A2
3B1
3C1
3C2
3D1
3D2
3E1
4A1
4A2
4A3
4A4
4B1,2
4B3,4
4B5,6
4B7,8
4C1
4C2
4C3
5A1
5A2
6A1,2
6A3
6A4,5
6A6,7
6B1,2
6C1
6C2
6D1
6D2
6D3,4,5,6
6D7
6E1
6F1,2
6F3,4,5,6
6G1
6G2,3
Orbit Size
1
2079
38016
62370
187110
997920
2395008
8448
221760
1261568
1892352
17031168
37847040
113541120
48660480
23950080
57480192
71850240
95800320
4790016
23950080
71850240
287400960
287400960
574801920
1724405760
43794432
1532805120
2128896
9123840
23950080
95800320
153280512
170311680
510935040
23950080
35925120
95800320
574801920
1532805120
85155840
766402560
2043740160
3065610240
26
(factored)
1
33 · 7 · 11
27 · 33 · 11
2 · 34 · 5 · 7 · 11
2 · 35 · 5 · 7 · 11
25 · 34 · 5 · 7 · 11
27 · 35 · 7 · 11
28 · 3 · 11
26 · 32 · 5 · 7 · 11
214 · 7 · 11
213 · 3 · 7 · 11
213 · 33 · 7 · 11
215 · 3 · 5 · 7 · 11
215 · 32 · 5 · 7 · 11
215 · 33 · 5 · 11
28 · 35 · 5 · 7 · 11
210 · 36 · 7 · 11
28 · 36 · 5 · 7 · 11
210 · 35 · 5 · 7 · 11
28 · 35 · 7 · 11
28 · 35 · 5 · 7 · 11
28 · 36 · 5 · 7 · 11
210 · 36 · 5 · 7 · 11
210 · 36 · 5 · 7 · 11
211 · 36 · 5 · 7 · 11
211 · 37 · 5 · 7 · 11
214 · 35 · 11
214 · 35 · 5 · 7 · 11
210 · 33 · 7 · 11
211 · 34 · 5 · 11
28 · 35 · 5 · 7 · 11
210 · 35 · 5 · 7 · 11
213 · 35 · 7 · 11
214 · 33 · 5 · 7 · 11
214 · 34 · 5 · 7 · 11
28 · 35 · 5 · 7 · 11
27 · 36 · 5 · 7 · 11
210 · 35 · 5 · 7 · 11
211 · 36 · 5 · 7 · 11
214 · 35 · 5 · 7 · 11
213 · 33 · 5 · 7 · 11
213 · 35 · 5 · 7 · 11
216 · 34 · 5 · 7 · 11
215 · 35 · 5 · 7 · 11
Disk
∆0
∆1
∆1
∆1
∆1
∆1
∆1
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆3
∆2
∆2
∆2
∆2
∆3
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆3
∆2
Table 11 (cont.)
CG (t)-orbit
6H1
6H2,3,4
6H5
6I1,2,3,4,5
6I6
6K1
7A1
7A2
7B1
8A1,2,3,4
8A5,6,7,8
8B1
8C1,2
9A1
9B1
9B2
9C1
9C2
9D1
9E1
9E2
9F1
9F2
9F3
10A1
10A2,3
10A4
10A5
10B1,2
10B3
11A1
12A1
12A2
12A3
12B1,2
12B3,4
12C1
12C2
12C3,4,5
12C6
12D1,2
12D3,4,5,6
12D7,8
12E1
12E2
Orbit Size
2043740160
3065610240
9196830720
766402560
6897623040
3065610240
1839366144
9196830720
10510663680
2299207680
6897623040
27590492160
9196830720
1362493440
340623360
3065610240
340623360
3065610240
2724986880
1362493440
12262440960
6131220480
12262440960
55180984320
919683072
4598415360
9196830720
13795246080
4598415360
27590492160
110361968640
191600640
574801920
4598415360
1532805120
4598415360
574801920
1149603840
1724405760
4598415360
153280512
2299207680
6897623040
3065610240
27590492160
27
(factored)
216 · 34 · 5 · 7 · 11
215 · 35 · 5 · 7 · 11
215 · 36 · 5 · 7 · 11
213 · 35 · 5 · 7 · 11
213 · 37 · 5 · 7 · 11
215 · 35 · 5 · 7 · 11
215 · 36 · 7 · 11
215 · 36 · 5 · 7 · 11
218 · 36 · 5 · 11
213 · 36 · 5 · 7 · 11
213 · 37 · 5 · 7 · 11
215 · 37 · 5 · 7 · 11
215 · 36 · 5 · 7 · 11
217 · 33 · 5 · 7 · 11
215 · 33 · 5 · 7 · 11
215 · 35 · 5 · 7 · 11
215 · 33 · 5 · 7 · 11
215 · 35 · 5 · 7 · 11
218 · 33 · 5 · 7 · 11
217 · 33 · 5 · 7 · 11
217 · 35 · 5 · 7 · 11
216 · 35 · 5 · 7 · 11
217 · 35 · 5 · 7 · 11
216 · 37 · 5 · 7 · 11
214 · 36 · 7 · 11
214 · 36 · 5 · 7 · 11
215 · 36 · 5 · 7 · 11
214 · 37 · 5 · 7 · 11
214 · 36 · 5 · 7 · 11
215 · 37 · 5 · 7 · 11
217 · 37 · 5 · 7 · 11
211 · 35 · 5 · 7 · 11
211 · 36 · 5 · 7 · 11
214 · 36 · 5 · 7 · 11
214 · 35 · 5 · 7 · 11
214 · 36 · 5 · 7 · 11
211 · 36 · 5 · 7 · 11
212 · 36 · 5 · 7 · 11
211 · 37 · 5 · 7 · 11
214 · 36 · 5 · 7 · 11
213 · 35 · 7 · 11
213 · 36 · 5 · 7 · 11
213 · 37 · 5 · 7 · 11
215 · 35 · 5 · 7 · 11
215 · 37 · 5 · 7 · 11
Disk
∆3
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆3
∆2
∆2
∆2
∆3
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆3
∆3
∆3
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆3
∆2
∆2
∆2
∆2
∆2
∆2
∆2
Table 11 (cont.)
CG (t)-orbit
12F1,2
12F3,4
12G1,2
12H1,2
12J1,2
12K1,2
12K3,4,5,6
12K7,8
12L1,2
12L3,4,5,6
13A1,2
14A1
14A2,3
14A4
14B1
15A1
15A2
15B1
15C1
15C2
17A1
18A1,2
18B1
18B2,3
18B4
18C1
18D1
18D2,3
18D4
18E1,2
18F1
18F2,3
18G1,2
18G3,4
20A1,2
20A3,4
20B1
21A1
21A2
21B1
22A1
24A1,2
24A3,4
24B1,2
24B3,4
24D1,2,3,4
Orbit Size
3065610240
9196830720
18393661440
27590492160
27590492160
3065610240
9196830720
27590492160
18393661440
27590492160
36787322880
18393661440
27590492160
55180984320
73574645760
613122048
27590492160
73574645760
12262440960
110361968640
220723937280
9196830720
3065610240
9196830720
27590492160
36787322880
6131220480
9196830720
55180984320
36787322880
36787322880
55180984320
36787322880
110361968640
18393661440
55180984320
110361968640
18393661440
55180984320
73574645760
110361968640
9196830720
27590492160
9196830720
27590492160
55180984320
28
(factored)
215 · 35 · 5 · 7 · 11
215 · 36 · 5 · 7 · 11
216 · 36 · 5 · 7 · 11
215 · 37 · 5 · 7 · 11
215 · 37 · 5 · 7 · 11
215 · 35 · 5 · 7 · 11
215 · 36 · 5 · 7 · 11
215 · 37 · 5 · 7 · 11
216 · 36 · 5 · 7 · 11
215 · 37 · 5 · 7 · 11
217 · 36 · 5 · 7 · 11
216 · 36 · 5 · 7 · 11
215 · 37 · 5 · 7 · 11
216 · 37 · 5 · 7 · 11
218 · 36 · 5 · 7 · 11
215 · 35 · 7 · 11
215 · 37 · 5 · 7 · 11
218 · 36 · 5 · 7 · 11
217 · 35 · 5 · 7 · 11
217 · 37 · 5 · 7 · 11
218 · 37 · 5 · 7 · 11
215 · 36 · 5 · 7 · 11
215 · 35 · 5 · 7 · 11
215 · 36 · 5 · 7 · 11
215 · 37 · 5 · 7 · 11
217 · 36 · 5 · 7 · 11
216 · 35 · 5 · 7 · 11
215 · 36 · 5 · 7 · 11
216 · 37 · 5 · 7 · 11
217 · 36 · 5 · 7 · 11
217 · 36 · 5 · 7 · 11
216 · 37 · 5 · 7 · 11
217 · 36 · 5 · 7 · 11
217 · 37 · 5 · 7 · 11
216 · 36 · 5 · 7 · 11
216 · 37 · 5 · 7 · 11
217 · 37 · 5 · 7 · 11
216 · 36 · 5 · 7 · 11
216 · 37 · 5 · 7 · 11
218 · 36 · 5 · 7 · 11
217 · 37 · 5 · 7 · 11
215 · 36 · 5 · 7 · 11
215 · 37 · 5 · 7 · 11
215 · 36 · 5 · 7 · 11
215 · 37 · 5 · 7 · 11
216 · 37 · 5 · 7 · 11
Disk
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆3
∆2
∆3
∆2
∆2
∆2
∆3
∆2
∆2
∆2
∆2
∆2
∆3
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆2
∆3
∆3
∆3
∆3
∆2
∆2
∆3
∆3
∆3
∆2
∆2
∆2
∆2
∆2
∆2
∆2
Table 11 (cont.)
CG (t)-orbit
26A1,2
27A1
28A1,2
30A1,2
30A3,4
30B1,2
33A1
33B1
35A1
36B1,2
36C1,2
39A1,2
42A1,2
42A3
60A1,2
Orbit Size
110361968640
73574645760
110361968640
27590492160
55180984320
110361968640
220723937280
220723937280
220723937280
36787322880
110361968640
73574645760
55180984320
110361968640
110361968640
29
(factored)
217 · 37 · 5 · 7 · 11
218 · 36 · 5 · 7 · 11
217 · 37 · 5 · 7 · 11
215 · 37 · 5 · 7 · 11
216 · 37 · 5 · 7 · 11
217 · 37 · 5 · 7 · 11
218 · 37 · 5 · 7 · 11
218 · 37 · 5 · 7 · 11
218 · 37 · 5 · 7 · 11
217 · 36 · 5 · 7 · 11
217 · 37 · 5 · 7 · 11
218 · 36 · 5 · 7 · 11
216 · 37 · 5 · 7 · 11
217 · 37 · 5 · 7 · 11
217 · 37 · 5 · 7 · 11
Disk
∆3
∆3
∆2
∆2
∆2
∆2
∆2
∆2
∆3
∆2
∆2
∆3
∆3
∆3
∆2
References
[1] C. Bates, D. Bundy, S. Perkins, P. Rowley, Commuting involution
graphs for symmetric groups. J. Algebra 266 (2003), no. 1, 133153.
[2] C. Bates, D. Bundy, S. Perkins, P. Rowley, Commuting involution
graphs for finite Coxeter groups. J. Group Theory 6 (2003), no. 4,
461476.
[3] C. Bates, D. Bundy, S. Hart, P. Rowley, Commuting involution graphs
for sporadic simple groups, J. Algebra 316 (2007), no. 2, 849-868.
[4] W. Bosma, J. Cannon and C. Playoust, The Magma Algebra System
I: The User Language, J. Symbolic Comput., 24, 235-265, 1997.
[5] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson,
Atlas of Finite Groups, Clarendon, Oxford, (1985).
[6] A. Everett, Commuting Involution Graphs of Certain Finite Classical
Groups. PhD Thesis, University of Manchester, 2010.
[7] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.3 ; 2002, (http://www.gap-system.org).
[8] P.Rowley, M monster graph. I, Proc. London Math. Soc. (3) 90 (2005),
no. 1, 4260.
[9] P. Taylor, Computational Investigation into Finite Groups, PhD. Thesis, The University of Manchester (2011).
[10] R. Wilson, Standard generators for sporadic simple groups, J. Algebra
184 (1996), 505-515.
[11] R. Wilson, P. Walsh, J. Tripp, I. Suleiman, S. Rogers, R . Parker, S.
Norton, S. Linton and J. Bray, Atlas of finite group representations,
http://brauer.maths.qmul.ac.uk/Atlas/v3/.
30