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1URAL MATHEMATICAL JOURNAL, Vol. 10, No. 1, 2024, pp. 28-43
DOI: 10.15826/umj.2024.1.003
ON A GROUP EXTENSION INVOLVING THE SPORADIC JANKO GROUP J2
Ayoub B. M. Basheer
School of Mathematical and Computer Sciences, University of Limpopo (Turfloop), P. Bag X1106, Sovenga 0727, South Africa
Mathematics Program, Faculty of Education and Arts, Sohar University,
Sohar, Oman
ayoubbasheer@gmail.com
Abstract: According to the electronic Atlas [23], the group J2 has an absolutely irreducible module of dimension 6 over I4. Therefore, a split extension group having the form 46:J2 := G exists. In this paper, we consider this group. Our purpose is to determine its conjugacy classes and character table using the methods of the coset analysis together with Clifford—Fischer theory. We determine the inertia factors of G by analyzing the maximal subgroups of J2 and the maximal of the maximal subgroups of J2 together with other various information. It turns out that the character table of G is a 53 X 53 real-valued matrix, while Fischer matrices of the extension are all integer-valued matrices with sizes ranging from 1 to 8.
Keywords: Group extensions, Janko sporadic simple group, Inertia groups, Fischer matrices, Character table.
1. Introduction
Visiting the history of the classification of finite simple groups, one can see that it was only a century after the establishment of the last Mathieu group that Z. Janko could construct a new sporadic simple group in 1964. This simple group has been named in his honor, is denoted by J1, and has order 175560. Then Janko predicted the existence of other sporadic simple groups; namely, J2, J3, and J4, which later are all proved to exist. According to Wilson [22], the original construction of the second Janko group J2 was due to Marshall Hall (and thus, in some other papers, this group is referred to as Hall-Janko group HJ but here we use the more familiar notation J2). Hall constructed this group as a permutation group acting on 100 points. Starting with the group U3(3), the group J2 appears as a maximal normal subgroup of index 2 of the automorphism group of a graph r associated with U3(3) (for further details on the vertices and how are they connected, see the description given on page 224 of [22]).
The group J2 has order 604800 = 27 x 33 x 52 x 7. It has Schur multiplier and outer automorphism groups both isomorphic to Z2. From the Atlas of Wilson [23], one can see that the group J2 has a 6-dimensional absolutely irreducible module over F4. Therefore, a split extension group of the form 46: J2 := G exists. The present paper focuses on the group G. Our purpose is to determine its conjugacy classes and the inertia factors of this extension with the fusions of their conjugacy classes into the classes of J2. We will also find the character tables of these inertia factors and, finally, the full character table of the extension G under consideration. The methods used here to achieve the previous purpose are the coset analysis technique and the theory of Clifford-Fischer matrices. The most interesting part of this paper is the process of determining the inertia factor groups, where there are three inertia factor groups; namely, H1 = J2, H2, and H3. The main technique used for determining the structures of H2 and H3 is the analysis of the maximal subgroups of J2 and the
maximal subgroups of these maximal subgroups. There are many possibilities for H2 and H3, and combining all of them leads to contradictions except for only one possibility where we find that H2 = 22+4:S3 and H3 = 22 x A5. We use a method of the coset analysis together with Clifford-Fischer theory to construct the character tables of H2 and H3, but we organize the columns of the character tables of these inertia factors according to the centralizers sizes. This paper determines all Fischer matrices of G; their sizes vary between f and 8. The character table of G is a 53 x 53 real-valued matrix, which will be divided into 63 parts corresponding to 3 inertia factors and 21 conjugacy classes of G = J2.
If G = N-G is a group extension (here, N is the kernel of the extension and G is isomorphic to G/N), then the character table of G produced using the coset analysis and Clifford-Fischer theory is in a special format that cannot be obtained by the direct computations using GAP [18] or Magma [15]. Another interesting point is the interplay between the coset analysis and Clifford-Fischer theory. This can be seen at the size of each Fischer matrix, where it is equal to the number of G-classes corresponding to [gi]a obtained via the coset analysis technique. In other words, computations of the conjugacy classes of G using the coset analysis technique will determine the sizes of all Fischer matrices.
From the Atlas [23], we can see that J2 has an absolutely irreducible module of dimension 6 over the field F4. With a being a generator of the field F4, the following two elements g1 and g2 are 6 x 6 matrices over F4 that generates J2:
gi =
a2 a2 0 0 0 0 \ / a 1 a2 1 a2 a2
1 a2 0 0 0 0 a 1 a 1 1 a
1 1 a2 a2 0 0 a a a2 a2 1 0
a 1 1 a2 0 0 , g2 = 0 0 0 0 1 1
0 a2 a2 a2 0 a a2 1 a2 a2 a a2
a2 1 a2 0 a2 0 / V a2 1 a2 a a2 a
\
where o(gi) = 2, o(g2) = 3, and o(gig2) = 7.
Using the above two generators of J2 together with few GAP commands, we were able to construct our split extension group G = 46:J2 in terms of 7 x 7 matrices over F4. With a being a generator of the field F4, the following elements g1 and g2 generate the group G:
0 1 a a a2 0 0 \ ( a 0 1 1 0 a2 0 \
a a2 0 a a a2 0 1 1 a2 a2 a2 1 0
0 0 a2 a2 a2 0 0 1 a 0 a 0 a2 0
1 a 0 a2 1 1 0 , 92 = 1 1 0 1 0 1 0
1 0 1 a 0 0 0 a2 0 1 0 1 0 0
a2 a 1 0 0 a2 0 a 1 a a 0 a2 0
a2 a2 a 0 a2 1 1 / \ a 1 a2 1 a 1 1 /
where o^) = 6, o(g2) = 12, and o{glg2) = 10.
To make the computations easier, we used a few GAP commands to convert the matrix representation of G into permutation representation. We represented G in terms of the set {1,2,..., 4096}.
Using GAP, we see that the group G possesses only one proper normal subgroup of order 4096. This normal subgroup is an elementary abelian group isomorphic to N. In GAP, one can check for the complements of N in G, where in our case we obtained four complements, all isomorphic to J2, and each of these four complements, together with N, gives the split extension in consideration.
For the notation used in this paper and how Clifford-Fischer theory and the coset analysis techniques are used, we follow [1-14, 17].
2. Conjugacy classes of G = 46: J2
Here we compute the conjugacy classes of G using the coset analysis technique (see [2] by Basheer, [5, 6, 8] by Basheer and Moori, or [20] and [21] by Moori for more details) since we are interested in organizing the classes of G corresponding to the classes of J2. Note that J2 has 21 conjugacy classes (see the Atlas [16] or Atlas of Wilson [23]). Corresponding to these 21 classes of J2, we obtained 53 classes in G.
In Table 1, we list the conjugacy classes of G, where in this table:
• ki represents the number of orbits Qn, Qi2,..., Qiki for the action of N on the coset AT/j = Ncji, where cji is a representative of a class of the complement J2 of A in G. In particular, the action of N on the identity coset N produces 4 096 orbits each consists of a single element. Therefore, for G, we have k\ = 4 096.
• fij is the number of orbits fused under the action of Cc(gi) on Qi,Q2,... ,Qk. In particular, the action of Cg(1g) = G = J2 on the orbits Qi,Q2,... ,Qk affords three orbits of lengths 1, 1575, and 2 520 (with corresponding point stabilizers J2, 22+4:S3, and 22 x A5. Thus, fii = 1, fi2 = 1 575, and fi3 = 2 520.
• niij are weights (attached to each class of G). These weights are computed by the formula
my = [Ndm : =
where A is the kernel of an extension G in consideration.
Table 1. The conjugacy classes of G.
[9I\G ki fij niij VMG o(9ij) \VMG\ \cc(9ij)\
/n = l mn = 1 gn 1 1 2 477260 800
<7i = 14 ki = 4 096 /12 = 1 575 m 12 = 1 575 912 2 1 575 1 570 864
/13 = 2 520 mis = 2 520 913 2 2 520 983 040
/21 = 1 m 21 = 16 921 2 5 040 491 520
g2 = 2A k2 = 256 /22 = 15 m22 = 240 922 2 75 600 32 768
/23 = 120 m2 3 = 1920 923 4 604 800 4 096
/24 = 120 77124 = 1 920 924 4 604 800 4 096
/31 = 1 777.31 = 64 931 2 161280 15 360
/32 = 1 77732 = 64 932 4 161280 15 360
/33 = 1 77733 = 64 933 4 161280 15 360
9s = 2B 4 6 3 /34 = 1 77734 = 64 934 4 161280 15 360
/35 = 15 77735 = 960 935 4 2 419 200 1024
/36 = 15 '77736 = 960 936 4 2 419 200 1024
/37 = 15 77737 = 960 937 4 2 419 200 1024
/38 = 15 77738 = 960 938 4 2 419 200 1024
94 = 3A kA = 1 /41 = 1 77741 = 4 096 941 3 2 293 760 1080
/51 = 1 777.51 = 256 951 3 4 300 800 576
g5 = 3B k5 = 16 /52 = 4 777.52 = 768 952 6 12 902 400 192
/53 = 12 777.53 = 3 072 953 6 51 609 600 48
/61 = 1 77761 = 256 961 4 1612 800 1536
continued on the next page
Table 1 (continued from the previous page)
M G ki fij niij \9Ü\G o{9ij) \VMG\ \Cc(9ij)\
96 = 4 A k6 = 16 /62 = 3 m&2 = 768 962 4 4 838 400 512
/63 = 3 W'63 = 768 963 4 4 838 400 512
CO m6 4 = 768 964 4 4 838 400 512
/65 = 6 m6 5 = 1536 965 4 9 676 800 256
97 = 5 A k7 = 16 /71 = 1 m71 = 256 971 5 516 096 4 800
/72 = 15 m7 2 = 3 840 972 10 7 741440 320
98 = 5 B k8 = 16 /81 = 1 m81 = 256 981 5 516 096 4 800
/82 = 15 W'82 = 3 840 982 10 7 741440 320
99 = 5 C kg = 1 /91 = 1 m91 = 4 096 991 5 49 545 216 50
9w = 5 D kw = = 1 /10,1 = 1 m 10,1 = 4 096 910,1 5 49 545 216 50
9n = 6 A ku = = 1 /11,1 = 1 = 4 096 911,1 6 103 219 200 24
/12,1 = 1 mi2,i = 1024 912,1 6 51 609 600 48
912 = 6 B ki2 = = 4 /12,2 = 1 mi2,2 = 1024 912,2 12 51 609 600 48
/l2,3 = 1 mi2,3 = 1024 512,3 12 51 609 600 48
/l2,4 = 1 mi2,4 = 1024 512,4 12 51 609 600 48
913 = 7 A hi = = 1 /13,1 = 1 W'13,1 = 4 096 5i3,i 7 353 894 400 7
/14,1 = 1 m 14,1 = 1024 514,1 8 77 414 400 32
914 = 8 A ku = = 4 /l4,2 = 1 W'14,2 = 1024 514,2 8 77 414 400 32
/14,3 = 1 W'14,3 = 1024 514,3 8 77 414 400 32
/l4,4 = 1 m 14,4 = 1024 514,4 8 77 414 400 32
/15,1 = 1 W'15,1 = 1024 915,1 10 30 965 760 80
915 = 10 A ki5 = = 4 /15,2 = 1 W'15,2 = 1024 515,2 20 30 965 760 80
/l5,3 = 1 W'15,3 = 1024 515,3 20 30 965 760 80
/l5,4 = 1 W'15,4 = 1024 515,4 20 30 965 760 80
/16,1 = 1 W'16,1 = 1024 516,1 10 30 965 760 80
916 = 10 B he = = 4 /l6,2 = 1 W'16,2 = 1024 516,2 20 30 965 760 80
/l6,3 = 1 W'16,3 = 1024 516,3 20 30 965 760 80
/l6,4 = 1 W'16,4 = 1024 516,4 20 30 965 760 80
9n = 10 C kn = = 1 /17,1 = 1 mi7,i = 4 096 517,1 10 247 726 080 10
918 = 10 D kis = = 1 /18,1 = 1 m 18,1 = 4 096 518,1 10 247 726 080 10
919 = 12 A hg = = 1 /19,1 = 1 W'19,1 = 4 096 519,1 12 206 438 400 12
920 = 15 A k-20 = = 1 /20,1 = 1 m20,1 = 4 096 520,1 15 165 150 720 15
921 = 15 B k-2i = = 1 /21,1 = 1 m2i,i = 4 096 521,1 15 165 150 720 15
3. Inertia factor groups of G = 46: J2
We have seen in Section 2 that the action of G on N produced three orbits of lengths 1, 1 575, and 2 520. By a theorem of Brauer (see, for example, [2, Theorem 5.1.1]), it follows that the action of G on Irr(iV) will also produce three orbits of lengths 1, r, and s, where
1 + r + s = |Irr(N)| = 4096;
that is
r + s = 4 095.
(3.1)
The values of r and s will be determined through deep investigation on the maximal subgroups of J2 or on the maximal of the maximal subgroups of J2 together with various information including sizes of the Fischer matrices, fusions of the conjugacy classes of some subgroups into the group J2, and other information. In Table 2, we supply the maximal subgroups of J2 (see the Atlas [16]). We will need these subgroups to determine H2 and H3.
Table 2. The maximal subgroups of G = J2.
Mi \Mi\ [J2 : Mi]
u3( 3) 6 048 100
(3"A6):2 2160 280
21+4:A5 1920 315
22+4:(3 x S3) 1152 525
A4 x A5 720 840
A5 x D10 600 1008
L3( 2):2 336 1800
52:D12 300 2016
a5 60 10 080
First, since 1, r, and s are the lengths of the orbits on the action of G on A (which can be reduced to the action of G on A), it follows that [G : Hi] = 1, [G : H2] = r, and [G : H3] = s, where H1, H2, and H3 are the inertia factors in G = J2. It follows that H1 = G = J2 and r, s | |G|; that is r,s|604 800. Now, 604 800 has 192 positive divisors, where 140 divisors are less than 4 095. Out of these 140 divisors, only four pairs (r, s) satisfy (3.1). These are the pairs
(r,s) € {(63, 4032), (315, 3780), (945, 3 150), (1 575, 2520)}. (3.2)
Here, we do not distinguish between the pairs (r, s) and (s,r) and therefore we exclude the other four pairs (4 032,63), (3 780,315), (3 150,945), and (2 520,1 575) from our consideration and restrict ourselves only to those in (3.2). Another point we put in mind is that since G is a split extension of 46 by J2 and 46 is an elementary abelian group, it follows that the three character tables of H1, H-2, and Hs, which we will use to construct the character table of G, are ordinary. From the Atlas and Table 1, we have |Irr(G)| = 53 and |Irr(iZ"i)| = |Irr(G)| = |Irr(J2)| = 21. Since
3
= |Irr(G)| = 53,
i=1
we have |Irr(ffi)| + |Irr(ii2)| + |Irr(ff3)| = |Irr(G)| = 53, that is
|Irr(H2)| + |Irr(H3)| =32. (3.3)
Our next task is to show that (r,s) = (1575,2 520) and the action of G on Irr(A) is dual to the action of G on classes of A. This will be achieved by excluding the other possible pairs by getting a contradiction to some fact in each case.
Proposition 1. (r,s) = (63, 4 032).
Proof. To obtain a contradiction, suppose that (r, s) = (63,4032), i.e., r = 63 and s = 4032 (or [J : H2] = 63 and [J2 : H3] = 4 032) and consequently |H2| = 9 600 and H = 150. Since |H2| = 9 600 and the maximal subgroups of J2 are given in Table 2, it follows that |H2| is bigger than the size of any maximal subgroup of J2, a contradiction. Thus, (r, s) cannot be (63,4 032). □
Proposition 2. (r,s) = (315, 3 780).
Proof. To obtain a contradiction, suppose that (r, s) = (315,3780), i.e., r = 315 and s = 3 780 (or [J2 : H2] = 315 and [J2 : H3] = 3 780) and, consequently, |H2| = 3 840 and H = 320. Since |H2| = 3 840 and the maximal subgroups of J2 are given in Table 2, it follows that H2 does not sit in any of the maximal subgroups of J2, a contradiction. Thus, (r, s) cannot be (315,3 780). □
Proposition 3. (r, s) = (945, 3150).
Proof. To obtain a contradiction, suppose that (r, s) = (945,3150), i.e., r = 945 and s = 3150 (or [J2 : H2] = 945 and [J2 : H3] = 3 150) and, consequently, |H2| = 1 280 and H = 384. Since |H2| = 1280 and the maximal subgroups of J2 are given in Table 2, we see that H2 is not among the maximal subgroups of J2 and does not sit in any of them. This contradiction proves that (r, s) cannot be (945,3 150). □
Proposition 4. The action of J2 on Irr(46) is dual to the action of J2 on the conjugacy classes of N = 46.
Proof. We have seen in Section 2 that the action of J2 on the conjugacy classes of N = 46 produced 3 orbits of lengths 1, 1 575, and 2 520. From (3.1), we have r+s = 4 095, where r and s are the lengths of the second the third orbits on the action of J2 on Irr(46). Further, by (3.2), we have (r, s) € {(63,4 032), (315,3 780), (945,3 150), (1 575,2 520)}. We also proved in Propositions 1, 2, and 3 that (r,s) € {(63,4 032), (315,3 780), (945,3 150)}. Therefore, (r,s) = (1575,2 520) and the action of J2 on Irr(46) is dual to the action of J2 on the conjugacy classes of N = 46, as claimed. □
Proposition 5. The inertia factor groups have the forms 22+4:S3 and 22 x A5.
Proof. From Proposition 4, we can see that the orbit lengths on the action of J2 on Irr(46) are 1, 1 575, and 2 520. It follows that [G : Hi] = 1, [G : H2] = 1575 and [G : H3] = 2 520 and, consequently, Hi = G = J2, H = 384, and H = 240. By (3.3), we also have |Irr(H2)| + |Irr(H3)| = 32. Now we investigate the maximal subgroups of J2 to locate H2 and H3. Since |H2| = 384 and the maximal subgroups of J2 are given in Table 2, it follows that H2 is either an index 5 subgroup of 2-+4:A5 or an index 3 subgroup of 22+4:(3 x S3). If H2 < 2-+4:A5 is such that [2-+4:A5 : H2] = 5, then H2 must be a maximal subgroup in it since the index is a prime number. Now, 2-+4:A5 has 4 maximal subgroups of orders 384, 320, 192, and 120. The maximal subgroup of order 384 has the structure 22+4:6 and 19 ordinary irreducible characters. Also, if H2 < 22+4:(3 x S3) is such that [22+4:(3 x S3) : H2] = 3, then H2 must be a maximal subgroup in it since the index is a prime number. Now, 22+4:(3 x S3) has 4 maximal subgroups of orders 576, 384 (twice), and 72. The two maximal subgroups of order 384 have structures 22+4:S3 and 21+4:A4, where |Irr(22+4:S3)| = 12 and |Irr(21+4:A)| = 15. Thus, we have
H2 € {22+4:6, 22+4:S3, 21+4:^4}, |Irr(22+4:6)| = 19, |Irr(22+4:S3)| = 12, |Irr(21+4:^4)| = 15. ^
Next, consider H3. Since |H3| = 240 and the maximal subgroups of J2 are given in Table 2, we deduce that H3 is either
• an index 9 subgroup of (3'A6):2,
• an index 8 subgroup of 2-+4:A5, or
• an index 3 subgroup of A4 x A5.
Consider each of these cases. Using GAP, one can see that the group (3^A6):2 has four maximal subgroups of orders 1080, 216, 60, and 48. Therefore, H3 cannot be a subgroup of (3^A6):2 since [(3^A6):2 : H3] = 9, which is impossible. Next, consider the case where H3 is an index 8 subgroup of 2-+4:A5. Checking the order of all maximal subgroups of 2-+4:A5, which can be done using GAP, shows that 2-+4:A5 has four maximal subgroups of orders 384, 320, 192, and 120. Therefore, H3 £ 2-+4:A5. Finally, we turn to the last case where we consider H3 to be a subgroup of A4 x A5 of index 3. The group A4 x A5 has five maximal subgroups of orders 240, 180, 144, 120, and 72. The maximal subgroup of order 240 has the structure 22 x A5 and 20 ordinary irreducible characters. We deduce that H3 has the structure 22 x A5 and |Irr(H3)| = 20. Using this together with (3.4), we conclude that (H2,H3) = (22+4:S3,22 x A5) is the required pair of inertia factor groups since it consists of (3.3), and all other possibilities are exhausted and each lead to a contradiction, except (H2, H3) = (22+4:S3,22 x A5). Hence, we have the result. □
Next, we construct the character tables of Hi, H2, and H3 and determine the fusions of the conjugacy classes of these groups into the classes of H1 = G = J2. The character table of the simple Janko group J2 can be found at the Atlas. As subgroups of G = J2 that generated by g1 and g2 given in Section 1, and a being a generator of F4, the two inertia factor groups H2 = 22+4:S3 and H3 = 22 x A5 are generated as follows: H2 = (a1 ,a2} and H3 = }, where
/ 1 a a a 0 \ / a 1 a 1 a2 a
a2 a2 1 a a a2 0 1 0 0 0 a
a2 a a2 a2 a2 a2 a 0 1 0 a 1
1 a 1 1 1 0 , a2 = a a 1 a2 a2 a
a2 a a a2 a a2 a a2 a a a a2
a2 a a a2 a2 a / V a 1 a2 a2 0 a2 )
1 a a2 0 1 1 \ ( a 0 a2 0 1 a2 \
0 0 a2 1 1 1 1 a2 a a a 0
0 a2 0 1 a2 a2 , A = a2 a 1 0 1 1
a a 1 1 1 1 0 a a2 a 0 a2
1 0 a2 a2 0 0 0 a a2 1 1 0
0 1 a2 0 a 0 / k 0 1 a a 0 a2
We recursively use Clifford-Fischer theory to construct the character table of H2. The action of S3 on the set Irr(22+4) produced 6 orbits of lengths 1, 3, 3, 3, 3, and 6 with the corresponding inertia factor groups S3, Z2 (four times), and the identity group. Also, H3 is the direct product of the elementary abelian group 22 by A5. Thus, the character table of H3 is easy to construct since we know the character tables of both 22 and A5. In this paper, we list the full character tables of H2 and H3 and organize the columns of the character tables according to the orders and the sizes of the centralizers.
Recall that H2 and H3 are not maximal subgroups of J2, but they are maximal of some maximal subgroups of J2 (H2 is a maximal subgroup of 22+4:(3 x S3) while H3 is a maximal subgroup of A4 x A5). We determined the fusions of the conjugacy classes of H2 and H3 into the classes J2 using the permutation characters of J2 on 22+4:(3 x S3) and A4 x A5; the permutation characters of 22+4:(3 x S3) and A4 x A5 on H2 and H3, respectively, together with the sizes of centralizers. The following proposition plays a great role in determining the fusions; its proof can be found in [2].
Proposition 6. Let Ki < K2 < K3, and let ^ be a class function on Ki. Then, (^tK\ )tK = ^tK\ ■ More generally, if Ki < K2 < ••• < Kn is a nested sequence of subgroups of Kn and ^ is a class function on Ki, then (^tK\ )tK ' " tK^-i = ^tK" ■
Proof. See Proposition 3.5.6 of [2]. □
We supply the full character tables of the inertia factor groups H2 and H3 together with the fusions of their conjugacy classes into the classes of J2 in Tables 3 and 4.
Table 3. The character table of H2 = 22+4:S3.
to la 2 a 2b 2c 3a 4a 4b 4c 4 d 8 a 8b 8c
\CHM\ 384 128 16 16 3 32 32 32 16 8 8 8
1A 2A 2 B 2A 3B 4A 4A 4A 4A 84 84 84
X\ 1 1 1 1 1 1 1 1 1 1 1 1
X'2 1 1 1 1 1 1 1
Xi 2 2 -1 2 2 2
X4 3 3 -1 -1 0 3 -1 -1 -1 1 1
X5 3 3 -1 -1 0 -1 3 -1 -1 1 1
X6 3 3 -1 1 0 -1 3 -1 1 -1 1
X" 3 3 -1 1 0 3 -1 -1 1 1 -1
X8 3 3 -1 0 -1 -1 3 1 -1 1
X9 3 3 -1 1 0 -1 -1 3 1 1
Xio 6 6 2 0 0 -2 -2 -2 0 0 0 0
Xn 12 -4 0 -2 0 0 0 0 2 0 0 0
Xl2 12 -4 0 2 0 0 0 0 -2 0 0 0
4. Fischer matrices of G = 46: J2
We now calculate the Fischer matrices of G = 46:J2. Following Section 3 of [5], we label the top and bottom of the columns of the Fischer matrix Fi corresponding to gi by the sizes of the cent.ralizers of gij, 1 < j < c(gi), in G and my, respectively.
The rows of Fi are partitioned into parts Fik, 1 < k < t, corresponding to the inertia factors H\,H2,... ,Ht, where each Fik consists of c(gik) rows corresponding to the a-1-regular classes (those are the Hk-classes that fuse to the class [gi]g). Thus, each row of Fi is labeled by a pair (k,m,), where 1 < k < t and 1 < m < c{gik). We list the values of \C-Q(gij)\ and my, 1 < i < 27, 1 < j < c(gi), in Table 1. The fusions of classes of H2 and H3 into classes of G are given in Tables 3 and 4, respectively. Since the size of the Fischer matrix Fi is c(gi), it follows from Table 1 that the sizes of the Fischer matrices of G = 46: J2 range between 1 and 8 for every i €{1,2,..., 21}.
The Fisher matrices have interesting arithmetic properties (see Proposition 3.6 in [5]). We used these properties to calculate some entries of these matrices and construct systems of algebraic equations. We solved these systems of equations using the symbolic mathematical package Maxima [19] and, hence, computed all of the Fisher matrices G that we list below.
F1
5i 512 513
o(gij) 1 2 2
Ш9г3)\ 2 477260 800 1 570 864 983 040
(к, m) \Снк{д\кт)\
(1Д) 604 800 1 1 1
(2,1) 384 1575 39 -25
(3,1) 240 2 520 -40 24
mij 1 1575 2 520
F2
52 521 522 523 524
0(92j) 2 2 4 4
l%(52,)l 491 520 32 768 4 096 4 096
(к, m) \Снк(д2кт)\
(1,1) 1920 1 1 1 1
(2,1) 128 15 15 -1 -1
(2,2) 16 120 -8 -8 8
(3,1) 16 120 -8 8 -8
m2j 16 240 1920 1920
F3
5з 531 532 5зз 534 535 536 537 538
o(gsj) 2 4 4 4 4 4 4 4
Шд3з)\ 15 360 15 360 15 360 15 360 1024 1024 1024 1024
(к, m) \Снк(дЗкт)\
(1,1) 240 1 1 1 1 1 1 1 1
(2,1) 16 15 15 15 15 -1 -1 -1 -1
(3,1) 240 1 -1 -1 1 -1 -1 1 1
(3,2) 240 1 -1 1 -1 -1 1 1 -1
(3,3) 240 1 1 -1 -1 -1 1 -1 1
(3,4) 16 15 -15 15 -15 1 -1 1 -1
(3,5) 16 15 -15 -15 15 1 -1 -1 1
(3,6) 16 15 15 -15 -15 1 1 -1 -1
m3j 64 64 64 64 960 960 960 960
F5
Л 55 551 552 553
54 541 0(95 j) 3 6 6
O(54J) 3 \Сс(9ы)\ 576 192 48
l%(54,)l 1080 (к, m) \Снк{д5кт)\
(k, m) \Снк(д4кт)\ (1,1) 36 1 1 1
(1,1) 1080 1 (2,1) 3 12 -4 0
ni4j 4 096 (3,1) 12 3 3 -1
m5j 256 768 3 072
F6
56 961 962 963 964 965
o(96j) 4 4 4 4 4
\CG(963)\ 1536 512 512 512 256
(k, m) \Снк{9бкт)\
(1Д) 96 1 1 1 1 1
(2,1) 32 3 -1 -1 3 -1
(2,2) 32 3 3 -1 -1 -1
(2,3) 32 3 -1 3 -1 -1
(2,4) 16 6 -2 -2 -2 2
m6j 256 768 768 768 1536
Ft
f8
97 971 972
0(97j) 5 10
Ш973)\ 4 800 320
(к, m) 1 CHk(g7km)\
(1,1) 300 1 15 1 -1
(3,1) 20
m7j 256 3 840
98 981 982
o(gsj) 5 10
Ш983)\ 4 800 320
(к, m) \СНк(д8кт)\
(1,1) 300 1 15 1 -1
(3,1) 20
m8j 256 3 840
Fq
Fio
99 991
о(99j) 5
\ОсШ 50
(к, m) 1 СНк(99кт)\
(1,1) 50 1
rriQj 4 096
910 9w,i
0(910j) 5
\Cc(9wj)\ 50
(к, m) \CHk(giokm)\
(1,1) 50 1
mWj 4 096
Fu
9n,i
o(gnj) 6
\Сс(дп3)\ 24
(к, m) \CHk(gukm)\
(1,1) 24 1
rriuj 4 096
fl2
912 512,1 512,2 512,3 512,4
o(g12j) 6 12 12 12
\Cc(gi23)\ 48 48 48 48
(k, m) \Снк(912кт)\
(1,1) 12 1 1 1 1
(3,1) 12 1 -1 1 -1
(3,2) 12 1 1 -1 -1
(3,3) 12 1 -1 -1 1
m12j 1024 1024 1024 1024
Fl3
513 5i3,i
o(gi3j) 7
I%(513,)| 7
(к, m) \CHk(gi3km)\
(1,1) 7 1
rriizj 4096
Fl4
514 514,1 514,2 514,3 514,4
o(gi43) 8 8 8 8
I%(514j)| 32 32 32 32
(k, m) \Снк(д14кт)\
(1,1) 8 1 1 1 1
(2,1) 12 1 -1 1 -1
(2,2) 12 1 1 -1 -1
(2,3) 12 1 -1 -1 1
mUj 1024 1024 1024 1024
F15
515 515,1 515,2 515,3 515,4
o{gi5j) 10 20 20 20
I%(515J)| 80 80 80 80
(fc, m) \CHk(gi5km)\
(1,1) 8 1 1 1 1
(3,1) 12 1 -1 1 -1
(3,2) 12 1 1 -1 -1
(3,3) 12 1 -1 -1 1
mi5j 1024 1024 1024 1024
f16
516 5i6,i 516,2 5i6,3 516,4
o(5i6j) 10 20 20 20
I%(5i6j)| 80 80 80 80
(к, m) |C#fe(5l6fcm)|
(1,1) 8 1 1 1 1
(3,1) 12 1 -1 1 -1
(3,2) 12 1 1 -1 -1
(3,3) 12 1 -1 -1 1
mi6j 1024 1024 1024 1024
F17
f18
517 517,1
o(gnj) 10
\Cd9i7j)\ 10
(к, m) \CHk(gnkm)\
(1,1) io 1
>m17j 4 096
518 g 18,1
o(gi8j) 10
Ш518,)| 10
(к, m) \CHk(gi8km)\
(1,1) 10 1
m18j 4 096
f19
f20
519 519,1
o{gi9j) 12
I<%(5i9j)| 12
(к, m) \CHk(gi9km)\
(1,1) 12 1
migj 4 096
520 520,1
o{g-20j) 15
l%(520i)| 15
(к, m) \СНк(920кт)\
(1,1) 15 1
nV20j 4 096
f21
521 521,1
0(521j) 15
l%(521,)| 15
(к, m) \Снк(д-21кт)\
(1,1) 15 1
m2ij 4 096
5. Character table of G = 46: J2
In Sections 2, 3, and 4, we have determined:
• the conjugacy classes of G = 46:J2 (Table 1);
• the inertia factors H1, H2, and H3;
• the character tables of all inertia factor groups of G (the Atlas together with Tables 3 and 4); in these two tables, we also supplied the fusions of the classes of the inertia factors H2 and H3 into classes of G;
• the Fischer matrices of G (see Section 4).
Following [2, 5], without any difficulties, one can construct the full character table of G in the format of Clifford-Fischer theory. The table will be composed of 63 parts corresponding to 21 cosets and three inertia factor groups. The full character table of G is a 53 x 53 R-valued matrix, and we give it in the format of Clifford-Fischer theory in Table 5. We conclude by remarking that the accuracy of this character table has been tested using GAP.
Acknowledgements
The author would like to thank Professor J.Moori, from whom he studied group theory and representation theory. The author would also like to thank the University of Limpopo for providing financial support.
REFERENCES
1. Ali F., Moori J. The Fischer-Clifford matrices and character table of a maximal subgroup of Fi24. Algebra Colloq., 2010. Vol. 17, No. 3. P. 389-414. DOI: 10.1142/S1005386710000386
2. Basheer A. B. M. Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups. PhD Thesis. Pietermaitzburg: University of KwaZulu-Natal, 2012.
3. Basheer A. B. M. On a group involving the automorphism of the Janko group J2. J. Indones. Math. Soc., 2023. Vol. 29, No. 2. P. 197-216. DOI: 10.22342/jims.29.2.1371.197-216
4. Basheer A. B. M. On a group extension involving the Suzuki group Sz(8). Afr. Mat., 2023. Vol. 34, No. 4. Art. no. 96. DOI: 10.1007/s13370-023-01130-z
5. Basheer A. B. M., Moori J. Fischer matrices of Dempwolff group 25 GL(5, 2). Int. J. Group Theory, 2012. Vol. 1, No. 4. P. 43-63. DOI: 10.22108/IJGT.2012.1590
6. Basheer A. B. M., Moori J. On the non-split extension group 26 Sp(6, 2). Bull. Iranian Math. Soc., 2013. Vol. 39, No. 6. P. 1189-1212.
7. Basheer A.B.M.,Moori J. On the non-split extension 22nSp(2n, 2). Bull. Iranian Math. Soc., 2015. Vol. 41, No. 2. P. 499-518.
8. Basheer A.B.M., Moori J. On a maximal subgroup of the Thompson simple group. Math. Commun., 2015. Vol. 20, No. 2. P. 201-218. URL: https://hrcak.srce.hr/149786
9. Basheer A. B.M., Moori J. A survey on Clifford-Fischer theory. In: Groups St Andrews 2013, C.M. Campbell, M.R. Quick, E.F. Robertson, C.M. Roney-Dougal (eds.) London Math. Soc. Lecture Note Ser., vol. 422. Cambridge University Press, 2015. P. 160-172. DOI: 10.1017/CBO9781316227343.009
10. Basheer A. B.M., Moori J. On a group of the form 37:Sp(6, 2). Int. J. Group Theory, 2016. Vol. 5, No. 2. P. 41-59. DOI: 10.22108/IJGT.2016.8047
11. Basheer A.B.M., Moori J. On two groups of the form 28:A9. Afr. Mat., 2017. Vol. 28. P. 1011-1032. DOI: 10.1007/s13370-017-0500-1
12. Basheer A.B.M., Moori J. On a group of the form 210:(U5(2):2). Ital. J. Pure Appl. Math., 2017. Vol. 37. P. 645-658. URL: https://ijpam.uniud.it/online_issue/201737/57-BasheerMoori.pdf
13. Basheer A. B. M., Moori J. Clifford-Fischer theory applied to a group of the form 2-+6:((31+2:8):2). Bull. Iraman Math. Soc., 2017. Vol. 43, No. 1. P. 41-52.
14. Basheer A. B.M., Moori J. On a maximal subgroup of the affine general linear group GL(6, 2). Adv. Group Theory Appl., 2021. Vol. 11. P. 1-30. DOI: 10.32037/agta-2021-001
15. Bosma W., Cannon J.J. Handbook of Magma Functions. Sydney: University of Sydney, 1994.
16. Conway J. H., Curtis R. T., Norton S. P., Parker R. A., Wilson R. A. ATLAS of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford: Clarendon Press, 1985. 250 p.
17. Fray R. L., Monaledi R. L., Prins A. L. Fischer-Clifford matrices of 28:(U4(2):2) as a subgroup of O+o(2). Afr. Mat., 2016. Vol. 27. P. 1295-1310. DOI: 10.1007/s13370-016-0410-7
18. GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra. Version 4.4.10, 2007. URL: http://www.gap-system.org
19. Maxima, A Computer Algebra System. Version 5.18.1, 2009. URL: http://maxima.sourceforge.net
20. Moori J. On the Groups G+ andG of the form. 210:M22 and 210:M22. PhD Thesis. Birmingham: University of Birmingham, 1975.
21. Moori J. On certain groups associated with the smallest Fischer group. J. London Math. Soc., 1981. Vol. 2. P. 61-67.
22. Wilson R. A. The Finite Simple Groups. London: Springer-Verlag, 2009. XV, 298 p. DOI: 10.1007/978-1-84800-988-2
23. Wilson R. A., et al. Atlas of Finite Group Representations. Version 3. URL: http://brauer.maths.qmul.ac.uk/Atlas/v3/
Table 4. The character table of H3 = 22xA5
o 1—1 o 1—1 o 1—1 1—1 1—1 1 1—1 1—1 1 * * 1 1 * * 1 1 1—1 1 1—1 1—1 1 1—1 o o o o
13 o i—i o 1—1 oq o 1—1 1—1 1—1 1—1 1 1—1 1 * * 1 * 1 1 * 1 1—1 1 1—1 1 1—1 1—1 o o o o
ts o 1—1 o fM oq o 1—1 1—1 1—1 1 1—1 1 1—1 * 1 * 1 1 * 1 * 1—1 1 1—1 1—1 1—1 1 o o o o
O i—I o fM o 1—1 1—1 1—1 1 1—1 1 1—1 * * 1 1 * 1 1 * 1—1 1 1—1 1—1 1—1 1 o o o o
o i—i o fM o 1—1 1—1 1—1 1—1 1 1—1 1 * * * 1 1 * 1 1 1—1 1 1—1 1 1—1 1—1 o o o o
e o i—i o fM oq o 1—1 1—1 1—1 1 1—1 1—1 1 * 1 * 1 * ^ ^ 1 * 1 1—1 1 1—1 1—1 1 1—1 o o o o
iO fM 1—1 oq iO 1—1 1—1 1 1—1 1—1 1 o O o o o o o o 1 1 1 1
iO fM 1—1 Dq iO 1—1 1—1 1—1 1 1—1 1 o o o o o o o o 1 1 1 1
e iO fM 1—1 Dq iO 1—1 1—1 1 1—1 1 1—1 o o o o o o o o 1 1 1 1
lO o fM ^ lO 1—1 1—1 1—1 1—1 * * * * 1—1 1 1—1 1 1—1 1 1—1 1 o o o o
e lO o fM Dq lO 1—1 1—1 1—1 1—1 * * * * 1—1 1 1—1 1 1—1 1 1—1 1 o o o o
e 00 fM 1—1 oq 00 1—1 1—1 1—1 1—1 o o o o o o o o 1 1 1 1
fM iO 1—1 ^ fM 1 1 1 1 1 1 1 1 o o o o 1—1 1—1 1—1 1—1
fM iO 1—1 oq fM 1 1 1 1 1 1 o o o o 1—1 1—1 1—1 1 1—1 1
13 fM iO 1—1 oq fM 1 1 1 1 1 1 o o o o 1—1 1—1 1 1—1 1 1—1
ts fM iO 1—1 oq fM 1 1 1 1 1 1 o o o o 1—1 1—1 1 1—1 1—1 1
u fM o fM oq fM 1—1 1—1 1 1—1 1 1—1 00 00 00 1 00 1 00 1 00 1 00 00 4 4 1 4 1 4 lO lO 1 lO 1 lO
^ fM o fM oq fM 1—1 1—1 1 1—1 1—1 1 00 00 00 1 00 1 00 00 00 1 00 1 4 4 1 4 4 1 lO lO 1 lO lO 1
e fM o fM oq fM 1—1 1—1 1—1 1 1—1 1 00 00 00 00 00 1 00 1 00 1 00 1 4 4 4 1 4 1 lO lO lO 1 lO 1
e i—i o fM 1—1 1—1 1—1 t-1 1—1 00 00 00 00 00 00 00 00 4 4 4 4 lO lO lO lO
CO ¡u 3 co £ T 'p. cs co lO 6 00 o £ £ cs £ co £ £ lO £ 6 £ £ 00 £ & o
where in Table 4, A = (1 - y/5)/2 and A* = (1 + y/5)/2.
to
l9i] J, 1A 2A 2 B 3A 3B
[9»iI?? la 2a 2b 2c 2d 4a 4 b 2e 4c id 4e 4/ 49 4h 4i 3a 3b 6a 6b
- ' 2477260800 1572864 983040 491520 32768 4096 4096 15360 15360 15360 15360 1024 1024 1024 1024 1080 576 192 48
XI 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X2 14 14 14 -2 -2 -2 -2 2 2 2 2 2 2 2 2 5 -1 -1 -1
X3 14 14 14 -2 -2 -2 -2 2 2 2 2 2 2 2 2 5 -1 -1 -1
X4 21 21 21 5 5 5 5 -3 -3 -3 -3 -3 -3 -3 -3 3 0 0 0
X5 21 21 21 5 5 5 5 -3 -3 -3 -3 -3 -3 -3 -3 3 0 0 0
X6 36 36 36 4 4 4 4 0 0 0 0 0 0 0 0 9 0 0 0
X7 63 63 63 15 15 15 15 -1 -1 -1 -1 -1 -1 -1 -1 0 3 3 3
X8 70 70 70 -10 -10 -10 -10 -2 -2 -2 -2 -2 -2 -2 -2 7 1 1 1
X9 70 70 70 -10 -10 -10 -10 -2 -2 -2 -2 -2 -2 -2 -2 7 1 1 1
X10 90 90 90 10 10 10 10 6 6 6 6 6 6 6 6 9 0 0 0
Xll 126 126 126 14 14 14 14 6 6 6 6 6 6 6 6 -9 0 0 0
X12 160 160 160 0 0 0 0 4 4 4 4 4 4 4 4 16 1 1 1
X13 175 175 175 15 15 15 15 -5 -5 -5 -5 -5 -5 -5 -5 -5 1 1 1
X14 189 189 189 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 0 0 0 0
X15 189 189 189 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 0 0 0 0
X16 224 224 224 0 0 0 0 -4 -4 -4 -4 -4 -4 -4 -4 8 -1 -1 -1
X17 224 224 224 0 0 0 0 -4 -4 -4 -4 -4 -4 -4 -4 8 -1 -1 -1
X18 225 225 225 -15 -15 -15 -15 5 5 5 5 5 5 5 5 0 3 3 3
X19 288 288 288 0 0 0 0 4 4 4 4 4 4 4 4 0 -3 -3 -3
X20 300 300 300 -20 -20 -20 -20 0 0 0 0 0 0 0 0 -15 0 0 0
X21 336 336 336 16 16 16 16 0 0 0 0 0 0 0 0 -6 0 0 0
X22 1 575 39 -25 135 7 -9 7 15 15 15 15 -1 -1 -1 -1 0 12 -4 0
X23 1 575 39 -25 -105 23 7 -9 15 15 15 15 -1 -1 -1 -1 0 12 -4 0
X24 3 150 78 -50 30 30 -2 -2 30 30 30 30 -2 -2 -2 -2 0 -12 4 0
X25 4 725 117 -75 -75 53 5 -11 -15 -15 -15 -15 1 1 1 1 0 0 0 0
X26 4 725 117 -75 165 37 -11 5 -15 -15 -15 -15 1 1 1 1 0 0 0 0
X27 4 725 117 -75 165 37 -11 5 -15 -15 -15 -15 1 1 1 1 0 0 0 0
X28 4 725 117 -75 165 37 -11 5 -15 -15 -15 -15 1 1 1 1 0 0 0 0
X29 4 725 117 -75 -75 53 5 -11 -15 -15 -15 -15 1 1 1 1 0 0 0 0
X30 4 725 117 -75 -75 53 5 -11 -15 -15 -15 -15 1 1 1 1 0 0 0 0
X31 9 450 234 -150 90 90 -6 -6 30 30 30 30 -2 -2 -2 -2 0 0 0 0
X32 18 900 468 -300 180 -76 -12 20 0 0 0 0 0 0 0 0 0 0 0 0
X33 18 900 468 -300 -300 -44 20 -12 0 0 0 0 0 0 0 0 0 0 0 0
X34 2 520 -40 24 120 -8 8 -8 48 -16 -16 -16 0 0 0 0 0 3 3 -1
X35 2 520 -40 24 120 -8 8 -8 -16 -16 -16 48 0 0 0 0 0 3 3 -1
X36 2 520 -40 24 120 -8 8 -8 -16 -16 48 -16 0 0 0 0 0 3 3 -1
X37 2 520 -40 24 120 -8 8 -8 -16 48 -16 -16 0 0 0 0 0 3 3 -1
X38 7 560 -120 72 -120 8 -8 8 -36 12 12 12 -4 12 -4 -4 0 0 0 0
X39 7 560 -120 72 -120 8 -8 8 -36 12 12 12 -4 12 -4 -4 0 0 0 0
X40 7 560 -120 72 -120 8 -8 8 12 12 12 -36 -4 -4 12 -4 0 0 0 0
X41 7 560 -120 72 -120 8 -8 8 12 12 12 -36 -4 -4 12 -4 0 0 0 0
X42 7 560 -120 72 -120 8 -8 8 12 12 -36 12 -4 -4 -4 12 0 0 0 0
X43 7 560 -120 72 -120 8 -8 8 12 12 -36 12 -4 -4 -4 12 0 0 0 0
X44 7 560 -120 72 -120 8 -8 8 12 -36 12 12 12 -4 -4 -4 0 0 0 0
X45 7 560 -120 72 -120 8 -8 8 12 -36 12 12 12 -4 -4 -4 0 0 0 0
X46 10 080 -160 96 0 0 0 0 12 -4 -4 -4 -4 12 -4 -4 0 3 3 -1
X47 10 080 -160 96 0 0 0 0 -4 -4 -4 12 -4 -4 12 -4 0 3 3 -1
X48 10 080 -160 96 0 0 0 0 -4 -4 12 -4 -4 -4 -4 12 0 3 3 -1
X49 10 080 -160 96 0 0 0 0 -4 12 -4 -4 12 -4 -4 -4 0 3 3 -1
X50 12 600 -200 120 120 -8 8 -8 60 -20 -20 -20 -4 12 -4 -4 0 -3 -3 1
X51 12 600 -200 120 120 -8 8 -8 -20 -20 -20 60 -4 -4 12 -4 0 -3 -3 1
X52 12 600 -200 120 120 -8 8 -8 -20 -20 60 -20 -4 -4 -4 12 0 -3 -3 1
X53 12 600 -200 120 120 -8 8 -8 -20 60 -20 -20 12 -4 -4 -4 0 -3 -3 1
Continued on next page
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