(vi) For each 1≤i≤r, the weights mij satisfy the relation
c(gi)
X
j=1
mij =|N|, (vii) Column Orthogonality Relation:
X
(k,m)∈Ji
|CHk(gikm)|a(k,m)ij a(k,m)
ij0 =δjj0|CG(gij)|, (viii) Row Orthogonality Relation:
c(gi)
X
j=1
mija(k,m)ij a(k
0,m0)
ij =δ(k,m)(k0,m0)a(k,m)i1 |N|.
PROOF. Proofs for many assertions of Proposition 5.2.2 can be founded in Moori’s students theses, for example see Ali [1] or Mpono [55] and some other assertions are provided in Schiffer [67] as well
as in Moori and Basheer [10] and Lux and Pahlings [58].
We conclude this section on Fischer matrices by remarking that the Fischer matrices of any exten- sion Gare unique up to the ordering of the columns and rows, provided that the character tables of the Hk have been determined. Permuting the columns of the Fischer matrices will result in a corresponding permutation of the columns of the character table ofG.
Chapter 5 – The Theory of Clifford-Fischer Matrices
the classes [gij]G, 1≤j≤c(gi),are given by the matrixKikFik,whereFik is the sub-matrix ofFi defined previously with rows correspond to the pairs (k, gik1),(k, gik2),· · ·,(k, gikc(gik)).Note that the size of Kik is|IrrProj(Hk, α−1k )| ×rik and the size ofFik is rik×c(gi).Therefore the character table of Gwill have the form
g1 g2 · · · gr
g11 g12 · · · g1c(g1) g21 g22 · · · g2c(g2) · · · gr1 gr2 · · · grc(gr)
K1 K11F11 K12F12 · · · K1rF1r
K2 K21F21 K22F22 · · · K2rF2r
... ... ... . .. ...
Kt Kt1Ft1 Kt2Ft2 · · · KtrFtr
Note 5.3.1. Observe that characters ofG contained in the blockK1 are just Irr(G) and therefore the size of K1iF1i,for each 1≤i≤r, is |Irr(G)| ×c(gi).In particular, columns of K11F11 are the degrees of irreducible characters ofGrepeated themselvesc(g1) times, where we know thatc(g1) is the number ofG−conjugacy classes obtained from the normal subgroupN.Also note that|Irr(G)|
satisfies the relation
|Irr(G)|=
t
X
k=1
|IrrProj(Hk, α−1k )|=|Irr(G)|+
t
X
k=2
|IrrProj(Hk, α−1k )|, (5.7) whereαk is a factor set ofM(Hk).
One of the biggest challenges in the theory of Clifford-Fischer matrices is the determination of the type of the character table of Hk (projective or ordinary), which is to be used in the construction of the character table of G. We may firstly assume that all the irreducible characters of N are extendible to their respective inertia groups and consequently all the character tables of theHkthat we need to use are the ordinary ones. However, in general, there is no reason guaranteing that one can work with the ordinary characters ofHk, 2≤k≤t(see Section 5.1 for some partial results on extendability of characters). Thus in practice making the right choice of the appropriate projective character table of Hk might be difficult unless the Schur multipliers of all the Hk are trivial.
Otherwise one have to test all the possible choices, and each choice will lead to a contradiction, except for one case that will be successful. Sometimes Eq. (5.7) might also be useful to prove that we need to use projective characters of some of the inertia factors of a specific extension. For
example in Section 7.1 we have found that the group G = 25·GL(5,2) has 41 conjugacy classes and we proved that the inertia factors are H1 = G = GL(5,2) and H2 = 24:GL(4,2), where we know that |Irr(H1)|= 27 and |Irr(H2)| = 25. Thus if we will use the ordinary character table of H2 we get that|Irr(G)|= 25 + 27 = 526= 41 = the number of conjugacy classes of G. This simple argument tells that we have to make use of projective characters ofH2.
We conclude this section by remarking that from Pahlings and Lux [58], the character table of G in the format of Clifford-Fischer theory can be re-written as a product of two matrices, where the first one is a block diagonal matrix with the (projective) character tables of the inertia factors Hk on the diagonal and the second matrix being built up by the Fischer matrices and zeros, arranged in a suitable way.
Chapter 6
A Group of the Form 3 7 :Sp(6, 2)
In this chapter we study a split extension group of the formG= 37:Sp(6,2),where in Section 6.1 we give a brief introduction and we generateGas a subgroup of SL(8,3).In Section 6.2 we list the conjugacy classes ofGobtained by applying the coset analysis technique and see that corresponding to the 30 conjugacy classes ofSp(6,2),we get 118 classes forG.In Section 6.3 through a long process we show that there are six inertia factors, namelyH1 =Sp(6,2), H2=U4(2), H3 = 25:A6, H4 =A7, H5= (3×A6):2 andH6 = (24:A5):2.In Section 6.4 we discuss the character tables of these groups and we supply these tables in the Appendix of this thesis. In Section 6.5 we calculate the 30 Fischer matrices of G and we see that the sizes of these matrices range between 2 and 10. In Section 6.6 we show how to obtain the character table ofG= 37:Sp(6,2) via Clifford-Fischer theory. The full character table ofGis given in the Appendix (see Table 11.8).
6.1. Introduction
LetG=Sp(6,2) be the symplectic group of order 1451520. By the electronic Atlas of Wilson [71], the group G has a 7−dimensional (absolutely) irreducible module over GF(3) = F3 = {0,1, ξ}, whereξis a primitive element of the fieldF3.Consequently a split extension of the form 37:Sp(6,2) does exist. Using the two 7×7 matrices overF3 that generate Sp(6,2),supplied by the electronic Atlas of Wilson, we were able to constructG and thenGinside GAP [30]. In fact we constructed the group G in GAP, in terms of 8×8 matrices over F3. The following three elementsg1, g2 and g3 are 8−dimensional matrices over F3 that generate G.
g1= 0 B B B B B B B B B B B B B B B B B
@
ξ 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 1 0 0 0 0 0 0
0 ξ ξ ξ 0 0 0 0
0 ξ ξ 0 ξ 0 0 0
0 ξ ξ 0 0 ξ 0 0
0 0 0 0 0 0 ξ 0
0 0 0 0 0 0 0 1
1 C C C C C C C C C C C C C C C C C A
, g2= 0 B B B B B B B B B B B B B B B B B
@
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
1 0 ξ ξ ξ 0 0 0
0 0 0 0 0 0 1 0
0 ξ 1 ξ 1 0 1 0
0 0 0 0 0 0 0 1
1 C C C C C C C C C C C C C C C C C A ,
g3= 0 B B B B B B B B B B B B B B B B B
@
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 1
1 C C C C C C C C C C C C C C C C C A ,
with o(g1) = 2, o(g2) = 7 and o(g3) = 3. However the group G can also be generated by the following two elements α1 and α2,where
α1= 0 B B B B B B B B B B B B B B B B B
@
ξ ξ 1 ξ ξ 1 1 0
0 1 0 ξ ξ ξ 1 0
1 ξ 1 1 1 ξ 1 0
ξ ξ 0 0 1 0 ξ 0 ξ 0 0 1 ξ 1 0 0
0 0 1 0 1 0 ξ 0
0 ξ ξ ξ 1 1 1 0
0 1 ξ 0 0 0 0 ξ
1 C C C C C C C C C C C C C C C C C A
, α2= 0 B B B B B B B B B B B B B B B B B
@
1 ξ ξ ξ 1 ξ 0 0
1 0 ξ ξ 0 0 1 0
1 0 0 ξ 1 0 1 0
0 1 0 0 1 1 1 0
1 0 0 1 ξ 0 1 0
1 1 0 ξ 1 ξ 0 0
1 ξ 1 1 1 0 ξ 0
ξ 0 0 0 0 0 1 ξ 1 C C C C C C C C C C C C C C C C C A ,
witho(α1) = 45, o(α2) = 45 ando(α1? α2) = 6.
Corollary 6.1.1. G≤SL(8,3)with index 302 954 487 010 019 919 360.
PROOF. This is readily verified since det(α1) = det(α2) = 1 and consequently det(g) = 1 for all
g∈G.
Using GAP, one can easily check all the normal subgroups of G. In fact the only proper normal subgroup of G is a group of order 2187 and thus must be isomorphic to the elementary abelian groupN = 37.The following elementsn1, n2,· · · , n7 are 8×8 matrices overF3 that are generators of N.
Chapter 6 – A Group of the Form 37:Sp(6,2)
n1= 0 B B B B B B B B B B B B B B B B B
@
ξ 0 0 0 0 0 0 0
0 ξ 0 0 0 0 0 0
0 0 ξ 0 0 0 0 0
0 0 0 ξ 0 0 0 0
0 0 0 0 ξ 0 0 0
0 0 0 0 0 ξ 0 0
0 0 0 0 0 0 ξ 0
0 0 1 ξ ξ 1 0 ξ
1 C C C C C C C C C C C C C C C C C A
, n2= 0 B B B B B B B B B B B B B B B B B
@
ξ 0 0 0 0 0 0 0
0 ξ 0 0 0 0 0 0
0 0 ξ 0 0 0 0 0
0 0 0 ξ 0 0 0 0
0 0 0 0 ξ 0 0 0
0 0 0 0 0 ξ 0 0
0 0 0 0 0 0 ξ 0
0 1 ξ 1 ξ 0 ξ ξ
1 C C C C C C C C C C C C C C C C C A ,
n3= 0 B B B B B B B B B B B B B B B B B
@
ξ 0 0 0 0 0 0 0
0 ξ 0 0 0 0 0 0
0 0 ξ 0 0 0 0 0
0 0 0 ξ 0 0 0 0
0 0 0 0 ξ 0 0 0
0 0 0 0 0 ξ 0 0
0 0 0 0 0 0 ξ 0
0 1 1 0 1 ξ 0 ξ
1 C C C C C C C C C C C C C C C C C A
, n4= 0 B B B B B B B B B B B B B B B B B
@
ξ 0 0 0 0 0 0 0
0 ξ 0 0 0 0 0 0
0 0 ξ 0 0 0 0 0
0 0 0 ξ 0 0 0 0
0 0 0 0 ξ 0 0 0
0 0 0 0 0 ξ 0 0
0 0 0 0 0 0 ξ 0
1 0 0 0 0 0 0 ξ
1 C C C C C C C C C C C C C C C C C A ,
n5= 0 B B B B B B B B B B B B B B B B B
@
ξ 0 0 0 0 0 0 0
0 ξ 0 0 0 0 0 0
0 0 ξ 0 0 0 0 0
0 0 0 ξ 0 0 0 0
0 0 0 0 ξ 0 0 0
0 0 0 0 0 ξ 0 0
0 0 0 0 0 0 ξ 0
1 0 1 0 0 ξ 1 ξ
1 C C C C C C C C C C C C C C C C C A
, n6= 0 B B B B B B B B B B B B B B B B B
@
ξ 0 0 0 0 0 0 0
0 ξ 0 0 0 0 0 0
0 0 ξ 0 0 0 0 0
0 0 0 ξ 0 0 0 0
0 0 0 0 ξ 0 0 0
0 0 0 0 0 ξ 0 0
0 0 0 0 0 0 ξ 0
1 ξ 0 1 ξ ξ ξ ξ
1 C C C C C C C C C C C C C C C C C A ,
n7= 0 B B B B B B B B B B B B B B B B B
@
ξ 0 0 0 0 0 0 0
0 ξ 0 0 0 0 0 0
0 0 ξ 0 0 0 0 0
0 0 0 ξ 0 0 0 0
0 0 0 0 ξ 0 0 0
0 0 0 0 0 ξ 0 0
0 0 0 0 0 0 ξ 0
1 1 1 1 ξ 1 0 ξ
1 C C C C C C C C C C C C C C C C C A .
In terms of 8×8 matrices over F3,the group Sp(6,2) is generated by the following two elements g1 and g2 :
g1= 0 B B B B B B B B B B B B B B B B B
@
ξ 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 ξ 0 0 0 0
0 0 ξ 0 0 0 0 0
0 0 1 1 1 0 0 0
0 0 1 1 0 1 0 0
0 0 1 1 0 0 1 0
0 0 0 0 0 0 0 1
1 C C C C C C C C C C C C C C C C C A
, g2= 0 B B B B B B B B B B B B B B B B B
@
ξ 0 0 0 0 0 0 0
0 0 ξ 0 0 0 0 0
0 0 0 0 ξ 0 0 0
0 0 0 0 0 ξ 0 0
0 0 0 0 0 0 ξ 0
0 ξ 0 1 1 1 0 0
0 0 0 0 0 0 0 ξ
0 0 1 ξ 1 ξ 0 ξ
1 C C C C C C C C C C C C C C C C C A ,
witho(g1) = 2, o(g2) = 7 ando(g1? g2) = 9.Note that the group Sp(6,2) =hg1, g2i together with the mentioned generators of N,gives the split extensionG= 37:Sp(6,2).