In the course of an
investigation ofsixdimensional complex linear groups, it was discovered that a central
extension of Z_{6} by PSU_{4}(3) has a representation of degree six. In fact, this
representation has as its image the unimodular subgroup X(G) of index 2 of the
following 6dimensional matrix group: all 6 by 6 permutaion matrices; all
unimodular diagonal matrices of order 3; I_{6} − Q∕3 where Q has all its entries
equal to one . This matrix group leaves the following lattice invariant:
where throughout this paper ω is a primitive third root of
unity; a_{i} − a_{j} ∈Z(ω) for all i,j; . The generators of the
matrix group are similar to the following generators for an 8dimensional
complex linear group with JordanHolder constituents Z_{2}, the nontrivial simple
constituent of 0_{8}(2),Z_{2}; all 8 by 8 permutation matrices, all unimodular
diagonal matrices of order 2, I_{8} − P∕4 where P has all entries equal to 1
.
The projective representation of PSU_{4}(3) can be used to construct a
12dimensional representation Y (H), a central extension of Z_{6} by the Suzuki group,
which leads to the known 24dimensional projective representation of the Conway
group. In fact, H has a subgroup K isomorphic to a central extension of
(Z_{6} × Z_{3}) by PSU_{4}(3). Also, Y H has two sixdimensional constituenls coming
from the above matrix group where the constiluents are related by an outer
automorphism of PSU_{4}(3) which does not lift to the central extension of Z_{6} by
PSU_{4}(3) with the sixdimensional representation. We obtain two commuting
automorphisms, α and β respectively, of G from I_{6} − Q∕3 and complex
conjugation. For PSU_{4}(3), the outer automorphism group is dihedral of order eight
with its center corresponding to complex conjugation of X(G). The entire
automorphism group lifls to K. We may take the center of K to be ⟨a,b,c⟩
with a and b of order 3 and c of order 2, with G≅K∕b, and with α(a) = a,
α(b) = b^{−1}, β(a) = a^{−1}, β(b) = b^{−1}. We can also find an automorphism γ of K
with γ(a) = b and γ(b) = a. We give the character table of K giving only
one representative of each family of algebraically conjugate characters and
classes. Irrational characters and classes are underlined. Only one class in each
coset of Z(K) is represented by the character tables. The characters in the
table U_{4}(3) give the characters with Z(K) in the kernel. The succeeding five
character tables in order give the following linear characters, respectively, on
Z(K) : 𝜃(a) = 𝜃(b) = 1,0(c) = −1; 𝜃(a) = ω, 𝜃(b) = 𝜃(c) = 1; 𝜃(a) = ω^{−1},
𝜃(b) = 1, 𝜃(c) = −1; 𝜃(a) = 𝜃(b) = ω, 𝜃(c) = 1; 𝜃(a) = 𝜃(b) = ω, 𝜃(c) = −1. The
characters with other actions are obtained by applying elements of the outer
automorphism group. The automorphism α transposes π_{7} with π_{7}^{−1}; and N_{1}
with N_{1}^{−1} in the character tables. The automorphism β transposes N_{1}
with N_{1}^{−1}; and N_{2} with N_{2}^{−1}. The automorphism γ transposes T_{1} with
T_{2}; JT_{1} with JT_{2}; N_{1} with N_{2}; N_{1}^{−1} with N_{2}^{−1}; and possibly π_{7} with
π_{7}^{−1}. As SU_{4}(3)∕Ω(ZSU_{4}(3)) has the centralizer of some central involution
isomorphic to the centralizer of some central involution J in G, presumably
SU_{4}(3)∕Ω_{1}(ZSU_{4}(3))≅G∕0_{3}(Z(G)).
