Two groups 2^3.PSL_2(7) and 2^3:PSL_2(7) of order 1344

Abstract

We analyze the group structures of two groups of order 1344 which are respectively non-split and split extensions of the elementary Abelian group of order 8 by its automorphism group PSL_2(7).They share the same character table. The group 2^3.PSL_2(7) is a finite subgroup of the Lie Group G_2 preserving the set of octonions \pm e_i , (i=1,2,...,7) representing a 7-dimensional octahedron.Its three maximal subgroups 2^3:7:3, 2^3.S_4 and 4.S_4:2 correspond to the finite subgroups of the Lie groups G_2, SO(4) and SU(3) respectively. The group 2^3:PSL_2(7) representing the split extension possesses five maximal subgroups 2^3:7:3, 2^3:S_4, 4:S_4:2 and two non-conjugate Klein's group PSL_2(7).The character tables of the groups and their maximal subgroups, tensor products and decompositions of the irreducible representations under the relevant maximal subgroups are identified. Possible implications in physics are discussed.