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We analyze the group structures of two groups of order 1344 which arerespectively non-split and split extensions of the elementary Abelian group oforder 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 theset of octonions \pm e_i , (i=1,2,...,7) representing a 7-dimensionaloctahedron.Its three maximal subgroups 2^3:7:3, 2^3.S_4 and 4.S_4:2 correspondto 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 maximalsubgroups 2^3:7:3, 2^3:S_4, 4:S_4:2 and two non-conjugate Klein's groupPSL_2(7).The character tables of the groups and their maximal subgroups, tensorproducts and decompositions of the irreducible representations under therelevant maximal subgroups are identified. Possible implications in physics arediscussed.

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