Two groups 23. psl2 (7)‎ and 23 : psl2 (7)‎ of order 1344

Abstract EN

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 ???2(7).

Two groups have the same number of conjugacy classes and the set of dimensions of irreducible representations is equal.

The group 23.???2(7) is a finite subgroup of the Lie Group ?2 preserving the set of octonions ±??,(?=1,2,…,7) representing a 7-dimensional octahedron.

Its three maximal subgroups 23:7:3, 23.?4 and 4.?4:2 correspond to the finite subgroups of the Lie groups ?2,??(4) and ??(3) respectively.

The group 23:???2(7) representing the split extension possesses five maximal subgroups 23:7:3, 23:?4, 4:?4:2 and two non-conjugate Klein’s group ???2(7).

The character tables of the groups and their maximal subgroups, tensor products and decompositions of their irreducible representations under the relevant maximal subgroups are identified.

Possible implications in physics are discussed.