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 𝑃𝑆𝐿2(7). Two groups have the same number of conjugacy classes and the set of dimensions of irreducible representations is equal. The group 2^3.𝑃𝑆𝐿2(7) is a finite subgroup of the Lie Group 𝐺2 preserving the set of octonions ±𝑒𝑖, (𝑖 = 1,2, … ,7) representing a 7-dimensional octahedron. Its three maximal subgroups 2^3: 7: 3, 2^3. 𝑆4 and 4. 𝑆4: 2 correspond to the finite subgroups of the Lie groups 𝐺2 , 𝑆𝑂(4) and 𝑆𝑈(3) respectively. The group 2^3 : 𝑃𝑆𝐿2(7) representing the split extension possesses five maximal subgroups 2^3: 7: 3, 2^3: 𝑆4, 4: 𝑆4: 2 and two non-conjugate Klein’s group 𝑃𝑆𝐿2(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.