Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| direct product | ΓL(1,𝔽8)×C4⋊C16 |
| unambiguous semidirect product | C4⋊(C16×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C7×C4⋊C16)⋊C3 |
| Fitting, Hall | C7×C4⋊C16 ⋊ C3 |
| Sylow | C7 ⋊ C3×C4⋊C16 |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,3,4)(5,7,6)(9,11)(12,13,14,17,15,18,20,23,16,19,21,24,22,25,26,27);(1,2,3,5,4,6,7)(8,9,10,11)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 380);