Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | ΓL(1,𝔽8)×#349 |
| unambiguous semidirect product | (C7×#349)⋊C3 |
| Fitting, Hall | C7×#349 ⋊ C3 |
| Sylow | C7 ⋊ C3×#349 |
116 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(2,3,4)(5,7,6)(8,9,10,14,11,15,17,21,12,13,16,19,18,20,22,23);(1,2,3,5,4,6,7)(9,13)(10,16)(15,20)(17,22)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 366);