Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | ΓL(1,𝔽8)×#327 |
| unambiguous semidirect product | (C7×#327)⋊C3 |
| Fitting, Hall | C7×#327 ⋊ C3 |
| Sylow | C7 ⋊ C3×#327 |
180 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 180 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)(9,13,14,21)(10,17)(15,22,16,18)(19,23);(1,2,3,5,4,6,7)(8,9)(10,18)(11,15)(12,16)(13,19)(14,20)(17,21)(22,23)95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 344);