Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | ΓL(1,𝔽8)×#432 |
| unambiguous semidirect product | (C7×#432)⋊C3 |
| Fitting, Hall | C7×#432 ⋊ C3 |
| Sylow | C7 ⋊ C3×#432 |
220 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 224 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,3,4)(5,7,6)(8,9)(10,13)(11,12)(14,15)(16,17,18,21,19,20,22,23);(1,2,3,5,4,6,7)(9,12)(13,15);(8,10,11,14)(9,13,12,15)(17,20)(21,23)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4252);