Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | ΓL(1,𝔽8)×#453 |
| unambiguous semidirect product | (C7×#453)⋊C3 |
| Fitting, Hall | C7×#453 ⋊ C3 |
| Sylow | C7 ⋊ C3×#453 |
336 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 15 points:
(2,3,4)(5,7,6)(9,12)(10,14)(13,15);(1,2,3,5,4,6,7)(8,9)(10,15)(11,13)(12,14);(10,14)(12,15)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4273);