Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | ΓL(1,𝔽8)×#449 |
| unambiguous semidirect product | (C7×#449)⋊C3 |
| Fitting, Hall | C7×#449 ⋊ C3 |
| Sylow | C7 ⋊ C3×#449 |
324 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,3,4)(5,7,6)(8,9)(10,11)(13,17)(14,15)(18,19);(1,2,3,5,4,6,7)(12,13)(14,17)(15,19)(16,18);(9,11)(12,14,16,15)(13,17,18,19)95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4269);