Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#89767 |
| Hall | C3×C7 ⋊ #428 |
| Sylow | C7 ⋊ C3×#428 |
236 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(4,5,7,9,10,8,6)(11,13,14,12)(15,16)(17,22)(18,20)(19,21);(16,20)(19,22);(5,6)(7,8)(9,10)(11,12,14,13)(16,19)(17,21)(20,22)246 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4720);