Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | ΓL(1,𝔽8)×#557 |
| Fitting, Hall | C7×#557 ⋊ C3 |
| Sylow | C7 ⋊ C3×#557 |
400 conjugacy classes of subgroups; only the 249 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 249 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) = 448 |
A minimal faithful permutation representation — 4 generators on 23 points:
(2,3,4)(5,7,6)(9,12)(13,14)(16,17,19,21)(18,22,23,20);(1,2,3,5,4,6,7)(8,9,11,12)(10,14,15,13);(16,18,19,23)(17,20,21,22);(8,10,11,15)(9,13,12,14)125 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10007);