Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×ΓL(1,𝔽8)×#123 |
| unambiguous semidirect product | (C2×C7×#123)⋊C3 |
| Fitting, Hall | C2×C7×#123 ⋊ C3 |
| Sylow | C7 ⋊ C2×C3×#123 |
684 conjugacy classes of subgroups; only the 267 normal subgroups are shown.
This group has 684 conjugacy classes of subgroups — too many to draw. Only its 267 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(16,17,18)(19,21,20);(4,6)(8,11)(9,13)(15,16,17,19,18,20,21);(7,8)(9,12)(10,11)(13,14);(3,4)(5,6)(7,9,10,13)(8,12,11,14)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 9974);