Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23 |
| direct product | C2×ΓL(1,𝔽8)×#97 |
| unambiguous semidirect product | (C2×C7×#97)⋊C3 |
| Fitting, Hall | C2×C7×#97 ⋊ C3 |
| Sylow | C7 ⋊ C2×C3×#97 |
292 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 195 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) = 448 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(16,17,18)(19,21,20);(7,8,9,12,10,11,13,14)(15,16,17,19,18,20,21);(3,4,5,6)(8,11)(12,14)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4223);