Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×C4⋊C4×ΓL(1,𝔽8) |
| unambiguous semidirect product | C4⋊(C42×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C4×C7×C4⋊C4)⋊C3 |
| Fitting, Hall | C4×C7×C4⋊C4 ⋊ C3 |
| Sylow | C7 ⋊ C3×C4×C4⋊C4 |
388 conjugacy classes of subgroups; only the 219 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 219 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) = 448 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,4)(14,15,16)(17,19,18);(6,8)(9,10,11,12)(13,14,15,17,16,18,19);(5,6,7,8)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4198);