Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | ΓL(1,𝔽8)×#379⋊C3 |
| Fitting, Hall | C7×#379 ⋊ C32 |
| derived | C24×C7 ⋊ C32×C4 |
| Sylow | #379 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×#379⋊C3 |
476 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 84 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,4)(5,6)(7,12,8,9,11,10)(13,14,15,17,16,18,19);(6,12)(9,10)(14,15,16)(17,19,18)100 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×AΣL(1,𝔽64), #72277843, AGL(1,𝔽7)×#379⋊C3 |
| maximal normal subgroup of | ΓL(1,𝔽8)×AΣL(1,𝔽64), #72277843, AGL(1,𝔽7)×#379⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4)(5,6)(7,12,8,9,11,10)(13,14,15,17,16,18,19), (6,12)(9,10)(14,15,16)(17,19,18));