Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | ΓL(1,𝔽8)×#5450 |
| Fitting, Hall | C7×H(3,𝔽4) ⋊ C32 |
| derived | C24×C7 ⋊ C22×C32 |
| Sylow | H(3,𝔽4) ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×#5450 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24×C7 | C22×C32 |
| Frattini | C22 | C22×A4×ΓL(1,𝔽8) |
| Fitting | C7×H(3,𝔽4) | C32 |
| socle | C22×C7 | C22×C3×A4 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 | |
344 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,4,6,3,5,7)(8,10)(9,13)(11,17,12,16,19,18)(14,22,23,21,15,20);(2,3,4)(5,7,6)(8,23,19,9)(10,22,17,21)(11,14,12,15)(13,18,20,16)85 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#11710731 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,6,3,5,7)(8,10)(9,13)(11,17,12,16,19,18)(14,22,23,21,15,20), (2,3,4)(5,7,6)(8,23,19,9)(10,22,17,21)(11,14,12,15)(13,18,20,16));