Finite Group DatabasebetaFinite Group Databasebeta| order | 8064 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | ΓL(1,𝔽8)×#3532⋊C3 |
| Fitting, Hall | C7×#3532 ⋊ C32 |
| derived | C7×H(3,𝔽4) ⋊ C2×C32 |
| Sylow | #3532 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×#3532⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×H(3,𝔽4) | C2×C32 |
| Frattini | C22 | C2×ΓL(1,𝔽8)×C22⋊A4 |
| Fitting | C7×#3532 | C32 |
| socle | C22×C7 | C2×C3×C22⋊A4 |
| LCS | G ▷ C7×H(3,𝔽4) | |
| UCS | C1 | |
1168 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 55 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 — 3 generators on 23 points:
(2,13,10,6,7,11)(3,15,14,9,8,16)(4,12,5)(17,18,20,22,19,21,23);(2,15)(3,9)(6,8)(7,14)(10,11)(13,16)(18,19,20)(21,23,22);(1,13,12,7)(2,9,15,11)(3,6,10,8)(4,16,5,14)90 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#3532⋊C3, ΓL(1,𝔽8)×#11710731 |
| maximal normal subgroup of | AGL(1,𝔽7)×#3532⋊C3, ΓL(1,𝔽8)×#11710731 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,13,10,6,7,11)(3,15,14,9,8,16)(4,12,5)(17,18,20,22,19,21,23), (2,15)(3,9)(6,8)(7,14)(10,11)(13,16)(18,19,20)(21,23,22), (1,13,12,7)(2,9,15,11)(3,6,10,8)(4,16,5,14));