Finite Group DatabasebetaFinite Group Databasebeta| order | 8064 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | ΓL(1,𝔽8)×#87522 |
| Fitting | C7×#585 ⋊ C3×D3 |
| derived | C7×#997 ⋊ C22×C3 |
| Sylow | C7 ⋊ C3×#87522 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×ΓL(1,𝔽8)×#1020 |
| commutator, cosocle, hypocenter | C7×#997 | C22×C3 |
| Fitting | C7×#585 | C3×D3 |
| socle | C2×C7 | C2×C3×#1020 |
| hypercenter | C4 | ΓL(1,𝔽8)×#1020 |
| LCS | G ▷ C7×#997 | |
| UCS | C1 ◁ C2 ◁ C4 | |
1128 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(1,10,3,12,5,11,9,4)(2,16,7,8,6,15,13,14)(18,19,20)(21,23,22);(1,2,5,6)(3,8,10,13,4,16,9,14,11,7,12,15)(17,18,19,21,20,22,23)115 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,10,3,12,5,11,9,4)(2,16,7,8,6,15,13,14)(18,19,20)(21,23,22), (1,2,5,6)(3,8,10,13,4,16,9,14,11,7,12,15)(17,18,19,21,20,22,23));