Finite Group DatabasebetaFinite Group Databasebeta| order | 12096 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | D3×ΓL(1,𝔽8)×#997 |
| Fitting, derived | C3×C7×H(5,𝔽2) ⋊ C2×C32 |
| Sylow | C7 ⋊ C3×D3×#997 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×ΓL(1,𝔽8)×C22⋊A4 |
| commutator, Fitting, cosocle, hypocenter | C3×C7×H(5,𝔽2) | C2×C32 |
| socle | C2×C3×C7 | C2×C3×C22⋊A4 |
| LCS | G ▷ C3×C7×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
1976 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1976 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 3 generators on 18 points:
(2,5,3)(6,7,8)(9,10,12,14,11,13,15)(17,18);(2,6)(3,8)(10,11,12)(13,15,14)(16,17,18);(1,7,4,5)(2,3,6,8)165 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((2,5,3)(6,7,8)(9,10,12,14,11,13,15)(17,18), (2,6)(3,8)(10,11,12)(13,15,14)(16,17,18), (1,7,4,5)(2,3,6,8));