Finite Group DatabasebetaFinite Group Databasebeta| order | 12096 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | ΓL(1,𝔽8)×C3⋊#5935 |
| Fitting | C3×C7×H(5,𝔽2) ⋊ C3×D3 |
| derived | C3×C7×#997 ⋊ C2×C3 |
| Sylow | C7 ⋊ C3×C3⋊#5935 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | ΓL(1,𝔽8)×C3⋊#1020 |
| commutator, cosocle, hypocenter | C3×C7×#997 | C2×C3 |
| Fitting | C3×C7×H(5,𝔽2) | C3×D3 |
| socle | C2×C3×C7 | C3×#1020 |
| LCS | G ▷ C3×C7×#997 | |
| UCS | C1 ◁ C2 | |
992 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 992 conjugacy classes of subgroups — too many to draw. Only its 51 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:
(1,3,4,7)(2,6)(5,8)(10,11,12)(13,15,14)(17,18);(9,10,12,14,11,13,15)(16,17,18);(2,3,8)(5,6,7)120 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((1,3,4,7)(2,6)(5,8)(10,11,12)(13,15,14)(17,18), (9,10,12,14,11,13,15)(16,17,18), (2,3,8)(5,6,7));