Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | ΓL(1,𝔽8)×#953 |
| Fitting | C24×C3×C7 ⋊ C2×C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ #120 |
| Sylow | C7 ⋊ C3×#953 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C22×C3×C7 | C23×C3 |
| Fitting | C24×C3×C7 | C2×C3 |
| hypercenter | C24 | D3×ΓL(1,𝔽8) |
| hypocenter | C3×C7 | C3×#120 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C24 | |
654 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 654 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(2,3,4)(5,7,6)(8,9,10)(16,18);(1,2,4,6,3,5,7)(9,10)(12,14)(15,16)(17,18);(11,12)(13,14)150 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((2,3,4)(5,7,6)(8,9,10)(16,18), (1,2,4,6,3,5,7)(9,10)(12,14)(15,16)(17,18), (11,12)(13,14));