Finite Group DatabasebetaFinite Group Databasebeta| order | 23040 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C2×#72218850 |
| Fitting, socle | C25 ⋊ S6 |
| derived | #72209614 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72218850 |
| commutator, cosocle, hypocenter | #72209614 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C25 | S6 |
| LCS | G ▷ #72209614 | |
| UCS | C1 ◁ C2 | |
2712 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 2712 conjugacy classes of subgroups — too many to draw. Only its 9 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) = 30 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(4,7,12)(5,10,17,6,18,11)(8,9,15,13,16,14);(1,2)(3,14,4,16,18)(5,7,12,8,11)(6,13,10,15,9)42 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72218858 |
| maximal normal subgroup of | C2×#72218858 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,7,12)(5,10,17,6,18,11)(8,9,15,13,16,14), (1,2)(3,14,4,16,18)(5,7,12,8,11)(6,13,10,15,9));