Finite Group DatabasebetaFinite Group Databasebeta| order | 11520 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C2×#72209614 |
| Fitting, socle | C25 ⋊ A6 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72209614 |
| commutator, hypocenter | #72209614 | C2 |
| Frattini | C1 | G |
| Fitting, socle | C25 | A6 |
| cosocle | C24 | C2×A6 |
| LCS | G ▷ #72209614 | |
| UCS | C1 ◁ C2 | |
513 conjugacy classes of subgroups; only the 6 normal subgroups are shown.
This group has 513 conjugacy classes of subgroups — too many to draw. Only its 6 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)(3,4,13,12,15,17,9,18)(5,6,10,11,14,16,7,8);(3,16,5,8,14)(4,10,7,12,11)(6,17,18,15,9)24 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72218848, C2×#72218850, C2×#72218849 |
| maximal normal subgroup of | C2×#72218848, C2×#72218850, C2×#72218849 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,13,12,15,17,9,18)(5,6,10,11,14,16,7,8), (3,16,5,8,14)(4,10,7,12,11)(6,17,18,15,9));