Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | S4×D45 |
| Fitting | C22×C5×C9 ⋊ C2×D3 |
| derived | C5×C9×A4 ⋊ C22 |
| Sylow | C5 ⋊ D9×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C5×C9×A4 | C22 |
| Frattini | C3 | S4×D15 |
| Fitting | C22×C5×C9 | C2×D3 |
| socle | C22×C3×C5 | D32 |
| LCS | G ▷ C5×C9×A4 | |
| UCS | C1 | |
214 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 29 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) = 180 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,4,3)(6,7)(8,9)(11,13)(12,16)(14,17)(15,18);(2,4)(5,6,8,9,7)(10,15,13,12,17,14,16,11,18)120 conjugacy classes — view the full character table.
| maximal subgroup of | S4×C5⋊AGL(1,ℤ/9), D5×D9×S4 |
| maximal normal subgroup of | S4×C5⋊AGL(1,ℤ/9), D5×D9×S4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3)(6,7)(8,9)(11,13)(12,16)(14,17)(15,18), (2,4)(5,6,8,9,7)(10,15,13,12,17,14,16,11,18));