Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | D5×A4×D9 |
| Fitting, derived | C22×C5×C9 ⋊ C22×C3 |
| Sylow | C5 ⋊ C2×A4×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C22×C5×C9 | C22×C3 |
| Frattini | C3 | D3×D5×A4 |
| socle | C22×C3×C5 | C2×C3×D3 |
| LCS | G ▷ C22×C5×C9 | |
| UCS | C1 | |
250 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 39 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:
(2,3)(4,5)(6,13,14,8,7,9,12,11,10)(16,18,17);(1,2,4,5,3)(7,9)(8,12)(10,13)(11,14)(15,18)(16,17)96 conjugacy classes — view the full character table.
| maximal subgroup of | D5×A4×AGL(1,ℤ/9), A4×D9×AGL(1,𝔽5), D5×D9×S4, … |
| maximal normal subgroup of | D5×A4×AGL(1,ℤ/9), A4×D9×AGL(1,𝔽5), D5×D9×S4, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,5)(6,13,14,8,7,9,12,11,10)(16,18,17), (1,2,4,5,3)(7,9)(8,12)(10,13)(11,14)(15,18)(16,17));