Finite Group DatabasebetaFinite Group Databasebeta| order | 4320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×D3×D5×S4 |
| Fitting, socle | C22×C32×C5 ⋊ C22×D3 |
| derived | C3×C5×A4 ⋊ C23×C3 |
| Sylow | C5 ⋊ C2×C3×D3×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D3×D5×S4 |
| commutator, cosocle, hypocenter | C3×C5×A4 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C5 | C22×D3 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 ◁ C3 | |
1210 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 1210 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,4,3)(6,7)(8,9)(10,11,12)(13,14,15);(5,6,8,9,7)(14,15);(2,4)180 conjugacy classes — view the full character table.
| maximal subgroup of | D5×S4×AGL(1,ℤ/9), C3×D3×AGL(1,𝔽5)×S4 |
| maximal normal subgroup of | C3×D3×AGL(1,𝔽5)×S4 |
| maximal quotient of | D5×S4×AGL(1,ℤ/9) |
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)(10,11,12)(13,14,15), (5,6,8,9,7)(14,15), (2,4));