Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×D3×D5×A4 |
| Fitting, socle | C22×C32×C5 ⋊ C22×C3 |
| derived | C22×C3×C5 ⋊ C22×C32 |
| Sylow | C5 ⋊ C2×C3×D3×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D3×D5×A4 |
| commutator, cosocle, hypocenter | C22×C3×C5 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C5 | C22×C3 |
| LCS | G ▷ C22×C3×C5 | |
| UCS | C1 ◁ C3 | |
464 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 2 generators on 15 points:
(2,3)(4,5)(9,10,11)(13,14,15);(1,2,4,5,3)(6,7,8)(10,11)(12,15)(13,14)144 conjugacy classes — view the full character table.
| maximal subgroup of | D5×A4×AGL(1,ℤ/9), C3×D3×A4×AGL(1,𝔽5), C3×D3×D5×S4, … |
| maximal normal subgroup of | C3×D3×A4×AGL(1,𝔽5), C3×D3×D5×S4, C3×D3×A4⋊AGL(1,𝔽5) |
| maximal quotient of | D5×A4×AGL(1,ℤ/9) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,5)(9,10,11)(13,14,15), (1,2,4,5,3)(6,7,8)(10,11)(12,15)(13,14));