Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×A4×A5 |
| Fitting | C22×C3 ⋊ C3×A5 |
| socle | C22×C3×A5 ⋊ C3 |
| derived | C22×A5 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | A4×A5 |
| commutator, hypocenter | C22×A5 | C32 |
| Frattini | C1 | G |
| Fitting | C22×C3 | C3×A5 |
| socle | C22×C3×A5 | C3 |
| cosocle | C22 | C32×A5 |
| LCS | G ▷ C22×A5 | |
| UCS | C1 ◁ C3 | |
205 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3)(4,7,5)(9,11,10);(1,3,2)(5,7,6)(8,10,9,12,11)60 conjugacy classes — view the full character table.
| derived subgroup of | #72247634, D3×A4⋊S5 |
| maximal subgroup of | A4×#11761914, C3×S4×A5, C3×A52, … |
| maximal normal subgroup of | C3×S4×A5, C3×A4⋊S5, D3×A4×A5, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,7,5)(9,11,10), (1,3,2)(5,7,6)(8,10,9,12,11));