Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C3⋊Dic3×A5 |
| derived | C32×A5 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊D3×A5 |
| commutator, hypocenter | C32×A5 | C4 |
| Fitting, cosocle | C2×C32 | C2×A5 |
| socle, prime radical | C2×C32×A5 | C2 |
| LCS | G ▷ C32×A5 | |
| UCS | C1 ◁ C2 | |
250 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 180 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(7,8)(9,10)(11,13,14);(1,2)(4,6)(7,9,8,10)(11,13,14);(2,3)(4,5)(7,9,8,10)(12,14)(13,15)60 conjugacy classes — view the full character table.
| maximal subgroup of | A5×#622 |
| maximal normal subgroup of | A5×#622 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(7,8)(9,10)(11,13,14), (1,2)(4,6)(7,9,8,10)(11,13,14), (2,3)(4,5)(7,9,8,10)(12,14)(13,15));