Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C22×A4×A5 |
| Fitting | C24 ⋊ C3×A5 |
| socle | C24×A5 ⋊ C3 |
| derived | C22×A5 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4×A5 |
| commutator, hypocenter | C22×A5 | C22×C3 |
| Frattini | C1 | G |
| Fitting | C24 | C3×A5 |
| socle | C24×A5 | C3 |
| cosocle | C22 | C22×C3×A5 |
| LCS | G ▷ C22×A5 | |
| UCS | C1 ◁ C22 | |
921 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 921 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2)(3,4)(5,8,7)(9,11)(10,12);(3,4)(5,6,8)(9,13,10,12,11)80 conjugacy classes — view the full character table.
| maximal subgroup of | A5×B(4,𝔽2)⋊C3, A5×GL(2,ℤ/4), C22×A52, … |
| maximal normal subgroup of | A5×GL(2,ℤ/4), C22×A4⋊S5 |
| maximal quotient of | #72277309 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4)(5,8,7)(9,11)(10,12), (3,4)(5,6,8)(9,13,10,12,11));