Finite Group DatabasebetaFinite Group Databasebeta| order | 5760 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| direct product | A5×#120⋊C3 |
| Fitting | #120 ⋊ C3×A5 |
| derived | C24×A5 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C24×A5 | C2×C3 |
| Frattini | C22 | C2×A4×A5 |
| Fitting | #120 | C3×A5 |
| socle | C22×A5 | C2×A4 |
| cosocle | C24 | C2×C3×A5 |
| LCS | G ▷ C24×A5 | |
| UCS | C1 | |
736 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 14 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) = 40 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,3,2,5)(4,8,6,7)(10,11)(12,13);(2,6,4)(5,7,8)(9,11,12)50 conjugacy classes — view the full character table.
| maximal quotient of | A5×B(4,𝔽2)⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,2,5)(4,8,6,7)(10,11)(12,13), (2,6,4)(5,7,8)(9,11,12));