Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C22×S6 |
| socle | C22×A6 ⋊ C2 |
| derived | A6 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C22 | S6 |
| commutator, cosocle, hypocenter | A6 | C23 |
| Frattini | C1 | G |
| socle | C22×A6 | C2 |
| LCS | G ▷ A6 | |
| UCS | C1 ◁ C22 | |
914 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 914 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 10 points:
(5,7,10,6,9,8);(1,2)(6,9,8)(7,10);(3,4)(6,9,8)(7,10)44 conjugacy classes — view the full character table.
| maximal quotient of | C2×#72218858 |
| outer automorphism group of | #3573, C3×#585 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,7,10,6,9,8), (1,2)(6,9,8)(7,10), (3,4)(6,9,8)(7,10));