Finite Group DatabasebetaFinite Group Databasebeta| order | 14400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | S52 |
| socle, derived | A52 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | A52 | C22 |
| LCS | G ▷ A52 | |
| UCS | C1 | |
809 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 809 conjugacy classes of subgroups — too many to draw. Only its 10 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) = 25 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,5,3,4,2)(6,8,9,10);(2,5)(6,7)(8,10)49 conjugacy classes — view the full character table.
| holomorph of | S5 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,5,3,4,2)(6,8,9,10), (2,5)(6,7)(8,10));