Finite Group DatabasebetaFinite Group Databasebeta| order | 7200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
| direct product | A5×S5 |
| socle, derived | A52 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, hypocenter | A52 | C2 |
| cosocle | A5 | C2×A5 |
| LCS | G ▷ A52 | |
| UCS | C1 | |
267 conjugacy classes of subgroups; only the 6 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 6 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:
(2,5)(3,4)(6,7,9)(8,10);(1,5,4)(6,9)(7,10)35 conjugacy classes — view the full character table.
| maximal subgroup of | S52, C2×A5×S5 |
| maximal normal subgroup of | S52, C2×A5×S5 |
| maximal quotient of | #72277237, #72277251, C2×A5×S5 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,5)(3,4)(6,7,9)(8,10), (1,5,4)(6,9)(7,10));