Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | AGL(1,𝔽5)×S5 |
| Fitting | C5 ⋊ C4×S5 |
| socle, derived | C5×A5 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, hypocenter | C5×A5 | C2×C4 |
| Fitting | C5 | C4×S5 |
| prime radical | D5×S5 | C2 |
| cosocle | D5×A5 | C22 |
| LCS | G ▷ C5×A5 | |
| UCS | C1 | |
199 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 199 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) = 10 |
| transitive degree | μt(G) = 25 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,3,4,5)(6,10,9,8,7);(2,4,5,3)(6,7)(8,9,10)35 conjugacy classes — view the full character table.
| maximal subgroup of | S52 |
| maximal quotient of | AGL(1,𝔽5)×A5⋊C4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,4,5)(6,10,9,8,7), (2,4,5,3)(6,7)(8,9,10));