Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×D5×S5 |
| Fitting | C2×C5 ⋊ C2×S5 |
| socle | C2×C5×A5 ⋊ C22 |
| derived | C5×A5 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D5×S5 |
| commutator, cosocle, hypocenter | C5×A5 | C23 |
| Frattini | C1 | G |
| Fitting | C2×C5 | C2×S5 |
| socle | C2×C5×A5 | C22 |
| LCS | G ▷ C5×A5 | |
| UCS | C1 ◁ C2 | |
455 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 455 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 50 |
A minimal faithful permutation representation — 3 generators on 12 points:
(3,6,7,4,5)(8,11,9,12);(3,5)(4,6)(8,12,10);(1,2)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A5×S5 |
| maximal quotient of | #72277310 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6,7,4,5)(8,11,9,12), (3,5)(4,6)(8,12,10), (1,2));