Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | S4×#1042 |
| Fitting, socle | C22×C52 ⋊ C4×D3 |
| derived | C52×A4 ⋊ C2×C4 |
| Sylow | C52 ⋊ C4×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C52×A4 | C2×C4 |
| Fitting, socle | C22×C52 | C4×D3 |
| prime radical | C5×D5×S4 | C2 |
| cosocle | C5×D5×A4 | C22 |
| LCS | G ▷ C52×A4 | |
| UCS | C1 | |
235 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,4,2,3)(5,6,8,9,7)(10,11,14,12,13);(2,3,4)(6,7)(8,9)(11,12,13,14)65 conjugacy classes — view the full character table.
| maximal subgroup of | #72258511, D5×AGL(1,𝔽5)×S4 |
| maximal normal subgroup of | D5×AGL(1,𝔽5)×S4 |
| maximal quotient of | #72258511 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,2,3)(5,6,8,9,7)(10,11,14,12,13), (2,3,4)(6,7)(8,9)(11,12,13,14));