Finite Group DatabasebetaFinite Group Databasebeta| order | 11520 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | AGL(1,𝔽5)×S42 |
| Fitting, socle | C24×C5 ⋊ C4×D32 |
| derived | C5×A42 ⋊ C22×C4 |
| Sylow | C5 ⋊ C4×S42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C5×A42 | C22×C4 |
| Fitting, socle | C24×C5 | C4×D32 |
| prime radical | D5×S42 | C2 |
| cosocle | D5×A42 | C23 |
| LCS | G ▷ C5×A42 | |
| UCS | C1 | |
5104 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 5104 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,5,3,4)(6,9,7,8)(10,12,11,13);(2,3,4,5)(7,8,9);(11,12)125 conjugacy classes — view the full character table.
| holomorph of | C5×S4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,5,3,4)(6,9,7,8)(10,12,11,13), (2,3,4,5)(7,8,9), (11,12));