Finite Group DatabasebetaFinite Group Databasebeta| order | 5760 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | AGL(1,𝔽5)×A4⋊S4 |
| Fitting, socle | C24×C5 ⋊ C4×C3⋊D3 |
| derived | C5×A42 ⋊ C2×C4 |
| Sylow | C5 ⋊ C4×A4⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C5×A42 | C2×C4 |
| Fitting, socle | C24×C5 | C4×C3⋊D3 |
| prime radical | D5×A4⋊S4 | C2 |
| cosocle | D5×A42 | C22 |
| LCS | G ▷ C5×A42 | |
| UCS | C1 | |
1196 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 1196 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 60 |
A minimal faithful permutation representation — 3 generators on 13 points:
(2,3,4,5)(6,9)(7,8)(10,13,12);(1,2,5,3,4)(7,9)(12,13);(7,8,9)(11,12,13)70 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4,5)(6,9)(7,8)(10,13,12), (1,2,5,3,4)(7,9)(12,13), (7,8,9)(11,12,13));