Finite Group DatabasebetaFinite Group Databasebeta| order | 5760 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | D5×S42 |
| Fitting, socle | C24×C5 ⋊ C2×D32 |
| derived | C5×A42 ⋊ C23 |
| Sylow | C5 ⋊ C2×S42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C5×A42 | C23 |
| Fitting, socle | C24×C5 | C2×D32 |
| LCS | G ▷ C5×A42 | |
| UCS | C1 | |
2742 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 2742 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(2,3)(4,5)(6,8,7)(10,12,11);(1,2,4,5,3)(8,9)(12,13);(12,13)100 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×S42, #72260294 |
| maximal normal subgroup of | AGL(1,𝔽5)×S42 |
| maximal quotient of | #72260294 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,5)(6,8,7)(10,12,11), (1,2,4,5,3)(8,9)(12,13), (12,13));