Finite Group DatabasebetaFinite Group Databasebeta| order | 8000 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C43 |
| direct product | AGL(1,𝔽5)3 |
| Fitting, socle, derived, Sylow | C53 ⋊ C43 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C53 | C43 |
| prime radical, cosocle | D53 | C23 |
| LCS | G ▷ C53 | |
| UCS | C1 | |
1128 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 15 points:
(2,3,4,5)(6,7,10,8,9)(11,12,15,13);(1,3,2,4,5)(7,8,9,10);(12,14,15,13)125 conjugacy classes — view the full character table.
| holomorph of | C5×AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4,5)(6,7,10,8,9)(11,12,15,13), (1,3,2,4,5)(7,8,9,10), (12,14,15,13));