Finite Group DatabasebetaFinite Group Databasebeta| order | 5120 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C42 |
| direct product | C4×C44⋊C5 |
| Fitting, Sylow | C45 ⋊ C5 |
| derived | C44 ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C44⋊C5 |
| commutator, hypocenter | C44 | C4×C5 |
| Frattini, socle | C25 | C2×C24⋊C5 |
| Fitting | C45 | C5 |
| prime radical | C2×C44⋊C5 | C2 |
| cosocle | C2×C44 | C2×C5 |
| LCS | G ▷ C44 | |
| UCS | C1 ◁ C4 | |
11214 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 11214 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,6,12,17,15,2,8,13,11,18,4,9,7,14,19,5,3,10,16,20);(1,5,4,2)(3,8)(6,9)(11,16)(14,17)(15,18,19,20)224 conjugacy classes — view the full character table.
| maximal subgroup of | #72237321, C4×#72209725 |
| maximal normal subgroup of | #72237321, C4×#72209725 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,6,12,17,15,2,8,13,11,18,4,9,7,14,19,5,3,10,16,20), (1,5,4,2)(3,8)(6,9)(11,16)(14,17)(15,18,19,20));