Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#87520 |
| Fitting | C4×#585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | #1020 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini, socle | C22 | C22×#1020 |
| Fitting | C4×#585 | D3 |
| prime radical | C2×#87520 | C2 |
| cosocle | C2×#997 | C23 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C42 | |
3041 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 3041 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,20,7,6,8,10,9,11,13,12,16,17)(14,19,15,18);(8,14,16,15)(11,18,20,19);(1,2,3,4)104 conjugacy classes — view the full character table.
| maximal subgroup of | #72227323, #72246076, #72246879, … |
| maximal normal subgroup of | #72227323, #72246076, #72246879, … |
| maximal quotient of | C2×C4×#87520 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,20,7,6,8,10,9,11,13,12,16,17)(14,19,15,18), (8,14,16,15)(11,18,20,19), (1,2,3,4));