Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25×C4 |
| direct product | C22×#85559 |
| Fitting | C22×H(3,𝔽4) ⋊ D3 |
| derived | #5464 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #85559 |
| commutator, cosocle, hypocenter | #5464 | C23 |
| Frattini | C22 | C22×#1020 |
| Fitting | C22×H(3,𝔽4) | D3 |
| socle | C24 | #1020 |
| LCS | G ▷ #5464 | |
| UCS | C1 ◁ C22 | |
3288 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 3288 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,20,15,14,7,16,11,9)(6,18,13,12,10,19,8,17);(3,4)(6,7,10)(8,9,16)(11,14,19)(12,20,15)(13,17,18);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#11715754, C22×#11715753, C2×C4×#85559, … |
| maximal normal subgroup of | C22×#11715754, C22×#11715753, C2×C4×#85559, … |
| maximal quotient of | C2×#72240890, C2×#72240888, C2×C4×#85559, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,20,15,14,7,16,11,9)(6,18,13,12,10,19,8,17), (3,4)(6,7,10)(8,9,16)(11,14,19)(12,20,15)(13,17,18), (1,2));