Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C22×#85557 |
| Fitting | C22×H(3,𝔽4) ⋊ D3 |
| derived | #5465 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #5934 |
| commutator, cosocle, hypocenter | #5465 | C23 |
| Frattini | C22 | C22×#1020 |
| Fitting | C22×H(3,𝔽4) | D3 |
| hypercenter | C24 | #1020 |
| LCS | G ▷ #5465 | |
| UCS | C1 ◁ C23 ◁ C24 | |
3624 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 3624 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(3,4)(5,8,15)(6,10,17)(7,13,19)(12,16,20);(1,2)(5,7)(10,16)(15,19)(17,20);(5,6,7,12)(8,11,13,18)(9,10,14,16)(15,17,19,20)64 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#72209772 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,8,15)(6,10,17)(7,13,19)(12,16,20), (1,2)(5,7)(10,16)(15,19)(17,20), (5,6,7,12)(8,11,13,18)(9,10,14,16)(15,17,19,20));