Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#87522 |
| Fitting | C4×#585 ⋊ D3 |
| derived | #997 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#1020 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini, socle | C22 | C22×#1020 |
| Fitting | C4×#585 | D3 |
| prime radical | C2×#87522 | C2 |
| cosocle | C2×#997 | C23 |
| hypercenter | C42 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
3046 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 3046 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,3,4)(7,13,16)(8,14,15)(11,18,20)(12,19,17);(5,17)(6,7)(8,19)(9,11)(10,15)(12,16)(13,20)(14,18);(6,10)(7,13)(8,15)(11,12)(14,16)(17,18)92 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#11715828, C4×#11714174, C4×#11714175, … |
| maximal normal subgroup of | C4×#11715828, C4×#11714174, C4×#11714175, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4)(7,13,16)(8,14,15)(11,18,20)(12,19,17), (5,17)(6,7)(8,19)(9,11)(10,15)(12,16)(13,20)(14,18), (6,10)(7,13)(8,15)(11,12)(14,16)(17,18));