Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#11714243 |
| Fitting | C2×#2273 ⋊ D3 |
| derived | #2273⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #85547 |
| commutator, cosocle, hypocenter | #2273⋊C3 | C22 |
| Frattini | C23 | C2×#1020 |
| Fitting | C2×#2273 | D3 |
| socle | C24 | #1020 |
| LCS | G ▷ #2273⋊C3 | |
| UCS | C1 ◁ C22 | |
1701 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 1701 conjugacy classes of subgroups — too many to draw. Only its 27 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 — 2 generators on 20 points:
(1,8,7,2,4,5,3,6)(9,14,10,12)(11,20,16,17,13,19,18,15);(2,3,8)(5,6,7)(9,13,17,10,11,15)(12,18,19,14,16,20)58 conjugacy classes — view the full character table.
| maximal quotient of | C2×#72228154 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,8,7,2,4,5,3,6)(9,14,10,12)(11,20,16,17,13,19,18,15), (2,3,8)(5,6,7)(9,13,17,10,11,15)(12,18,19,14,16,20));