Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#2273⋊C3 |
| Fitting, Sylow | C4×#2273 ⋊ C3 |
| derived | #2273 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C22⋊(C42⋊C3) |
| commutator, hypocenter | #2273 | C3×C4 |
| Frattini, socle | C24 | C2×C22⋊A4 |
| Fitting | C4×#2273 | C3 |
| prime radical | C2×#2273⋊C3 | C2 |
| cosocle | C2×#2273 | C2×C3 |
| LCS | G ▷ #2273 | |
| UCS | C1 ◁ C2×C4 | |
2525 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 2525 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 96 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3,5)(6,7,8)(9,14,20,10,16,19,12,17,15,13,11,18);(2,6)(5,8);(1,5)(2,3)(4,8)(6,7)(9,13,12,10)(11,16)(14,17)(15,19,20,18)112 conjugacy classes — view the full character table.
| maximal subgroup of | #72267247, C4×#10935⋊C3 |
| maximal normal subgroup of | C4×#10935⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,5)(6,7,8)(9,14,20,10,16,19,12,17,15,13,11,18), (2,6)(5,8), (1,5)(2,3)(4,8)(6,7)(9,13,12,10)(11,16)(14,17)(15,19,20,18));