Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#32875⋊C3 |
| Fitting, Sylow | C2×#32875 ⋊ C3 |
| derived | #338 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2842⋊C3 |
| commutator, cosocle, hypocenter | #338 | C23×C3 |
| Frattini | C22×C4 | C23×A4 |
| Fitting | C2×#32875 | C3 |
| hypercenter | C2×C4 | #560⋊C3 |
| LCS | G ▷ #338 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
1180 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 1180 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(4,9,18,8,5,12)(10,17,15,16,11,14);(1,2)(3,16,6,18,7,10,13,12)(4,15,17,5,8,14,11,9);(4,8)(5,9)(6,13)(10,16)72 conjugacy classes — view the full character table.
| maximal subgroup of | #72268881 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,9,18,8,5,12)(10,17,15,16,11,14), (1,2)(3,16,6,18,7,10,13,12)(4,15,17,5,8,14,11,9), (4,8)(5,9)(6,13)(10,16));