Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C3⋊D3×D9⋊C9 |
| Fitting, Sylow | C32×C9⋊C9 ⋊ C22 |
| derived | C32×C9 ⋊ C22×C9 |
622 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 622 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(5,6)(7,11,15,9,14,10,12,8,13)(16,18,21,19,22,23,17,20,24);(4,5,6)(17,19)(18,21)(20,23)(22,24);(1,2,3)180 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#92265, C3⋊D3×AGL(1,ℤ/27), D32×D9⋊C9, … |
| maximal normal subgroup of | C3⋊D3×#92265, D32×D9⋊C9, #72254045, … |
| maximal quotient of | C3⋊D3×AGL(1,ℤ/27) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(5,6)(7,11,15,9,14,10,12,8,13)(16,18,21,19,22,23,17,20,24), (4,5,6)(17,19)(18,21)(20,23)(22,24), (1,2,3));