Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×#66431⋊C3 |
| Fitting, Sylow | C33×3−1+2 ⋊ C22 |
| derived | C32×C9 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #66431⋊C3 |
| commutator, cosocle, hypocenter | C32×C9 | C22×C32 |
| Frattini | C3 | C32×#1113 |
| Fitting | C33×3−1+2 | C22 |
| socle | C34 | C2×C3×D3 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 ◁ C3 | |
1672 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 1672 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 36 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3)(5,6)(7,8)(10,14,13,12,18,15,16,11,17);(4,5,6)(8,9)(11,17,18,13,14,15)(12,16)153 conjugacy classes — view the full character table.
| maximal subgroup of | #72243379, C3×#72214299, C3×D32×AGL(1,ℤ/9), … |
| maximal normal subgroup of | C3×D32×AGL(1,ℤ/9), #72259340, D3×#66431⋊C3, … |
| maximal quotient of | #72243379, C3×#72214299, #72254081, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(5,6)(7,8)(10,14,13,12,18,15,16,11,17), (4,5,6)(8,9)(11,17,18,13,14,15)(12,16));