Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C33 |
| direct product | C3×C3⋊D3×(C3⋊D3)⋊C3 |
| Fitting, Sylow | C33×H(3,𝔽3) ⋊ C22 |
| derived | C34 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊D3×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C34 | C22×C32 |
| Frattini | C3 | C32×D3×C3⋊D3 |
| Fitting | C33×H(3,𝔽3) | C22 |
| socle | C34 | C2×C3×D3 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C3 | |
2824 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 2824 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 108 |
A minimal faithful permutation representation — 3 generators on 18 points:
(5,6)(8,9)(10,11,13,15,14,16)(12,18);(1,2,3)(7,8,9)(10,13)(11,17)(12,16)(15,18);(4,5,6)180 conjugacy classes — view the full character table.
| maximal subgroup of | C3×(C3⋊D3)⋊C32, #72243969, (C3⋊D3)⋊C3×C3⋊AGL(1,ℤ/9), … |
| maximal normal subgroup of | C3×(C3⋊D3)⋊C32, C3×#14519181, C3×D32×(C3⋊D3)⋊C3 |
| maximal quotient of | #72243728, #72243729, #72243831, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(8,9)(10,11,13,15,14,16)(12,18), (1,2,3)(7,8,9)(10,13)(11,17)(12,16)(15,18), (4,5,6));