Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C33 |
| direct product | AGL(1,ℤ/9)×C3⋊(C3⋊D3) |
| Fitting, Sylow | C33×3−1+2 ⋊ C22 |
| derived | C33×C9 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C33×C9 | C22×C3 |
| Frattini | C3 | C3×D3×C3⋊(C3⋊D3) |
| Fitting | C33×3−1+2 | C22 |
| socle | C34 | C2×C3×D3 |
| LCS | G ▷ C33×C9 | |
| UCS | C1 | |
2560 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 2560 conjugacy classes of subgroups — too many to draw. Only its 205 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) = 162 |
A minimal faithful permutation representation — 4 generators on 18 points:
(2,3)(5,6)(8,9)(10,15,12,17,13,11)(16,18);(7,8,9)(10,12)(11,18)(14,17)(15,16);(4,5,6);(1,2,3)150 conjugacy classes — view the full character table.
| maximal subgroup of | #72254059, AGL(1,ℤ/9)×C3⋊((C3⋊D3)⋊C3), D3×C3⋊D3×AGL(1,ℤ/9) |
| maximal normal subgroup of | AGL(1,ℤ/9)×C3⋊((C3⋊D3)⋊C3), D3×C3⋊D3×AGL(1,ℤ/9) |
| maximal quotient of | #72254059 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(5,6)(8,9)(10,15,12,17,13,11)(16,18), (7,8,9)(10,12)(11,18)(14,17)(15,16), (4,5,6), (1,2,3));