Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C34 |
| direct product | D3×C3⋊(C3⋊((C3⋊D3)⋊C3)) |
| Fitting, Sylow | C33×H(3,𝔽3) ⋊ C22 |
| derived | C35 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C35 | C22×C3 |
| Frattini | C3 | C3×D3×C3⋊(C3⋊D3) |
| Fitting | C33×H(3,𝔽3) | C22 |
| socle | C34 | C2×C3×D3 |
| LCS | G ▷ C35 | |
| UCS | C1 | |
4380 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 4380 conjugacy classes of subgroups — too many to draw. Only its 242 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)(7,8)(11,16,18,13,14,17)(12,15);(4,5,6)(8,9);(10,11,13)(12,14,17)(15,18,16);(1,2,3)162 conjugacy classes — view the full character table.
| maximal subgroup of | #72243931, D3×#13091644, D32×C3⋊((C3⋊D3)⋊C3) |
| maximal normal subgroup of | D3×#13091644, D32×C3⋊((C3⋊D3)⋊C3) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(5,6)(7,8)(11,16,18,13,14,17)(12,15), (4,5,6)(8,9), (10,11,13)(12,14,17)(15,18,16), (1,2,3));