Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C34 |
| direct product | C32×D3×(C3⋊D3)⋊C3 |
| Fitting, Sylow | C33×H(3,𝔽3) ⋊ C22 |
| derived | C33 ⋊ C22×C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | D3×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C33 | C22×C33 |
| Frattini | C3 | C33×D32 |
| Fitting | C33×H(3,𝔽3) | C22 |
| socle | C34 | C2×C3×D3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C32 | |
3328 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 3328 conjugacy classes of subgroups — too many to draw. Only its 298 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:
(8,9)(10,11,12,15,13,16)(14,18);(4,5,6)(7,8,9)(10,12)(11,18,16,17,15,14);(1,2,3)270 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((8,9)(10,11,12,15,13,16)(14,18), (4,5,6)(7,8,9)(10,12)(11,18,16,17,15,14), (1,2,3));