Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2 |
| direct product | 3−1+2×#1113 |
| Fitting, Sylow | C33×3−1+2 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×#1113 |
| commutator, socle, cosocle | C34 | C22×C32 |
| Fitting | C33×3−1+2 | C22 |
| prime radical | C32×#1113 | C3 |
| hypercenter | 3−1+2 | #1113 |
| hypocenter | C33 | C22×3−1+2 |
| LCS | G ▷ C34 ▷ C33 | |
| UCS | C1 ◁ C3 ◁ 3−1+2 | |
1574 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 1574 conjugacy classes of subgroups — too many to draw. Only its 105 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:
(2,3)(4,5,6)(7,8)(10,16,17,15,13,14,12,11,18);(1,2,3)(5,6)(8,9)(11,13,16)(14,18,17)165 conjugacy classes — view the full character table.
| maximal subgroup of | #72259340, D33×3−1+2, AGL(1,ℤ/9)×#1113, … |
| maximal normal subgroup of | D33×3−1+2, AGL(1,ℤ/9)×#1113, #72259581 |
| maximal quotient of | #72259340 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,5,6)(7,8)(10,16,17,15,13,14,12,11,18), (1,2,3)(5,6)(8,9)(11,13,16)(14,18,17));