Finite Group DatabasebetaFinite Group Databasebeta| order | 5832 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | D3×#66431⋊C3 |
| Fitting, Sylow | C33×3−1+2 ⋊ C23 |
| derived | C33×C9 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C33×C9 | C23×C3 |
| Frattini | C3 | C3×D3×#1113 |
| Fitting | C33×3−1+2 | C23 |
| socle | C34 | C22×C3×D3 |
| LCS | G ▷ C33×C9 | |
| UCS | C1 | |
2277 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 2277 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 72 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3)(5,6)(7,8,9)(10,17,13,11,12,15)(14,16);(2,3)(4,5,6)(8,9)(10,12)(11,18)(14,15)(16,17);(8,9)153 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72214299, #72259533 |
| maximal quotient of | D3×#72214299, #72259533 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(5,6)(7,8,9)(10,17,13,11,12,15)(14,16), (2,3)(4,5,6)(8,9)(10,12)(11,18)(14,15)(16,17), (8,9));