Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C32 |
| direct product | C3×#11758168 |
| Fitting, socle, Sylow | C36 ⋊ C22 |
| derived | C35 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #11758168 |
| commutator, cosocle, hypocenter | C35 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C36 | C22 |
| LCS | G ▷ C35 | |
| UCS | C1 ◁ C3 | |
16648 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 16648 conjugacy classes of subgroups — too many to draw. Only its 174 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 — 3 generators on 18 points:
(1,2,3)(5,6)(7,8)(10,11,12)(14,15)(17,18);(5,6)(8,9)(11,12)(16,17,18);(4,5,6)(13,14,15)234 conjugacy classes — view the full character table.
| maximal subgroup of | #72253800, C3×#72214065, #72253819 |
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)(10,11,12)(14,15)(17,18), (5,6)(8,9)(11,12)(16,17,18), (4,5,6)(13,14,15));