Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C32 |
| direct product | C32×C3⋊#1113 |
| Fitting, socle, Sylow | C36 ⋊ C22 |
| derived | C34 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C3⋊#1113 |
| commutator, cosocle, hypocenter | C34 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C36 | C22 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C32 | |
16648 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 16648 conjugacy classes of subgroups — too many to draw. Only its 210 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:
(4,5,6)(8,9)(11,12)(13,14,15)(16,17);(1,2,3)(10,11,12)(14,15)(17,18);(7,8,9)297 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×#72253805, C3×#72224516, C3×#72224543, … |
| maximal normal subgroup of | C3×#14519523 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5,6)(8,9)(11,12)(13,14,15)(16,17), (1,2,3)(10,11,12)(14,15)(17,18), (7,8,9));