Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C32 |
| direct product | C3×(C3⋊#1113)⋊C3 |
| Fitting, Sylow | C33×H(3,𝔽3) ⋊ C22 |
| derived | C34 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | (C3⋊#1113)⋊C3 |
| commutator, cosocle, hypocenter | C34 | C22×C32 |
| Frattini | C3 | C32×#1113 |
| Fitting | C33×H(3,𝔽3) | C22 |
| socle | C34 | C2×C3×D3 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C3 | |
2394 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 2394 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(4,5,6)(7,8)(11,16,18,13,14,17)(12,15);(1,2,3)(5,6)(8,9)(10,11,13)(12,14,17)(15,18,16)153 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72214370, C3×#14519181, #72254081, … |
| maximal normal subgroup of | C3×#14519181, C3×D32×(C3⋊D3)⋊C3, C3×#14519184 |
| maximal quotient of | #72243944, #72243945, #72243949, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5,6)(7,8)(11,16,18,13,14,17)(12,15), (1,2,3)(5,6)(8,9)(10,11,13)(12,14,17)(15,18,16));