Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | #2722 |
| Fitting, derived, Sylow | H(3,𝔽3)2 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C32 | C3⋊D32 |
| commutator, Fitting, cosocle, hypocenter | H(3,𝔽3)2 | C22 |
| LCS | G ▷ H(3,𝔽3)2 | |
| UCS | C1 ◁ C32 | |
1591 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 1591 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 81 |
A minimal faithful permutation representation — 3 generators on 18 points:
(2,5,6)(4,8,9)(10,11)(12,14)(15,18);(2,4)(5,9)(6,8)(11,13)(14,17)(16,18);(1,2,4)(3,6,8)(5,9,7)(11,14,18)(13,16,17)100 conjugacy classes — view the full character table.
| maximal subgroup of | #72228645, #72228655, #272×((C3⋊D3)⋊C3)⋊C2, … |
| maximal normal subgroup of | #72228645, #72228655, #272×((C3⋊D3)⋊C3)⋊C2, … |
| maximal quotient of | #72277736 |
| outer automorphism group of | #6887 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,5,6)(4,8,9)(10,11)(12,14)(15,18), (2,4)(5,9)(6,8)(11,13)(14,17)(16,18), (1,2,4)(3,6,8)(5,9,7)(11,14,18)(13,16,17));