Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | (C3⋊D3)⋊C3×#272 |
| Fitting, Sylow | H(3,𝔽3)2 ⋊ C22 |
| derived | C32×H(3,𝔽3) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊D3×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C32×H(3,𝔽3) | C22×C3 |
| Frattini, socle | C32 | C3×D3×C3⋊D3 |
| Fitting | H(3,𝔽3)2 | C22 |
| LCS | G ▷ C32×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
1107 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 1107 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 54 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,8)(3,5,7)(4,6,9)(11,13)(14,18)(15,17);(2,4)(3,6)(5,7)(8,9)(11,14,15)(13,17,18);(10,11,13)(12,15,17)(14,18,16)100 conjugacy classes — view the full character table.
| maximal subgroup of | #272×((C3⋊D3)⋊C3)⋊C2, (C3⋊D3)⋊C3×((C3⋊D3)⋊C3)⋊C2, #72228697, … |
| maximal normal subgroup of | #272×((C3⋊D3)⋊C3)⋊C2, (C3⋊D3)⋊C3×((C3⋊D3)⋊C3)⋊C2, #72228697 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,8)(3,5,7)(4,6,9)(11,13)(14,18)(15,17), (2,4)(3,6)(5,7)(8,9)(11,14,15)(13,17,18), (10,11,13)(12,15,17)(14,18,16));