Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#11758092 |
| Fitting, Sylow | C33×H(3,𝔽3) ⋊ C22 |
| derived | C32×H(3,𝔽3) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C3⋊#1113 |
| commutator, cosocle, hypocenter | C32×H(3,𝔽3) | C22×C3 |
| Frattini | C3 | C3×C3⋊#1113 |
| Fitting | C33×H(3,𝔽3) | C22 |
| socle | C34 | C2×C3⋊D3 |
| LCS | G ▷ C32×H(3,𝔽3) | |
| UCS | C1 ◁ C32 | |
2886 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 2886 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 108 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3)(4,5,6)(7,8)(11,13)(14,18)(15,17);(5,6)(8,9)(11,14,15)(13,17,18);(10,11,13)(12,15,17)(14,18,16)153 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72214354, C3×#14519181, #72277736, … |
| maximal normal subgroup of | C3×#14519181, #72277879 |
| maximal quotient of | #72243842, #72243843, #72243878, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,5,6)(7,8)(11,13)(14,18)(15,17), (5,6)(8,9)(11,14,15)(13,17,18), (10,11,13)(12,15,17)(14,18,16));