Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×#66432⋊C3 |
| Fitting, Sylow | C3×H(5,𝔽3) ⋊ C22 |
| derived | C3×H(3,𝔽3) ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #66432⋊C3 |
| commutator, cosocle, hypocenter | C3×H(3,𝔽3) | C22×C32 |
| Frattini | C3 | C32×#1113 |
| Fitting | C3×H(5,𝔽3) | C22 |
| socle | C32 | C3×#1113 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
1591 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 1591 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 30 points:
(4,5,7,10,15,11)(6,29,26,30,14,19)(8,20)(9,24)(12,17,22,27,23,28)(13,21)(18,25);(1,2,3)(4,9,17)(5,6,13,8,12,19)(7,18,28)(10,26,24,20,23,30)(11,14,25,16,22,29)(15,21,27)141 conjugacy classes — view the full character table.
| maximal subgroup of | #72243811, #72243812, C3×#72221953, … |
| maximal normal subgroup of | C3×#14519185, D3×#66432⋊C3, #72277801, … |
| maximal quotient of | #72277801 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5,7,10,15,11)(6,29,26,30,14,19)(8,20)(9,24)(12,17,22,27,23,28)(13,21)(18,25), (1,2,3)(4,9,17)(5,6,13,8,12,19)(7,18,28)(10,26,24,20,23,30)(11,14,25,16,22,29)(15,21,27));