Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C33 |
| direct product | C2×#13091501 |
| Fitting | C2×H(3,𝔽3)2 ⋊ C2 |
| derived | C32×H(3,𝔽3) ⋊ C22×C3 |
| Sylow | H(3,𝔽3)2 ⋊ C22 |
2247 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 2247 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,8)(3,5,7)(4,6,9)(19,20);(2,4)(3,6)(5,7)(8,9)(11,14)(15,18)(16,17);(11,15,16)(14,17,18);(1,2,4)(3,5,8)(6,9,7)(10,11,14)(12,15,18)(13,16,17)148 conjugacy classes — view the full character table.
| maximal subgroup of | #72229131, #72229139, #72229150, … |
| maximal normal subgroup of | #72229131, #72229139, #72229150, … |
| maximal quotient of | #72229150 |
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)(19,20), (2,4)(3,6)(5,7)(8,9)(11,14)(15,18)(16,17), (11,15,16)(14,17,18), (1,2,4)(3,5,8)(6,9,7)(10,11,14)(12,15,18)(13,16,17));