Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C35 |
| direct product | C2×H(3,𝔽3)×(C3⋊D3)⋊C3 |
| Fitting | C2×H(3,𝔽3)2 ⋊ C2 |
| Sylow | H(3,𝔽3)2 ⋊ C22 |
1767 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 1767 conjugacy classes of subgroups — too many to draw. Only its 238 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 — 3 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)(13,17,18);(2,5,9)(4,7,8)(10,11,13)(12,15,17)(14,18,16)220 conjugacy classes — view the full character table.
| maximal subgroup of | H(3,𝔽3)×#638⋊C3, #72229124, #72229160, … |
| maximal normal subgroup of | H(3,𝔽3)×#638⋊C3, #72229124, #72229160, … |
| maximal quotient of | H(3,𝔽3)×#638⋊C3 |
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)(13,17,18), (2,5,9)(4,7,8)(10,11,13)(12,15,17)(14,18,16));