Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
| direct product | 3−1+2×C3⋊((C3⋊D3)⋊C3) |
| Fitting, Sylow | C3×H(3,𝔽3)×3−1+2 ⋊ C2 |
4645 conjugacy classes of subgroups; only the 252 normal subgroups are shown.
This group has 4645 conjugacy classes of subgroups — too many to draw. Only its 252 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 3 generators on 21 points:
(2,3)(5,10,12,7,8,11)(6,9)(13,16,21,15,19,17,18,14,20);(1,2,3)(5,8,12)(7,10,11);(4,5,7)(6,8,11)(9,12,10)(14,16,19)(17,21,20)231 conjugacy classes — view the full character table.
| maximal subgroup of | #72243422, #72243916, #72253809, … |
| maximal normal subgroup of | #72253809, #72253826, D3×3−1+2×(C3⋊D3)⋊C3, … |
| maximal quotient of | #72243422, #72243916 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(5,10,12,7,8,11)(6,9)(13,16,21,15,19,17,18,14,20), (1,2,3)(5,8,12)(7,10,11), (4,5,7)(6,8,11)(9,12,10)(14,16,19)(17,21,20));