Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C36 |
| direct product | C3×3−1+2×(C3⋊D3)⋊C3 |
| Fitting, Sylow | C3×H(3,𝔽3)×3−1+2 ⋊ C2 |
5356 conjugacy classes of subgroups; only the 734 normal subgroups are shown.
This group has 5356 conjugacy classes of subgroups — too many to draw. Only its 734 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 — 4 generators on 21 points:
(1,2,3)(5,7)(6,9)(8,10)(11,12);(4,5,11)(6,8,10)(7,9,12)(13,16,21,15,19,17,18,14,20);(5,8,12)(7,10,11);(14,16,19)(17,21,20)330 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #72243516, #72243705, (C3⋊D3)⋊C3×C3⋊AGL(1,ℤ/9), … |
| maximal normal subgroup of | #72243516, #72243705, (C3⋊D3)⋊C3×C3⋊AGL(1,ℤ/9), … |
| maximal quotient of | #72243516, #72243705 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(5,7)(6,9)(8,10)(11,12), (4,5,11)(6,8,10)(7,9,12)(13,16,21,15,19,17,18,14,20), (5,8,12)(7,10,11), (14,16,19)(17,21,20));