Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C36 |
| direct product | D3×H(3,𝔽3)×3−1+2 |
| Fitting, Sylow | C3×H(3,𝔽3)×3−1+2 ⋊ C2 |
5869 conjugacy classes of subgroups; only the 777 normal subgroups are shown.
This group has 5869 conjugacy classes of subgroups — too many to draw. Only its 777 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:
(2,3)(13,16,21,15,19,17,18,14,20);(1,2,3)(5,8,9)(7,11,12);(4,5,7)(6,9,11)(8,12,10);(14,16,19)(17,21,20)363 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #72243516, #72243853, D3×H(3,𝔽3)×AGL(1,ℤ/9), … |
| maximal normal subgroup of | D3×H(3,𝔽3)×AGL(1,ℤ/9), D3×3−1+2×(C3⋊D3)⋊C3, D3×#13091513 |
| maximal quotient of | #72243516, #72243853 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(13,16,21,15,19,17,18,14,20), (1,2,3)(5,8,9)(7,11,12), (4,5,7)(6,9,11)(8,12,10), (14,16,19)(17,21,20));