Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C34 |
| direct product | D3×3−1+22 |
| Fitting, Sylow | C3×3−1+22 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C34×D3 |
| commutator, socle, cosocle | C33 | C2×C34 |
| Fitting | C3×3−1+22 | C2 |
| prime radical | C34×D3 | C32 |
| hypercenter | 3−1+22 | D3 |
| hypocenter | C3 | C2×3−1+22 |
| LCS | G ▷ C33 ▷ C3 | |
| UCS | C1 ◁ C32 ◁ 3−1+22 | |
3790 conjugacy classes of subgroups; only the 777 normal subgroups are shown.
This group has 3790 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)(4,7,12,6,10,8,9,5,11);(5,7,10)(8,12,11);(14,16,19)(17,21,20)363 conjugacy classes. The character table is too large to display.
| maximal subgroup of | 3−1+2×#92265, D3×3−1+2×AGL(1,ℤ/9), D3×#13091521, … |
| maximal normal subgroup of | D3×3−1+2×AGL(1,ℤ/9), D3×#13091521 |
| maximal quotient of | 3−1+2×#92265, #72243574, #72243705 |
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)(4,7,12,6,10,8,9,5,11), (5,7,10)(8,12,11), (14,16,19)(17,21,20));