Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C33 |
| direct product | C3×#13091038 |
| Fitting, Sylow | C3×#10624730 ⋊ C2 |
| derived | C3×AΣL(1,𝔽27) ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #13091038 |
| commutator, cosocle, hypocenter | C3×AΣL(1,𝔽27) | C2×C32 |
| Frattini | C33 | C32×C3⋊D3 |
| Fitting | C3×#10624730 | C2 |
| socle | C33 | C3×#272 |
| LCS | G ▷ C3×AΣL(1,𝔽27) | |
| UCS | C1 ◁ C3 | |
11452 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 11452 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 54 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3)(5,7)(6,9)(8,10)(11,12)(14,19,20,17,18,21)(15,16);(5,8,12)(7,10,11)(14,20,18)(17,19,21);(4,5,7)(6,8,11)(9,12,10)(13,20,17,15,18,19,16,14,21)198 conjugacy classes — view the full character table.
| maximal subgroup of | #72277595, C3×#72277602 |
| maximal normal subgroup of | #72277595, C3×#72277602 |
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)(14,19,20,17,18,21)(15,16), (5,8,12)(7,10,11)(14,20,18)(17,19,21), (4,5,7)(6,8,11)(9,12,10)(13,20,17,15,18,19,16,14,21));