Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C35 |
| direct product | C3×#13091513 |
| Fitting, Sylow | C3×H(3,𝔽3)×3−1+2 ⋊ C2 |
| derived | C32×C9 ⋊ C2×C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #13091513 |
| commutator, cosocle, hypocenter | C32×C9 | C2×C33 |
| Frattini | C32 | C33×C3⋊D3 |
| Fitting | C3×H(3,𝔽3)×3−1+2 | C2 |
| socle | C33 | C32×C3⋊D3 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 ◁ C3 | |
4816 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 4816 conjugacy classes of subgroups — too many to draw. Only its 250 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:
(5,10,12,7,8,11)(6,9)(14,16)(15,19)(17,20)(18,21);(1,2,3)(13,17,16,15,21,18,19,14,20);(4,5,7)(6,8,11)(9,12,10)(14,17,21)(16,20,18)222 conjugacy classes — view the full character table.
| maximal subgroup of | #72243916, #72253773, #72253826, … |
| maximal normal subgroup of | #72243916, #72253826, C3×(C3⋊D3)⋊C3×AGL(1,ℤ/9), … |
| maximal quotient of | #72243486, #72243916 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,10,12,7,8,11)(6,9)(14,16)(15,19)(17,20)(18,21), (1,2,3)(13,17,16,15,21,18,19,14,20), (4,5,7)(6,8,11)(9,12,10)(14,17,21)(16,20,18));