Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C33 |
| direct product | C3×#13091511 |
| Fitting, Sylow | C3×H(3,𝔽3)×3−1+2 ⋊ C2 |
| derived | C9×H(3,𝔽3) ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C3⋊(C3⋊AGL(1,ℤ/9)) |
| commutator, cosocle, hypocenter | C9×H(3,𝔽3) | C2×C32 |
| Frattini | C32 | C32×C3⋊(C3⋊D3) |
| Fitting | C3×H(3,𝔽3)×3−1+2 | C2 |
| socle | C33 | C3×C3⋊(C3⋊D3) |
| LCS | G ▷ C9×H(3,𝔽3) | |
| UCS | C1 ◁ C32 | |
5476 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 5476 conjugacy classes of subgroups — too many to draw. Only its 266 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)(6,10)(8,11)(9,12)(14,16)(17,20)(19,21);(4,8,9,6,12,11,10,5,7)(14,17,21)(16,19,20);(13,14,16)(15,17,20)(18,21,19);(14,17,21)(16,19,20)222 conjugacy classes — view the full character table.
| maximal subgroup of | #72243459, #72253809, C3×#72253807 |
| maximal normal subgroup of | #72253809, C3×#72253807 |
| maximal quotient of | #72253773 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(5,7)(6,10)(8,11)(9,12)(14,16)(17,20)(19,21), (4,8,9,6,12,11,10,5,7)(14,17,21)(16,19,20), (13,14,16)(15,17,20)(18,21,19), (14,17,21)(16,19,20));