Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C34 |
| direct product | C3×#13091501 |
| Fitting, Sylow | C3×H(3,𝔽3)2 ⋊ C2 |
| derived | C32×H(3,𝔽3) ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C3⋊(C3⋊((C3⋊D3)⋊C3)) |
| commutator, cosocle, hypocenter | C32×H(3,𝔽3) | C2×C32 |
| Frattini | C32 | C32×C3⋊(C3⋊D3) |
| Fitting | C3×H(3,𝔽3)2 | C2 |
| socle | C33 | C3×C3⋊(C3⋊D3) |
| LCS | G ▷ C32×H(3,𝔽3) | |
| UCS | C1 ◁ C32 | |
8662 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 8662 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:
(5,10,12,7,8,11)(6,9)(14,16)(17,21)(18,20);(1,2,3)(4,5,7)(6,8,11)(9,12,10);(14,17,18)(16,20,21);(4,5,7)(6,8,11)(9,12,10)(13,14,16)(15,18,20)(17,21,19)222 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72214354, C3×#72214370, C3×#72214385 |
| maximal normal subgroup of | C3×#72214354, C3×#72214370, C3×#72214385 |
| maximal quotient of | #72243693, #72243695, #72243761, … |
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)(17,21)(18,20), (1,2,3)(4,5,7)(6,8,11)(9,12,10), (14,17,18)(16,20,21), (4,5,7)(6,8,11)(9,12,10)(13,14,16)(15,18,20)(17,21,19));