Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of C3⋊(C3⋊D3) |
| automorphism | Aut(G) order 606528 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C33 |
| direct product | C3×C3⋊(C3⋊D3) |
| unambiguous semidirect product | C3⋊(C3×C3⋊D3) |
| Fitting, socle, Sylow | C34 ⋊ C2 |
| derived | C33 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊(C3⋊D3) |
| commutator, cosocle, hypocenter | C33 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C34 | C2 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C3 | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2,3)(5,6)(8,9)(11,12);(10,11,12);(7,8,9);(4,5,6)45 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×C3⋊D3, D3×C3⋊(C3⋊D3), C3⋊#1113, … |
| maximal normal subgroup of | C3×D3×C3⋊D3, D3×C3⋊(C3⋊D3), C3⋊#1113, … |
| maximal quotient of | C3×C3⋊(C3⋊Dic3), C2×C3×C3⋊(C3⋊D3), C3×C3⋊(C3⋊D9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 53);