Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of C3⋊S4 |
| automorphism | Aut(G) = C2×((C3⋊S4)⋊C3)⋊C2 (order 864) · Inn(G) = C3⋊S4 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C3⋊S4 |
| unambiguous semidirect product | (C2×C3)⋊S4 |
| unambiguous semidirect product | (C2×A4)⋊D3 |
| Fitting, socle | C23×C3 ⋊ D3 |
| derived | C3×A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊S4 |
| commutator, cosocle, hypocenter | C3×A4 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3 | D3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C2 | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 3 generators on 9 points:
(1,2)(4,5)(6,7,9,8);(1,2)(3,4,5);(7,8,9)18 conjugacy classes — view the full character table.
| maximal subgroup of | (C3⋊S4)⋊C4, #64207, #64210, … |
| maximal normal subgroup of | (C3⋊S4)⋊C4, #64207, #64210, … |
| maximal quotient of | #64259, C4×C3⋊S4, C4⋊(C3⋊S4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 189);