Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of D3×A4 |
| automorphism | Aut(G) = C3×D3×S4 (order 432) · Inn(G) = D3×A4 · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C2 |
| direct product | D3×C22⋊C9 |
| unambiguous semidirect product | C3⋊(C2×C22⋊C9) |
| unambiguous semidirect product | (C22×D3)⋊C9 |
| derived | C22×C3 ⋊ C2×C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | D3×A4 |
| commutator, hypocenter | C22×C3 | C2×C9 |
| Fitting, socle, cosocle | C22×C32 | C2×C3 |
| prime radical | C22×C3×D3 | C3 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C3 | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3,4)(6,7)(8,12,16,10,15,11,13,9,14);(1,4)(2,3)(5,6,7)36 conjugacy classes — view the full character table.
| maximal subgroup of | D3×(C22⋊C9)⋊C2, C2×D3×C22⋊C9, D9⋊(C22⋊C9), … |
| maximal normal subgroup of | D3×(C22⋊C9)⋊C2, C2×D3×C22⋊C9, D3×C9×A4, … |
| maximal quotient of | C3⋊(#41⋊C9), D3×Q8⋊C9, Dic3×C22⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 98);