Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of D3×C3⋊D3 |
| automorphism | Aut(G) = C22×D3×AGL(2,𝔽3) (order 10368) · Inn(G) = D3×C3⋊D3 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | D3×C3⋊Dic3 |
| unambiguous semidirect product | (C3×D3)⋊Dic3 |
| derived, Sylow | C33 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×C3⋊D3 |
| commutator, hypocenter | C33 | C2×C4 |
| Fitting, socle, cosocle | C2×C33 | C22 |
| prime radical | C2×C32×D3 | C2 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2 | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,3,4)(5,6,7)(9,10)(12,13);(6,7)(11,12,13);(8,9,10)36 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#622, #88925, D32×Dic3, … |
| maximal normal subgroup of | D3×#622, #88925, D32×Dic3, … |
| maximal quotient of | D3×C3⋊(C3⋊C8), #88786, Dic3×C3⋊Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 124);