Finite Group DatabasebetaFinite Group Databasebeta| order | 462 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×C3×C5×D3 (order 360) · Inn(G) = D3 · Out(G) = C22×C3×C5 |
| homology | Abelianization (H1) = C2×C7×C11 · Schur multiplier (H2) = C1 |
| direct product | D3×C7×C11 |
| unambiguous semidirect product | C3⋊(C2×C7×C11) |
| Fitting, socle, Hall | C3×C7×C11 ⋊ C2 |
| derived, Sylow | C3 ⋊ C2×C7×C11 |
| Hall | C3×C11 ⋊ C2×C7 |
| Hall | C3×C7 ⋊ C2×C11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C11 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C7×C11 |
| Frattini | C1 | G |
| Fitting, socle | C3×C7×C11 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C7×C11 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 231 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,3)(4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21);(1,2,3)231 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C7×D11, D3×C11×D7, D3×D77, … |
| maximal normal subgroup of | D3×C7×D11, D3×C11×D7, D3×D77, … |
| maximal quotient of | C7×C11×Dic3, C2×D3×C7×C11, C7×C11×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(462, 8);