Finite Group DatabasebetaFinite Group Databasebeta| order | 414 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×D3×C11 (order 264) · Inn(G) = D3 · Out(G) = C22×C11 |
| homology | Abelianization (H1) = C2×C3×C23 · Schur multiplier (H2) = C1 |
| direct product | C3×D3×C23 |
| unambiguous semidirect product | C3⋊(C2×C3×C23) |
| Fitting, socle, Hall | C32×C23 ⋊ C2 |
| derived | C3 ⋊ C2×C3×C23 |
| Sylow | C32 ⋊ C2×C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C23 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C3×C23 |
| Frattini | C1 | G |
| Fitting, socle | C32×C23 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C23 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 138 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3)(5,6)(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29);(4,5,6)207 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D23, D32×C23, D3×D69, … |
| maximal normal subgroup of | C3×D3×D23, D32×C23, D3×D69, … |
| maximal quotient of | C3×Dic3×C23, C2×C3×D3×C23, C3×D9×C23, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(414, 6);