Finite Group DatabasebetaFinite Group Databasebeta| order | 1386 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C11 · Schur multiplier (H2) = C3 |
| direct product | C7×C11×C3⋊D3 |
| unambiguous semidirect product | C3⋊(D3×C7×C11) |
| Fitting, socle, Hall | C32×C7×C11 ⋊ C2 |
| derived, Sylow | C32 ⋊ C2×C7×C11 |
| Hall | C32×C11 ⋊ C2×C7 |
| Hall | C32×C7 ⋊ C2×C11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C11 | C3⋊D3 |
| commutator, cosocle, hypocenter | C32 | C2×C7×C11 |
| Frattini | C1 | G |
| Fitting, socle | C32×C7×C11 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C7×C11 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 693 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18)(20,21)(23,24);(22,23,24);(19,20,21)462 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C7×C3⋊D3×D11, C11×D7×C3⋊D3, C3⋊D3×D77, … |
| maximal normal subgroup of | C7×C3⋊D3×D11, C11×D7×C3⋊D3, C3⋊D3×D77, … |
| maximal quotient of | C7×C11×C3⋊Dic3, C2×C7×C11×C3⋊D3, C7×C3⋊D3×C121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1386, 28);