Finite Group DatabasebetaFinite Group Databasebeta| order | 15678 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C67 · Schur multiplier (H2) = C1 |
| direct product | D3×C13⋊C3×C67 |
| unambiguous semidirect product | C3⋊(C2×C13⋊C3×C67) |
| unambiguous semidirect product | (D3×C13)⋊(C3×C67) |
| Fitting, socle | C3×C13×C67 ⋊ C2×C3 |
| derived | C3×C13 ⋊ C2×C3×C67 |
| Hall | C3×C13⋊C3×C67 ⋊ C2 |
| Hall | C13×C67 ⋊ C3×D3 |
| Hall | C3×C13⋊C3 ⋊ C2×C67 |
| Sylow | C13 ⋊ C3×D3×C67 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C67 | D3×C13⋊C3 |
| commutator, cosocle, hypocenter | C3×C13 | C2×C3×C67 |
| Frattini | C1 | G |
| Fitting, socle | C3×C13×C67 | C2×C3 |
| LCS | G ▷ C3×C13 | |
| UCS | C1 ◁ C67 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2613 |
1407 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D3×C67×D13⋊C3, D3×(C13⋊C3)⋊D67, D3×C13⋊C3×D67, … |
| maximal normal subgroup of | D3×C67×D13⋊C3, D3×(C13⋊C3)⋊D67, D3×C13⋊C3×D67, … |
| maximal quotient of | Dic3×C13⋊C3×C67, C2×D3×C13⋊C3×C67 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15678, 50);