Finite Group DatabasebetaFinite Group Databasebeta| order | 15678 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊D2613 |
| unambiguous semidirect product | C13⋊(C3⋊D201) |
| unambiguous semidirect product | C67⋊(C3⋊D39) |
| Fitting, socle, derived, Hall | C32×C13×C67 ⋊ C2 |
| Hall | C13×C67 ⋊ C3⋊D3 |
| Hall | C32×C67 ⋊ D13 |
| Hall | C32×C13 ⋊ D67 |
| Sylow | C67 ⋊ C3⋊D39 |
| Sylow | C13 ⋊ C3⋊D201 |
| Sylow | C32 ⋊ D871 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C13×C67 | C2 |
| LCS | G ▷ C32×C13×C67 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 7839 |
Polycyclic presentation:
2,3^2,13,67|b^a=b^2,c^a=c^2,d^a=d^12,e^a=e^663921 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D3×D2613, D39×D201, D67×C3⋊D39, … |
| maximal normal subgroup of | D3×D2613, D39×D201, D67×C3⋊D39, … |
| maximal quotient of | C3⋊Dic2613, C2×C3⋊D2613 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15678, 62);