Finite Group DatabasebetaFinite Group Databasebeta| order | 15678 |
| isoclinism | isoclinic class of D67 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C13 · Schur multiplier (H2) = C3 |
| direct product | C32×C13×D67 |
| Fitting, socle, Hall | C32×C13×C67 ⋊ C2 |
| derived, Sylow | C67 ⋊ C2×C32×C13 |
| Hall | C13×C67 ⋊ C2×C32 |
| Hall | C32×C67 ⋊ C2×C13 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C13 | D67 |
| commutator, cosocle, hypocenter | C67 | C2×C32×C13 |
| Frattini | C1 | G |
| Fitting, socle | C32×C13×C67 | C2 |
| LCS | G ▷ C67 | |
| UCS | C1 ◁ C32×C13 | |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 7839 |
4095 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C32×D13×D67, C3×D3×C13×D67, C3×D39×D67, … |
| maximal normal subgroup of | C32×D13×D67, C3×D3×C13×D67, C3×D39×D67, … |
| maximal quotient of | C32×C13×Dic67, C2×C32×C13×D67 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15678, 72);