Finite Group DatabasebetaFinite Group Databasebeta| order | 6732 |
| automorphism | Aut(G) order 3231360 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic1683 |
| unambiguous semidirect product | C9⋊Dic187 |
| unambiguous semidirect product | C11⋊Dic153 |
| derived, Hall | C9×C11×C17 ⋊ C4 |
| Hall | C11×C17 ⋊ Dic9 |
| Hall | C9×C17 ⋊ Dic11 |
| Hall | C9×C11 ⋊ Dic17 |
| Sylow | C17 ⋊ Dic99 |
| Sylow | C11 ⋊ Dic153 |
| Sylow | C9 ⋊ Dic187 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D1683 |
| commutator, hypocenter | C9×C11×C17 | C4 |
| Frattini | C2×C3 | D561 |
| Fitting | C2×C9×C11×C17 | C2 |
| socle | C2×C3×C11×C17 | D3 |
| LCS | G ▷ C9×C11×C17 | |
| UCS | C1 ◁ C2 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,11,17,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^10,d^a=d^16,f^a=f^21686 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C5×Dic1683, Dic8415, C9⋊((C11⋊C5)⋊Dic17), … |
| maximal normal subgroup of | C5×Dic1683, C9⋊((C11⋊C5)⋊Dic17), Dic3366 |
| maximal quotient of | C5×Dic1683, Dic8415, C9⋊(C11⋊#66774), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6732, 8);