Finite Group DatabasebetaFinite Group Databasebeta| order | 7524 |
| automorphism | Aut(G) order 4062960 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic1881 |
| unambiguous semidirect product | C9⋊Dic209 |
| unambiguous semidirect product | C11⋊Dic171 |
| unambiguous semidirect product | C19⋊Dic99 |
| derived, Hall | C9×C11×C19 ⋊ C4 |
| Hall | C11×C19 ⋊ Dic9 |
| Hall | C9×C19 ⋊ Dic11 |
| Hall | C9×C11 ⋊ Dic19 |
| Sylow | C19 ⋊ Dic99 |
| Sylow | C11 ⋊ Dic171 |
| Sylow | C9 ⋊ Dic209 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D1881 |
| commutator, hypocenter | C9×C11×C19 | C4 |
| Frattini | C2×C3 | D627 |
| Fitting | C2×C9×C11×C19 | C2 |
| socle | C2×C3×C11×C19 | D3 |
| LCS | G ▷ C9×C11×C19 | |
| UCS | C1 ◁ C2 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,11,19,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^10,d^a=d^18,f^a=f^21884 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C5×Dic1881, Dic9405, C9⋊(C19⋊(Dic11⋊C5)), … |
| maximal normal subgroup of | C5×Dic1881, C9⋊(C19⋊(Dic11⋊C5)), Dic3762 |
| maximal quotient of | C5×Dic1881, Dic9405, C9⋊(C11⋊(C19⋊AGL(1,𝔽5))), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7524, 12);