Finite Group DatabasebetaFinite Group Databasebeta| order | 7542 |
| automorphism | Aut(G) order 9457668 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D3771 |
| unambiguous semidirect product | C9⋊D419 |
| unambiguous semidirect product | C419⋊D9 |
| Fitting, derived, Hall | C9×C419 ⋊ C2 |
| Sylow | C419 ⋊ D9 |
| Sylow | C9 ⋊ D419 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C9×C419 | C2 |
| Frattini | C3 | D1257 |
| socle | C3×C419 | D3 |
| LCS | G ▷ C9×C419 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,419,3|b^3=d^2,b^a=b^2d,c^a=c^418,d^a=d^21887 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D9×D419, C2×D3771, C5×D3771, … |
| maximal normal subgroup of | D9×D419, C2×D3771, C5×D3771 |
| maximal quotient of | Dic3771, C2×D3771, C5×D3771, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7542, 4);