Finite Group DatabasebetaFinite Group Databasebeta| order | 8670 |
| automorphism | Aut(G) order 9432960 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D4335 |
| unambiguous semidirect product | C3⋊D1445 |
| unambiguous semidirect product | C5⋊D867 |
| unambiguous semidirect product | C289⋊D15 |
| Fitting, derived, Hall | C3×C5×C289 ⋊ C2 |
| Hall | C5×C289 ⋊ D3 |
| Hall | C3×C289 ⋊ D5 |
| Sylow | C289 ⋊ D15 |
| Hall | C3×C5 ⋊ D289 |
| Sylow | C5 ⋊ D867 |
| Sylow | C3 ⋊ D1445 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C3×C5×C289 | C2 |
| Frattini | C17 | D255 |
| socle | C3×C5×C17 | D17 |
| LCS | G ▷ C3×C5×C289 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,5,17^2|d^17=e^16,b^a=b^2,c^a=c^4,d^a=d^16e,e^a=e^162169 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D15×D289, D5×D867, D3×D1445, … |
| maximal normal subgroup of | D15×D289, D5×D867, D3×D1445, … |
| maximal quotient of | Dic4335, C2×D4335, D13005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8670, 8);