Finite Group DatabasebetaFinite Group Databasebeta| order | 8670 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C17 |
| unambiguous semidirect product | C5⋊((C17⋊D17)⋊C3) |
| unambiguous semidirect product | (C5⋊(C17⋊D17))⋊C3 |
| Fitting, socle, derived, Hall | C5×C172 ⋊ C2×C3 |
| Hall | C5×C172⋊C3 ⋊ C2 |
| Hall | C5⋊(C17⋊D17) ⋊ C3 |
| Hall | C172⋊C3 ⋊ D5 |
| Sylow | C172 ⋊ C3×D5 |
| Sylow | C5 ⋊ (C17⋊D17)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C172 | C2×C3 |
| LCS | G ▷ C5×C172 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,5,17^2|c^a=c^4,d^a=d^16,e^a=e^16,d^b=d^2e^16,e^b=d^7e^14252 conjugacy classes — view the full character table.
| maximal subgroup of | #31773247, #19153613⋊C3, #19153612⋊C3, … |
| maximal normal subgroup of | #31773247, #19153613⋊C3, #19153612⋊C3, … |
| maximal quotient of | #19153601⋊C3, C2×C5⋊((C17⋊D17)⋊C3), C5⋊#19104514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8670, 13);