Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C5×C7×Q8 ⋊ C2 |
| Hall | C3×C5×C7 ⋊ #41 |
| Hall | C5×C7 ⋊ #239 |
| Hall | C3×C7 ⋊ #713 |
| Hall | C3×C5 ⋊ #1161 |
| Sylow | C7 ⋊ #6821 |
| Sylow | C5 ⋊ #66732 |
| Sylow | C3 ⋊ #10587991 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D105 |
| commutator, socle, cosocle | C2×C3×C5×C7 | C23 |
| Fitting | C3×C5×C7×Q8 | C2 |
| hypercenter | Q8 | D105 |
| hypocenter | C3×C5×C7 | #41 |
| LCS | G ▷ C2×C3×C5×C7 ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
160 conjugacy classes of subgroups.
| transitive degree | μt(G) = 840 |
Polycyclic presentation:
2^4,3,5,7|b^2=d,c^2=d,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^6,c^b=cd270 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 917);