Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D3×D35 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Hall | C3×C5×C7 ⋊ C2×C8 |
| Hall | C5×C7 ⋊ D3×C8 |
| Hall | C3×C7 ⋊ C8×D5 |
| Hall | C3×C5 ⋊ C8×D7 |
| Sylow | C7 ⋊ #6648 |
| Sylow | C5 ⋊ D21⋊C8 |
| Sylow | C3 ⋊ C8×D35 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D3×D35 |
| commutator, hypocenter | C3×C5×C7 | C2×C8 |
| Fitting, cosocle | C3×C4×C5×C7 | C22 |
| socle | C2×C3×C5×C7 | C2×C4 |
| prime radical | C2×D105 | C4 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C4 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 840 |
Polycyclic presentation:
2^4,3,5,7|a^2=c,b^2=c,c^2=d,f^a=f^4,g^a=g^6,e^b=e^2228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 219);