Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×C5×C7 ⋊ D8 |
| Hall | C5×C7 ⋊ #213 |
| Hall | C3×C7 ⋊ #686 |
| Hall | C3×C5 ⋊ D56 |
| Sylow | C7 ⋊ #6715 |
| Sylow | C5 ⋊ #66561 |
| Sylow | C3 ⋊ #10587842 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11719758 |
| commutator, Fitting, cosocle | C3×C4×C5×C7 | C22 |
| Frattini, hypercenter | C4 | D7×D15 |
| socle | C2×C3×C5×C7 | D4 |
| hypocenter | C3×C5×C7 | D8 |
| LCS | G ▷ C3×C4×C5×C7 ▷ C2×C3×C5×C7 ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 840 |
Polycyclic presentation:
2^4,3,5,7|b^2=cd,c^2=d,b^a=bc,c^a=cd,g^a=g^6,e^b=e^2,f^b=f^4150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 149);