Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C3×C5×C7 ⋊ M16 |
| Hall | C5×C7 ⋊ #208 |
| Hall | C3×C7 ⋊ #676 |
| Hall | C3×C5 ⋊ #1131 |
| Sylow | C7 ⋊ #6649 |
| Sylow | C5 ⋊ #66557 |
| Sylow | C3 ⋊ #10587893 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D3×D35 |
| commutator, socle | C2×C3×C5×C7 | C2×C4 |
| Fitting, cosocle | C3×C4×C5×C7 | C22 |
| prime radical | C2×C3×D35 | C4 |
| hypocenter | C3×C5×C7 | M16 |
| LCS | G ▷ C2×C3×C5×C7 ▷ C3×C5×C7 | |
| UCS | C1 ◁ C4 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 840 |
Polycyclic presentation:
2^4,3,5,7|b^2=c,c^2=d,b^a=bd,f^a=f^4,g^a=g^6,e^b=e^2222 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 220);