Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 7 / 1 / vi

Choose a representative Sylow subgroup with generatorsOn the first four points, $r^4=s^2=1$ and $srs=r^{-1}$, so $⟨r,s⟩$ is the dihedral group of order eight. The disjoint transposition $t$ commutes with it and generates an independent factor of order two. Thus $G=⟨r,s,t⟩\cong D_8\times C_2$ has order 16. Since the power of two dividing $7!$ is $2^{\lfloor 7/2\rfloor +\lfloor 7/4\rfloor }=2^4$, this is a Sylow 2-subgroup of $S_7$. Any other choice is conjugate to it and has conjugate series.

The dihedral relation gives $[r,s]=r^2$, while $t$ commutes with both generators. Modulo $⟨r^2⟩$ all three generators commute, so the commutator subgroup is exactly $⟨r^2⟩$, and $r^2$ is central. Hence the lower central series isWithin the dihedral factor only $1,r^2$ commute with both $r$ and $s$. The center of $G$ is therefore $⟨r^2,t⟩\cong C_2\times C_2$. Its quotient has order four and is abelian, so the upper central series isBoth series show that the nilpotency class is two.