TITLE: Presentation of wreath product $G=S_3 \wr S_3$ of symmetric groups. What is the isomorphism type of $G/[G,G]$?
QUESTION [5 upvotes]: I'm trying to answer the first part of a group theory question as revision for an exam that goes as follows;

Let $G = S_3 \wr S_3$, the permutational wreath product of two symmetric groups of degree three. Give a presentation for $G$ and determine the isomorphism type of $G/[G, G]$.

I'm not sure how to go about finding generators for the wreath product itself.

Is there a method for combining the generators of the symmetric groups to form generators for the wreath prouct?

Any pointers would be much appreciated, thanks in advance!

REPLY [4 votes]: Here is a presentation of $S_3 \wr S_3$. Note that $a,b$ generate the top $S_3$ factor, and $x,y$ generate one of the three factors of the base group, the other two being $\langle x^a,y^a \rangle$ and $\langle x^{a^2},y^{a^2}\rangle$, where $x^a = a^{-1}xa$.
$$\langle a,b,x,y,\mid  a^3,b^2,(ab)^2,x^3,y^2,(xy)^2,
 [x,x^a],[y,x^a],[x,x^{a^2}],[y,x^{a^2}],
 [x,y^a],[y,y^a],[x,y^{a^2}],[y,y^{a^2}],  
 [x,b], [y,b] \rangle.$$
A routine calculation from the presentation shows that $G/[G,G] \cong C_2 \times C_2$, but it is not hard to show that directly.
PS: In fact the four  relators $[x,x^{a^2}],[y,x^{a^2}],[x,y^{a^2}],[y,y^{a^2}]$ are redundant. The fact that $\langle x,y \rangle$ commutes with $\langle x^{a^2},y^{a^2} \rangle$ follows from the fact that $\langle x,y \rangle$ commutes with $\langle x^{a},y^{a} \rangle$ by conjugating by $b$.