Determine a normal subgroup $N$ of $G$ and determine the multiplication table for $G/N$

Question: Write down a multiplication table for $G$, the group of symmetries of a square. Determine a normal subgroup $N$ of $G$ and determine the multiplication table for $G/N$.

My problem is doing the multiplication table for symmetries of a square and determining the normal subgroup $N$ of $G$ right now. Once I know that, I can find the last part of the question. Need help

• Well, there is always a lazy option: Take $N=G$, then $N/G$ is the trivial group. – Sebastian Schulz Apr 4 '17 at 14:07
• To start a multiplication table, you need to know how many elements there are, and decide a name for each one. Have you done that? – Arthur Apr 4 '17 at 14:07
• @Arthur no because I do not how many elements should be in there like you just said – behold Apr 4 '17 at 14:09

The symmetry group of the square is the dihedral group $D_4$ with $8$ elements. Every subgroup of index $2$ is normal, so we could take the subgroup $C_4$ generated by a rotation of the sqaure. Then the quotient group $D_4/C_4\cong C_2$ has two elements.

References:

Table of dihedral group D4

How to describe all normal subgroups of the dihedral group Dn?

• the table of dihedral looks confusing because the comments pointed out many errors on that table – behold Apr 4 '17 at 19:59
• Yes, you are right. But the answers there correct this, and I am sure you will find several other tables for $D_4$, if you google. Did you find this one? – Dietrich Burde Apr 4 '17 at 20:00