# A group of order $16$.

Let $G$ be the group generated by $a,b,c$ with relations \begin{align} a^2=1, b^2=1, c^2=1, ac=ca, bc=cb, abab=baba. \end{align} What is this group? I think that the group has order $16$. Is it correct?

• Why do you think that the order is $16$? – Arnaud Mortier Jul 8 '18 at 12:20

Clearly $c$ is central, so $G=\left< a,b\right>\times\left<c\right>$. But the relations between $a$ and $b$ are $a^2=b^2=(ab)^4=1$ so that $\left<a,b\right>$ is dihedral of order $8$.