Find an element $a$ from some group such that $|a|=6$ and $C(a) \neq C(a^3)$ Find an element $a$ from some group $G$ such that $|a|=6$ and $C(a) \neq C(a^3)$.
$|a|$ is the order of $a$.
$C(a)$ is the Centraliser of $a$ in $G$.
This question is from Contemprary Abstract Algebra by Joseph A Gallian 4th edition,
Chapter 3, Question 24.

The simplest example that i see is the following. Take
$$G=S_5,$$ 
where $S_5$ denotes the group of all the permutations of the set $\{1,2,3,4,5\}$, and take 
$$a=(123)(45).$$
So we have that 
$$a^3=(45).$$
The centralizer of $a$ is given by
$$ C(a)=\left\{\sigma\tau \;| \; \sigma \in \langle(123)\rangle, \tau \in \langle(45)\rangle \right\} \simeq \mathbb Z_3 \times\mathbb Z_2,$$
while the centralizer of $a^3$
$$C(a^3)=\left\{\sigma\tau \;| \; \sigma \in S_3, \tau \in \langle(45)\rangle \right\} \simeq S_3 \times \mathbb Z_2.$$
