<s>[INST] Find an isomorphism from the octic group $G$ to the group $G'$ Find an isomorphism from the octic group $G$ to the group $G'$:
$G = \{e, s, s^2, s^3, b, g, d, t\}$
$$e =  (1)\quad s =  (1234)\quad s^2= (13)(24)\quad s^3= (1432)\quad b =  (14)(23)\quad g =  (24)\quad d =  (12)(34)\quad t =  (13)$$  
$G' = \{I_2, R, R^2, R^3, H, D, V, T\}$
$$\begin{align*}I_2&=\begin{pmatrix}1 & 0\\ 0& 1 \end{pmatrix}& R&=\begin{pmatrix} 0& -1\\1 &0  \end{pmatrix}& R^2&=\begin{pmatrix}-1 & 0\\ 0&-1  \end{pmatrix}& R^3&=\begin{pmatrix} 0& 1\\-1 & 0\end{pmatrix}\\\\
H&=\begin{pmatrix} 1& 0\\0 & -1 \end{pmatrix}& V&=\begin{pmatrix} -1& 0\\0 & 1 \end{pmatrix}& D&=\begin{pmatrix} 0& 1\\1& 0\end{pmatrix}& T&=\begin{pmatrix} 0& -1\\ -1& 0\end{pmatrix}\end{align*}$$
 [/INST] Jessica: Preserving the order of all elements isn't sufficient; you have to show that it preserves the group operation.
Here's a hint though.  $G^\prime$ is the set of symmetries of a square, right?  Try labelling the square's corners.  Then each symmetry is just a permutation of its corners...
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