= Solution
The relations in (a) reduce every word in the generators to $U^aV^b$, with $0\leq a<4$ and $b\in\{0,1\}$. These eight <matrices> are distinct: the $b=0$ elements have third diagonal entry $1$ and the $b=1$ elements have entry $-1$, while the four planar blocks are distinct in each family. Their multiplication is
$$
\boxed{(U^aV^b)(U^cV^d)=U^{a+(-1)^bc}V^{b+d},}
$$
with exponents reduced modulo four and two. This proves closure. The <identity element> is $U^0V^0$, the inverse of $U^a$ is $U^{-a}$, and each $U^aV$ is its own inverse. <Associativity> is inherited from matrix multiplication, proving the <group axioms>. This is the eight-element <dihedral group> $D_8$.
Set $g_j=U^j$ and $g_{j+4}=U^jV$ for $j=0,1,2,3$, consistently with the two labels specified in the question. The row-times-column <multiplication table> is
$$
\boxed{\begin{array}{c|cccccccc}
&g_0&g_1&g_2&g_3&g_4&g_5&g_6&g_7\\\hline
g_0&g_0&g_1&g_2&g_3&g_4&g_5&g_6&g_7\\
g_1&g_1&g_2&g_3&g_0&g_5&g_6&g_7&g_4\\
g_2&g_2&g_3&g_0&g_1&g_6&g_7&g_4&g_5\\
g_3&g_3&g_0&g_1&g_2&g_7&g_4&g_5&g_6\\
g_4&g_4&g_7&g_6&g_5&g_0&g_3&g_2&g_1\\
g_5&g_5&g_4&g_7&g_6&g_1&g_0&g_3&g_2\\
g_6&g_6&g_5&g_4&g_7&g_2&g_1&g_0&g_3\\
g_7&g_7&g_6&g_5&g_4&g_3&g_2&g_1&g_0
\end{array}}
$$
The rotation powers commute with one another, and $U^a$ commutes with $U^cV$ exactly when $a$ is even. Two elements $U^aV,U^cV$ commute exactly when $a-c$ is even. These statements also check the symmetric portions of the table.
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