The symmetry group of the interval $[-1,1]$ is $\mathbb Z_2$, since it consists only of the identity and the reflection at the origin.
Consider now the square $[-1,1]^2$. Obviously, its symmetry group contains the product $\mathbb Z_2 \times \mathbb Z_2$, since we reflect the square across the x-axis and the y-axis. But we can additionally rotate the square, so we recognize $\mathbb Z_4$ as an additional factor.
Question: How can you calculate the symmetry group of the product of two polytopes from the symmetry group of the factors?