What are some Group representation of the rubik's cube group?

Solution 1:

I think $20$ is the smallest degree of a faithful representation of the Rubik cube group, certainly in characteristic $0$ or characteristic coprime to the group order, and probably over any field. As Henning Makholm commented, there exist faithful representations of degree $20$, so we just need to show that this is the smallest degree possible.

The Rubik cube group contains a subgroup $H = H_1 \times H_2$, where $H_1$ and $H_2$ have the structures $H_1 = 2^{11}:A_{12}$ and $H_2 =3^7:A_8$.

Now the only nontrivial proper normal subgroups of $H_1$ are its centre $M$ of order $2$, and an elementary abelian group $N$ of order $2^{11}$. In particular, $M$ is its unique minimal normal subgroup, so a minimal degree faithful representation of $H_1$ must be irreducible. Its restriction to $N$ cannot be homogeneous (since $N$ is abelian but not cyclic), and its homogeneous components are permuted by $A_{12}$, so there must be at least $12$ of them.

So the smallest degree of a faithful representation of $H_1$ is $12$ and similarly it is $8$ for $H_2$. By the theory of representations of direct products, the smallest degree of a faithful irreducible representation of $H$ is $12 \times 8 = 96$. Since $H$ has exactly two minimal normal subgroups, the only way we could improve on that is with a representation with two constituents having different minimal normal subgroups in their kernels, and doing that results in a faithful representation of degree (at least) $20$.