What are some Group representation of the rubik's cube group? The Rubik's cube corresponds to valid sequences of moves of the Rubik's cube. What are some group representations of this group (with respect to finite dimensional vector spaces on finite fields)?
Ideally, I am looking for embeddings. I know that you can make a representation of the symmetric group 48 on a 48 dimensional vector space, and then embed the rubik's cube group into that. Can you make a representation based on a lower dimensional vector space?
(My group theory and linear algebra are a little rusty. Feel free to edit this to make more sense.)
 A: 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) $20$.
