Inspired by this question, in particular by the indeed elegant description of the Mathieu group $M_{23}$ it starts with, I am wondering about the following:

Instead of $C$, defined as the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}=23\cdot89+1$ elements, let us take the "complementary" subgroup $D$ of order $89$. Knowing that $M_{23}$ is the group of additive maps of $F$ to itself which permute the set $C$, *what about the group $G$ of additive maps of $F$ to itself which permute the set $D$ instead?* Naively, I would expect this group $G$ to be also a simple group by "duality", but it cannot be a sporadic one because of the divisor $89$.

What can be said about $G$?

Of course, any composite pernicious Mersenne number which is a product of two primes defines two "dual" groups in a similar way. Is there any interesting relationship between the groups of such a pair? Disclaimer: I am not a group theorist.