Faithful irreducible representation of M16

We use the group with presentation (here $$e$$ denotes the identity element):

$$G = M_{16} := \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^5 \rangle$$

Summary
This is actually a collection of two faithful irreducible two-dimensional representations of the group $$M_{16}$$, which form a single orbit under the action of the automorphism group, and also form a single orbit under the action of Galois automorphisms in the field of realization (at least in characteristic zero).

Representation table


Below are the matrices for concrete realizations of these two representation. Here $$i$$ denotes a square root of $$-1$$. The same representations can be realized over any field containing a square root of -1, if $$i$$ is replaced by that square root. Note that the two representations can be obtained from each other by sending $$i$$ to $$-i$$. This automorphism makes sense in characteristic zero, and more generally when the square root of $$-1$$ does not live in the prime field.

First version:

Second version: