Element structure of general linear group of degree two over a finite discrete valuation ring

This article discusses the element structure of the general linear group of degree two over a finite discrete valuation ring (i.e., local principal ideal ring) with residue field having size $$q$$, underlying prime characteristic of residue field $$p$$, and length $$l$$. The size of the whole ring is $$q^l$$, and each successive power of the unique maximal ideal has index $$q$$ in its predecessor.

Examples are $$\mathbb{Z}/(p^l\mathbb{Z})$$ (here $$p = q$$), $$\mathbb{F}_q[t]/(t^l)$$, and Galois rings.

The Galois ring with residue field of size $$q$$ and length $$l$$ is the unique (up to isomorphism) discrete valuation ring obtained as a degree $$\log_pq$$ extension of $$\mathbb{Z}/p^l\mathbb{Z}$$, and hence generalizes both $$\mathbb{Z}/(p^l\mathbb{Z})$$ and $$\mathbb{F}_q$$. The Galois ring has characteristic $$p^l$$.

Particular cases
The size of the group is $$q^{4l - 3}(q - 1)(q^2 - 1)$$ and the number of conjugacy classes is $$q^{2l} - q^{l-1}$$. Some particular cases for small values of $$l$$ are given below.