Linear encoding of a group

Definition

A linear encoding of a group $G$ is the following data:

An integer $n$ , a field $F_{q}$ and the standard encoding for $G L_{n} (F_{q})$ (as described in encoding of symmetric groups)
A specification of the given group $G$ as a subgroup of the general linear group by means of a generating set $A$ (preferably a small one)
A membership test for $G$ as a subgroup of the general linear group

A linear encoding can also be viewed as a concrete, or explicit, description using a failthful linear representation.

Permutation encoding is the notion for encoding as a subgroup of a symmetric group.