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To focus just on the question of giving a presentation of a finite general linear group over the prime field (or other finite field), leaving aside the fuzzy connection with representations of finite groups: This can be looked at profitably in the broader setting of finite groups of Lie type as studied by Steinberg in the early 1960s, even though there is a little distance between general linear and special linear groups. For a "universal" Chevalley group such as a special linear group over an arbitrary field, Steinberg gives a nice presentation in section 6 of his 1967-68 Yale lecture notes, building on his earlier Brussels conference note (published in French). None of this material seems to be readily available online, unfortunately, though Steinberg's papers are collected in an affordable AMS volume. Attempts over the years to publish the Yale lectures more formally fell through.

Steinberg's presentation is based on the BN-pair structure, but with some refinements. Treating the finite general linear group doesn't require too much modification, but perhaps isn't written down explicitly (?) In any case, this type of presentation is transparent and has the added merit of leading (over commutative rings) to fundamental ideas in algebraic K-theory involving Steinberg symbols, etc. Meanwhile finite group theorists involved with the classification of simple groups have made their own good use of BN-pair ideas following Tits and Steinberg.

ADDED: Notice at the end of Steinberg's section 6 (page 72) the simple explicit presentation of $SL_n(\mathbb{F}_q)$ given by his method. You could get this by taking shortcuts in his proof, but keep in mind that the case $n=2$ requires extra care. To get instead the general linear group you just have to add some generators corresponding to scalar matrices along with suitable relations. Nothing is gained in this approach by working only over the prime field, which anyway would be inadequate for most applications.

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To focus just on the question of giving a presentation of a finite general linear group over the prime field (or other finite field), leaving aside the fuzzy connection with representations of finite groups: This can be looked at profitably in the broader setting of finite groups of Lie type as studied by Steinberg in the early 1960s, even though there is a little distance between general linear and special linear groups. For a "universal" Chevalley group such as a special linear group over an arbitrary field, Steinberg gives a nice presentation in section 6 of his 1967-68 Yale lecture notes, building on his earlier Brussels conference note (published in French). None of this material seems to be readily available online, unfortunately, though Steinberg's papers are collected in an affordable AMS volume. Attempts over the years to publish the Yale lectures more formally fell through.

Steinberg's presentation is based on the BN-pair structure, but with some refinements. Treating the finite general linear group doesn't require too much modification, but perhaps isn't written down explicitly (?) In any case, this type of presentation is transparent and has the added merit of leading (over commutative rings) to fundamental ideas in algebraic K-theory involving Steinberg symbols, etc. Meanwhile finite group theorists involved with the classification of simple groups have made their own good use of BN-pair ideas following Tits and Steinberg.