Construction of the faithful irreducible representation for the subgroup \(G\) contained in \(S_7\) (Q1239256)

Every element in the transitive subgroup \(G\) of order 42 contained in the symmetric group of degree 7 is expressed as an ordered product of powers of the basic permutations \(g2 \equiv (1234567) \text{ and } g_8 \equiv (1)(243756)\). The only faithful irreducible representation for \(G\) is then given explicitly in terms of the six-dimensional matrices \(D(g_2)\) and \(D(g_8)\), obtained here with the aid of 'es. simple computational algorithm.