//This file gives two 248x248 matrices a,b over GF(3) such that Order(a) = 2, Order(b) = 3, Order(a*b) eq 41 and a,b generate a subgroup of the adjoint representation of E_8(3) isomorphic to PSL(2,81). This is defined to be a subgroup of GL(248,3). This presentation is a standard presentation for PSL(2,81).

//Also given is another 248x248 matrix c over GF(3) such that Order(c) = 2 and the group generated by a,b,c is isomorphic to PGL(2,81).



a := 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b := 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c := 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Q := GL(248,3);
H := sub<Q|a,b>; //H is a subgroup of the adjoint representation of E_8(3) isomorphic to PSL(2,81) (defined to be a subgroup of GL(248,3)).
OH := sub<Q|a,b,c>; //OH is an overgroup of H, isomorphic to PGL(2,81), which is also a subgroup of E_8(3) but defined over GL(248,3).