Let V := FullRowSpace(GF(2),2) m1:=One(GF(2))*[[0,1,0],[1,0,0],[0,0,1]]; m2:=One(GF(2))*[[0,1,0],[0,0,1],[1,0,0]]; m:=Group(m1,m2); this is a matrix representation of $S3$ in GF(2) If we say OrbitLengths(m,V) we get [2,4,4,4] that is we show that this has four orbits and of the lenghts given that is the conjugacy classes here. Moving to irreducible characters by Brauer we know that the number of orbits is four, my problem is how do we get the orbitlengths of the orbits of irreducible characters ? Yours faithfully TT Seretlo
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