//Q8.A4.D6 //A4^2.C2^2 FScale:=CyclotomicField(24:Sparse:=true); F:=FScale; Z4:=F.1^2; sq2:=F.1+F.1^7; G:=MatrixGroup<4,F| [-Z4/2,-Z4/2,-Z4/2,-Z4/2, -1/2,1/2,-1/2,1/2, -1/2,-1/2,1/2,1/2, Z4/2,-Z4/2,-Z4/2,Z4/2], [-1/sq2,0,-1/sq2,0, 0,Z4/sq2,0,Z4/sq2, Z4/sq2,0,-Z4/sq2,0, 0,1/sq2,0,-1/sq2], [-1/sq2,-1/sq2,0,0, Z4/sq2,-Z4/sq2,0,0, 0,0,Z4/sq2,Z4/sq2, 0,0,1/sq2,-1/sq2], [Z4,0,0,0, 0,1,0,0, 0,0,1,0, 0,0,0,-Z4]>; 1 1 C1 A4^2.C2^2 3 [] 1 4 C2 C2*S4 2 <<1>> 1 5 C2 C2*S4 2 <<1>> 1 6 C2 C4:D4 2 <<1>> 1 7 C2 C2*D4 2 <<1>> 1 8 C2 C2*D4 2 <<1>> 1 9 C2 C2^3 2 <<1>> 1 10 C3 C3*S4 2 <<1>> 1 11 C3 C3*S4 2 <<1>> 1 12 C3 C3^2 1 <<1>, <1>> 1 12 C3 C3^2 1 <<2>, <2>> 1 15 C4 C4*S4 2 <<3>> 1 17 C4 C4*S4 2 <<1>> 1 29 C4 C4^2 1 <<1>, <2>> 1 29 C4 C4^2 1 <<1>, <1>> 1 51 C6 C6 0 <<1>, <3>, <4>> 1 52 C6 C6 0 <<1>, <3>, <4>> 1 56 C3^2 C3^2 0 <<1, 0>, <2, 1>, <2, 2>> 1 56 C3^2 C3^2 0 <<1, 0>, <1, 2>, <2, 2>> 1 86 C2*C4 C2*C4 0 <<0, 1>, <1, 0>, <1, 1>> 1 96 C2*C4 C2*C4 0 <<0, 1>, <1, 0>, <1, 1>> 1 117 C12 C12 0 <<4>, <5>, <9>> 1 117 C12 C12 0 <<4>, <7>, <9>> 1 118 C12 C12 0 <<3>, <5>, <8>> 1 118 C12 C12 0 <<3>, <7>, <8>> 1 144 C4^2 C4^2 0 <<0, 3>, <1, 1>, <1, 2>> 1 144 C4^2 C4^2 0 <<0, 1>, <1, 1>, <1, 2>> <2, <[ <<1>> ]>> <3, <[ <<1>> ]>> <4, <[ <<1>> ]>> <5, <[ <<1>> ]>> <6, <[ <<1>> ]>> <7, <[ <<1>> ]>> <8, <[ <<1>>, <<2>> ]>> <9, <[ <<1>>, <<2>> ]>> <10, <[ <<1>, <1>>, <<2>, <2>> ]>> <11, <[ <<2>, <2>>, <<1>, <1>> ]>> <12, <[ <<3>>, <<1>> ]>> <13, <[ <<1>>, <<3>> ]>> <14, <[ <<1>, <2>>, <<2>, <3>> ]>> <15, <[ <<1>, <1>>, <<3>, <3>> ]>> <16, <[ <<1>, <3>, <4>>, <<2>, <3>, <5>> ]>> <17, <[ <<1>, <3>, <4>>, <<2>, <3>, <5>> ]>> <18, <[ <<1, 0>, <2, 1>, <2, 2>>, <<0, 2>, <1, 0>, <1, 1>>, <<1, 1>, <1, 2>, <2, 0>>, <<0, 1>, <2, 0>, <2, 2>> ]>> <19, <[ <<1, 0>, <1, 2>, <2, 2>>, <<0, 1>, <1, 0>, <1, 1>>, <<1, 1>, <2, 0>, <2, 1>>, <<0, 2>, <2, 0>, <2, 2>> ]>> <20, <[ <<0, 1>, <1, 0>, <1, 1>>, <<0, 3>, <1, 0>, <1, 3>> ]>> <21, <[ <<0, 1>, <1, 0>, <1, 1>>, <<0, 3>, <1, 0>, <1, 3>> ]>> <22, <[ <<4>, <5>, <9>>, <<3>, <4>, <11>>, <<3>, <7>, <8>>, <<1>, <8>, <9>> ]>> <23, <[ <<4>, <7>, <9>>, <<1>, <3>, <4>>, <<3>, <5>, <8>>, <<8>, <9>, <11>> ]>> <24, <[ <<3>, <5>, <8>>, <<8>, <9>, <11>>, <<4>, <7>, <9>>, <<1>, <3>, <4>> ]>> <25, <[ <<3>, <7>, <8>>, <<1>, <8>, <9>>, <<4>, <5>, <9>>, <<3>, <4>, <11>> ]>> <26, <[ <<0, 3>, <1, 1>, <1, 2>>, <<1, 1>, <2, 3>, <3, 2>>, <<0, 1>, <3, 2>, <3, 3>>, <<1, 2>, <2, 1>, <3, 3>> ]>> <27, <[ <<0, 1>, <1, 1>, <1, 2>>, <<1, 1>, <2, 1>, <3, 2>>, <<0, 3>, <3, 2>, <3, 3>>, <<1, 2>, <2, 3>, <3, 3>> ]>> Reduced: 1 1 C1 A4^2.C2^2 3 [] 1 4 C2 C2*S4 2 <<1>> 1 5 C2 C2*S4 2 <<1>> 1 6 C2 C4:D4 2 <<1>> 1 7 C2 C2*D4 2 <<1>> 1 8 C2 C2*D4 2 <<1>> 1 9 C2 C2^3 2 <<1>> 1 10 C3 C3*S4 2 <<1>> 1 11 C3 C3*S4 2 <<1>> 1 12 C3 C3^2 1 <<1>, <1>> 1 12 C3 C3^2 1 <<2>, <2>> 1 15 C4 C4*S4 2 <<3>> 1 16 C4 C4*S4 2 <<1>> 1 29 C4 C4^2 1 <<1>, <2>> 1 29 C4 C4^2 1 <<1>, <1>> 1 56 C3^2 C3^2 0 <<1, 0>, <2, 1>, <2, 2>> 1 56 C3^2 C3^2 0 <<1, 0>, <1, 2>, <2, 2>> 1 118 C12 C12 0 <<4>, <5>, <9>> 1 118 C12 C12 0 <<4>, <7>, <9>> 1 124 C12 C12 0 <<3>, <5>, <8>> 1 124 C12 C12 0 <<3>, <7>, <8>> 1 166 C4^2 C4^2 0 <<1, 0>, <2, 3>, <3, 1>> 1 166 C4^2 C4^2 0 <<1, 0>, <2, 1>, <3, 1>> <2, <[ <<1>> ]>> <3, <[ <<1>> ]>> <4, <[ <<1>> ]>> <5, <[ <<1>> ]>> <6, <[ <<1>> ]>> <7, <[ <<1>> ]>> <8, <[ <<1>>, <<2>> ]>> <9, <[ <<1>>, <<2>> ]>> <10, <[ <<1>, <1>>, <<2>, <2>> ]>> <11, <[ <<2>, <2>>, <<1>, <1>> ]>> <12, <[ <<3>>, <<1>> ]>> <13, <[ <<1>>, <<3>> ]>> <14, <[ <<1>, <2>>, <<2>, <3>> ]>> <15, <[ <<1>, <1>>, <<3>, <3>> ]>> <16, <[ <<1, 0>, <2, 1>, <2, 2>>, <<0, 2>, <1, 0>, <1, 1>>, <<1, 1>, <1, 2>, <2, 0>>, <<0, 1>, <2, 0>, <2, 2>> ]>> <17, <[ <<1, 0>, <1, 2>, <2, 2>>, <<0, 1>, <1, 0>, <1, 1>>, <<1, 1>, <2, 0>, <2, 1>>, <<0, 2>, <2, 0>, <2, 2>> ]>> <18, <[ <<4>, <5>, <9>>, <<3>, <4>, <11>>, <<3>, <7>, <8>>, <<1>, <8>, <9>> ]>> <19, <[ <<4>, <7>, <9>>, <<1>, <3>, <4>>, <<3>, <5>, <8>>, <<8>, <9>, <11>> ]>> <20, <[ <<3>, <5>, <8>>, <<8>, <9>, <11>>, <<4>, <7>, <9>>, <<1>, <3>, <4>> ]>> <21, <[ <<3>, <7>, <8>>, <<1>, <8>, <9>>, <<4>, <5>, <9>>, <<3>, <4>, <11>> ]>> <22, <[ <<1, 0>, <2, 3>, <3, 1>>, <<1, 3>, <2, 1>, <3, 0>>, <<0, 3>, <3, 0>, <3, 1>>, <<0, 1>, <1, 0>, <1, 3>> ]>> <23, <[ <<1, 0>, <2, 1>, <3, 1>>, <<1, 3>, <2, 3>, <3, 0>>, <<0, 1>, <3, 0>, <3, 1>>, <<0, 3>, <1, 0>, <1, 3>> ]>>