var zn = new Group256(20);var G = zn.Generate("Z", 1);var H = zn.Generate("5Z", 5);G.DisplayHead();H.DisplayHead();var Qg = new QuotientGroup<byte, Integer256>(G, H);Qg.Details();Qg.DisplayClasses();

|Z| = 20IsGroup      : TrueIsCommutative: True|5Z| = 4IsGroup      : TrueIsCommutative: True|Z/5Z| = 5 with |Z| = 20 and |5Z| = 4, OpBothIsGroup      : TrueIsCommutative: True@ = (  0)[  1]a = (  1)[ 20]b = (  2)[ 10]c = (  3)[ 20]d = (  4)[  5]|Z/5Z| = 5 with |Z| = 20 and |5Z| = 4, OpBoth *|@ a b c d--|---------- @|@ a b c d a|a b c d @ b|b c d @ a c|c d @ a b d|d @ a b cClass of : (  0)[  1]    Represent    (  0)[  1]    (  5)[  4]    ( 10)[  2]    ( 15)[  4]Class of : (  1)[ 20]    Represent    (  1)[ 20]    (  6)[ 10]    ( 11)[ 20]    ( 16)[  5]Class of : (  2)[ 10]    Represent    (  2)[ 10]    (  7)[ 20]    ( 12)[  5]    ( 17)[ 20]Class of : (  3)[ 20]    Represent    (  3)[ 20]    (  8)[  5]    ( 13)[ 20]    ( 18)[ 10]Class of : (  4)[  5]    Represent    (  4)[  5]    (  9)[ 20]    ( 14)[ 10]    ( 19)[ 20]

Simple S3, the group of permutations

var sn = new Sigma(3);var H0 = sn.Generate("H0", (1, 2, 3), (1, 2));var H1 = sn.Generate("H1", (1, 2, 3));H0.DisplayElements();H1.DisplayElements();var Qg0 = new QuotientGroup<byte, Permutation>(H0, H1);Qg0.Details();Qg0.DisplayClasses();

|H0| = 6IsGroup      : TrueIsCommutative:False@ = ( 1  2  3)[ 1+]a = ( 1  3  2)[ 2-]b = ( 2  1  3)[ 2-]c = ( 3  2  1)[ 2-]d = ( 2  3  1)[ 3+]e = ( 3  1  2)[ 3+]|H1| = 3IsGroup      : TrueIsCommutative: True@ = ( 1  2  3)[ 1+]a = ( 2  3  1)[ 3+]b = ( 3  1  2)[ 3+]|H0/H1| = 2 with |H0| = 6 and |H1| = 3, OpBothIsGroup      : TrueIsCommutative: True@ = ( 1  2  3)[ 1+]a = ( 1  3  2)[ 2-]|H0/H1| = 2 with |H0| = 6 and |H1| = 3, OpBoth *|@ a--|---- @|@ a a|a @Class of : ( 1  2  3)[ 1+]    Represent    ( 1  2  3)[ 1+]    ( 2  3  1)[ 3+]    ( 3  1  2)[ 3+]Class of : ( 1  3  2)[ 2-]    Represent    ( 1  3  2)[ 2-]    ( 2  1  3)[ 2-]    ( 3  2  1)[ 2-]

Then S5

var H0 = sn.Generate("H0", (1, 2, 3), (4, 5));var H1 = sn.Generate("H1", (4, 5));H0.DisplayElements();H1.DisplayElements();var Qg0 = new QuotientGroup<byte, Permutation>(H0, H1);Qg0.Details();Qg0.DisplayClasses();

|H0| = 6IsGroup      : TrueIsCommutative: True@ = ( 1  2  3  4  5)[ 1+]a = ( 1  2  3  5  4)[ 2-]b = ( 2  3  1  4  5)[ 3+]c = ( 3  1  2  4  5)[ 3+]d = ( 2  3  1  5  4)[ 6-]e = ( 3  1  2  5  4)[ 6-]|H1| = 2IsGroup      : TrueIsCommutative: True@ = ( 1  2  3  4  5)[ 1+]a = ( 1  2  3  5  4)[ 2-]|H0/H1| = 3 with |H0| = 6 and |H1| = 2, OpBothIsGroup      : TrueIsCommutative: True@ = ( 1  2  3  4  5)[ 1+]a = ( 2  3  1  4  5)[ 3+]b = ( 3  1  2  4  5)[ 3+]|H0/H1| = 3 with |H0| = 6 and |H1| = 2, OpBoth *|@ a b--|------ @|@ a b a|a b @ b|b @ aClass of : ( 1  2  3  4  5)[ 1+]    Represent    ( 1  2  3  4  5)[ 1+]    ( 1  2  3  5  4)[ 2-]Class of : ( 2  3  1  4  5)[ 3+]    Represent    ( 2  3  1  4  5)[ 3+]    ( 2  3  1  5  4)[ 6-]Class of : ( 3  1  2  4  5)[ 3+]    Represent    ( 3  1  2  4  5)[ 3+]    ( 3  1  2  5  4)[ 6-]