**Generators:***g1=g1*,*g2a=g2a*,*g2b=g2b***Representations:**dim3, dim1a, dim1b.**Conjugacy Class 1**of Group A4 Graph1 - 3 elements in conjugacy class including Example1_1**Cable 1:[4, 2, 1, 3].**- Orders of Disjoint Cycles: [3].
**Cable 2:[1, 4, 2, 3].**- Orders of Disjoint Cycles: [3].

**Conjugacy Class 2**of Group A4 Graph2 - 3 elements in conjugacy class including Example2_1**Cable 1:[2, 4, 3, 1].**- Orders of Disjoint Cycles: [3].
**Cable 2:[4, 3, 2, 1].**- Orders of Disjoint Cycles: [2, 2].

**Generators:***g1=[2, 1, 3, 4]*,*g2=[2, 3, 1, 4]*,*g3=[2, 3, 4, 1]***Representations:**dim1, dim2, dim3a, dim3b.**Conjugacy Class 1**of Group S4 Graph1 - 9 elements in conjugacy class including Example1_1**Cable 1:[4, 3, 1, 2].**- Orders of Disjoint Cycles: [4].
**Cable 2:[1, 4, 2, 3].**- Orders of Disjoint Cycles: [3].

**Conjugacy Class 2**of Group S4 Graph2 - 9 elements in conjugacy class including Example2_1**Cable 1:[2, 1, 3, 4].**- Orders of Disjoint Cycles: [2].
**Cable 2:[4, 2, 1, 3].**- Orders of Disjoint Cycles: [3].

**Conjugacy Class 3**of Group S4 Graph3 - 10 elements in conjugacy class including Example3_1**Cable 1:[3, 2, 1, 4].**- Orders of Disjoint Cycles: [2].
**Cable 2:[4, 3, 1, 2].**- Orders of Disjoint Cycles: [4].

**Conjugacy Class 4**of Group S4 Graph4 - 6 elements in conjugacy class including Example4_1**Cable 1:[3, 1, 4, 2].**- Orders of Disjoint Cycles: [4].
**Cable 2:[4, 3, 1, 2].**- Orders of Disjoint Cycles: [4].

**Generators:***g1=[2, 3, 1, 4, 5, 6]*,*g2a=[2, 1, 4, 3, 5, 6]*,*g2b=[3, 4, 1, 2, 5, 6]*,*ref=[1, 2, 3, 4, 6, 5]***Representations:**dim3, dim1a, dim1b, dim3r, dim1ar, dim1br.**Conjugacy Class 1**of Group A4Z2 Graph1 - 9 elements in conjugacy class including Example1_1**Cable 1:[3, 4, 1, 2, 6, 5].**- Orders of Disjoint Cycles: [2, 2, 2].
**Cable 2:[2, 4, 3, 1, 5, 6].**- Orders of Disjoint Cycles: [3].

**Conjugacy Class 2**of Group A4Z2 Graph2 - 9 elements in conjugacy class including Example2_1**Cable 1:[4, 2, 1, 3, 6, 5].**- Orders of Disjoint Cycles: [3, 2].
**Cable 2:[2, 4, 3, 1, 5, 6].**- Orders of Disjoint Cycles: [3].

**Conjugacy Class 3**of Group A4Z2 Graph3 - 9 elements in conjugacy class including Example3_1**Cable 1:[1, 3, 4, 2, 6, 5].**- Orders of Disjoint Cycles: [3, 2].
**Cable 2:[2, 1, 4, 3, 5, 6].**- Orders of Disjoint Cycles: [2, 2].

**Conjugacy Class 4**of Group A4Z2 Graph4 - 9 elements in conjugacy class including Example4_1**Cable 1:[1, 3, 4, 2, 6, 5].**- Orders of Disjoint Cycles: [3, 2].
**Cable 2:[2, 1, 4, 3, 6, 5].**- Orders of Disjoint Cycles: [2, 2, 2].

**Conjugacy Class 5**of Group A4Z2 Graph5 - 8 elements in conjugacy class including Example5_1**Cable 1:[1, 3, 4, 2, 6, 5].**- Orders of Disjoint Cycles: [3, 2].
**Cable 2:[4, 2, 1, 3, 6, 5].**- Orders of Disjoint Cycles: [3, 2].

**Generators:***g1=[2, 1, 3, 4, 5, 6]*,*g2=[2, 3, 1, 4, 5, 6]*,*g3=[2, 3, 4, 1, 5, 6]*,*ref=[1, 2, 3, 4, 6, 5]***Representations:**dim1, dim2, dim3a, dim3b, dim1r, dim2r, dim3ar, dim3br.**Conjugacy Class 1**of Group S4Z2 Graph1 - 23 elements in conjugacy class including Example1_1**Cable 1:[2, 4, 1, 3, 6, 5].**- Orders of Disjoint Cycles: [4, 2].
**Cable 2:[1, 2, 4, 3, 5, 6].**- Orders of Disjoint Cycles: [2].

**Conjugacy Class 2**of Group S4Z2 Graph2 - 21 elements in conjugacy class including Example2_1**Cable 1:[2, 3, 1, 4, 6, 5].**- Orders of Disjoint Cycles: [3, 2].
**Cable 2:[4, 2, 3, 1, 6, 5].**- Orders of Disjoint Cycles: [2, 2].

**Conjugacy Class 3**of Group S4Z2 Graph3 - 23 elements in conjugacy class including Example3_1**Cable 1:[4, 2, 3, 1, 6, 5].**- Orders of Disjoint Cycles: [2, 2].
**Cable 2:[4, 1, 2, 3, 5, 6].**- Orders of Disjoint Cycles: [4].

**Conjugacy Class 4**of Group S4Z2 Graph4 - 19 elements in conjugacy class including Example4_1**Cable 1:[3, 4, 2, 1, 6, 5].**- Orders of Disjoint Cycles: [4, 2].
**Cable 2:[4, 1, 2, 3, 5, 6].**- Orders of Disjoint Cycles: [4].

**Conjugacy Class 5**of Group S4Z2 Graph5 - 21 elements in conjugacy class including Example5_3**Cable 1:[3, 4, 2, 1, 6, 5].**- Orders of Disjoint Cycles: [4, 2].
**Cable 2:[4, 1, 3, 2, 6, 5].**- Orders of Disjoint Cycles: [3, 2].

**Conjugacy Class 6**of Group S4Z2 Graph6 - 21 elements in conjugacy class including Example6_1**Cable 1:[2, 3, 1, 4, 6, 5].**- Orders of Disjoint Cycles: [3, 2].
**Cable 2:[4, 2, 3, 1, 5, 6].**- Orders of Disjoint Cycles: [2].

**Conjugacy Class 7**of Group S4Z2 Graph7 - 21 elements in conjugacy class including Example7_1**Cable 1:[4, 3, 1, 2, 5, 6].**- Orders of Disjoint Cycles: [4].
**Cable 2:[2, 3, 1, 4, 6, 5].**- Orders of Disjoint Cycles: [3, 2].

**Generators:***g1=g1*,*g2a=g2a*,*g2b=g2b*,*g3=g3***Representations:**dim3a, dim3b, dim4, dim5.**Conjugacy Class 1**of Group A5 Graph1 - 21 elements in conjugacy class including Example1_1**Cable 1:[4, 2, 1, 3, 5].**- Orders of Disjoint Cycles: [3].
**Cable 2:[1, 3, 2, 5, 4].**- Orders of Disjoint Cycles: [2, 2].

**Conjugacy Class 2**of Group A5 Graph2 - 21 elements in conjugacy class including Example2_1**Cable 1:[2, 3, 5, 1, 4].**- Orders of Disjoint Cycles: [5].
**Cable 2:[5, 4, 3, 2, 1].**- Orders of Disjoint Cycles: [2, 2].

**Conjugacy Class 3**of Group A5 Graph3 - 21 elements in conjugacy class including Example3_1**Cable 1:[4, 2, 1, 3, 5].**- Orders of Disjoint Cycles: [3].
**Cable 2:[2, 3, 5, 1, 4].**- Orders of Disjoint Cycles: [5].

**Conjugacy Class 4**of Group A5 Graph4 - 21 elements in conjugacy class including Example4_1**Cable 1:[5, 2, 3, 1, 4].**- Orders of Disjoint Cycles: [3].
**Cable 2:[2, 3, 5, 1, 4].**- Orders of Disjoint Cycles: [5].

**Conjugacy Class 5**of Group A5 Graph5 - 14 elements in conjugacy class including Example5_1**Cable 1:[4, 3, 5, 2, 1].**- Orders of Disjoint Cycles: [5].
**Cable 2:[2, 3, 5, 1, 4].**- Orders of Disjoint Cycles: [5].

**Conjugacy Class 6**of Group A5 Graph6 - 21 elements in conjugacy class including Example6_1**Cable 1:[5, 4, 1, 3, 2].**- Orders of Disjoint Cycles: [5].
**Cable 2:[5, 2, 4, 3, 1].**- Orders of Disjoint Cycles: [2, 2].

**Conjugacy Class 7**of Group A5 Graph7 - 9 elements in conjugacy class including Example7_1**Cable 1:[4, 1, 3, 2, 5].**- Orders of Disjoint Cycles: [3].
**Cable 2:[1, 2, 5, 3, 4].**- Orders of Disjoint Cycles: [3].

**Conjugacy Class 8**of Group A5 Graph8 - 15 elements in conjugacy class including Example8_1**Cable 1:[4, 5, 1, 2, 3].**- Orders of Disjoint Cycles: [5].
**Cable 2:[4, 5, 2, 3, 1].**- Orders of Disjoint Cycles: [5].

**Generators:***g1=[2, 3, 1, 4, 5, 6, 7]*,*g2a=[2, 1, 4, 3, 5, 6, 7]*,*g2b=[3, 4, 1, 2, 5, 6, 7]*,*g3=[2, 3, 4, 5, 1, 6, 7]*,*ref=[1, 2, 3, 4, 5, 7, 6]***Representations:**dim3a, dim3b, dim4, dim5, dim3ar, dim3br, dim4r, dim5r.**Conjugacy Class 1**of Group A5Z2 Graph1 - 10 examples without vertex overlap including Example1_2**Cable 1:[1, 3, 2, 5, 4, 6, 7].**- Orders of Disjoint Cycles: [2, 2].
**Cable 2:[3, 5, 4, 2, 1, 7, 6].**- Orders of Disjoint Cycles: [5, 2].

**Conjugacy Class 2**of Group A5Z2 Graph2 - 1 example without vertex overlap including Example2_16**Cable 1:[5, 1, 3, 4, 2, 7, 6].**- Orders of Disjoint Cycles: [3, 2].
**Cable 2:[1, 3, 2, 5, 4, 6, 7].**- Orders of Disjoint Cycles: [2, 2].

**Conjugacy Class 3**of Group A5Z2 Graph3 - 10 examples without vertex overlap including Example3_1**Cable 1:[3, 5, 4, 2, 1, 7, 6].**- Orders of Disjoint Cycles: [5, 2].
**Cable 2:[1, 3, 2, 5, 4, 7, 6].**- Orders of Disjoint Cycles: [2, 2, 2].

**Conjugacy Class 4**of Group A5Z2 Graph4 - 11 examples without vertex overlap including Example4_3**Cable 1:[2, 4, 3, 1, 5, 6, 7].**- Orders of Disjoint Cycles: [3].
**Cable 2:[1, 3, 2, 5, 4, 7, 6].**- Orders of Disjoint Cycles: [2, 2, 2].

**Conjugacy Class 5**of Group A5Z2 Graph5 - 18 examples without vertex overlap including Example5_3**Cable 1:[2, 4, 5, 3, 1, 6, 7].**- Orders of Disjoint Cycles: [5].
**Cable 2:[1, 3, 2, 5, 4, 7, 6].**- Orders of Disjoint Cycles: [2, 2, 2].

**Conjugacy Class 6**of Group A5Z2 Graph6 - 4 examples without vertex overlap including Example6_1**Cable 1:[3, 4, 2, 5, 1, 7, 6].**- Orders of Disjoint Cycles: [5, 2].
**Cable 2:[1, 3, 2, 5, 4, 7, 6].**- Orders of Disjoint Cycles: [2, 2, 2].

**Conjugacy Class 7**of Group A5Z2 Graph7 - 1 example without vertex overlap including Example7_10**Cable 1:[4, 3, 1, 5, 2, 7, 6].**- Orders of Disjoint Cycles: [5, 2].
**Cable 2:[1, 3, 2, 5, 4, 6, 7].**- Orders of Disjoint Cycles: [2, 2].

**Conjugacy Class 8**of Group A5Z2 Graph8 - 2 examples without vertex overlap including Example8_25**Cable 1:[3, 5, 4, 2, 1, 6, 7].**- Orders of Disjoint Cycles: [5].
**Cable 2:[2, 4, 3, 1, 5, 7, 6].**- Orders of Disjoint Cycles: [3, 2].

**Conjugacy Class 9**of Group A5Z2 Graph9 - 5 examples without vertex overlap including Example9_7**Cable 1:[3, 4, 5, 1, 2, 6, 7].**- Orders of Disjoint Cycles: [5].
**Cable 2:[4, 5, 3, 1, 2, 7, 6].**- Orders of Disjoint Cycles: [2, 2, 2].

**Conjugacy Class 10**of Group A5Z2 Graph10 - 0 examples without vertex overlap.**Cable 1:[4, 2, 3, 5, 1, 7, 6].**- Orders of Disjoint Cycles: [3, 2].
**Cable 2:[3, 1, 2, 4, 5, 7, 6].**- Orders of Disjoint Cycles: [3, 2].

**Conjugacy Class 11**of Group A5Z2 Graph11 - 5 examples without vertex overlap including Example11_2**Cable 1:[2, 3, 5, 1, 4, 6, 7].**- Orders of Disjoint Cycles: [5].
**Cable 2:[3, 1, 2, 4, 5, 7, 6].**- Orders of Disjoint Cycles: [3, 2].

**Conjugacy Class 12**of Group A5Z2 Graph12 - 1 example without vertex overlap including Example12_30**Cable 1:[1, 5, 4, 3, 2, 7, 6].**- Orders of Disjoint Cycles: [2, 2, 2].
**Cable 2:[3, 1, 2, 4, 5, 7, 6].**- Orders of Disjoint Cycles: [3, 2].

**Conjugacy Class 13**of Group A5Z2 Graph13 - 2 examples without vertex overlap including Example13_8**Cable 1:[3, 1, 2, 4, 5, 7, 6].**- Orders of Disjoint Cycles: [3, 2].
**Cable 2:[1, 2, 4, 5, 3, 6, 7].**- Orders of Disjoint Cycles: [3].

**Conjugacy Class 14**of Group A5Z2 Graph14 - 2 examples without vertex overlap including Example14_7**Cable 1:[3, 1, 2, 4, 5, 7, 6].**- Orders of Disjoint Cycles: [3, 2].
**Cable 2:[3, 1, 4, 5, 2, 7, 6].**- Orders of Disjoint Cycles: [5, 2].

**Conjugacy Class 15**of Group A5Z2 Graph15 - 10 examples without vertex overlap including Example15_3**Cable 1:[2, 4, 5, 3, 1, 7, 6].**- Orders of Disjoint Cycles: [5, 2].
**Cable 2:[3, 2, 5, 4, 1, 6, 7].**- Orders of Disjoint Cycles: [3].

**Conjugacy Class 16**of Group A5Z2 Graph16 - 1 example without vertex overlap including Example16_8**Cable 1:[2, 4, 5, 3, 1, 7, 6].**- Orders of Disjoint Cycles: [5, 2].
**Cable 2:[4, 5, 1, 2, 3, 7, 6].**- Orders of Disjoint Cycles: [5, 2].

**Conjugacy Class 17**of Group A5Z2 Graph17 - 2 examples without vertex overlap including Example17_19**Cable 1:[5, 1, 3, 4, 2, 7, 6].**- Orders of Disjoint Cycles: [3, 2].
**Cable 2:[4, 5, 1, 2, 3, 7, 6].**- Orders of Disjoint Cycles: [5, 2].

**Conjugacy Class 18**of Group A5Z2 Graph18 - 16 examples without vertex overlap including Example18_2**Cable 1:[3, 5, 2, 1, 4, 6, 7].**- Orders of Disjoint Cycles: [5].
**Cable 2:[4, 5, 1, 2, 3, 7, 6].**- Orders of Disjoint Cycles: [5, 2].

**Conjugacy Class 19**of Group A5Z2 Graph19 - 8 examples without vertex overlap including Example19_1**Cable 1:[3, 5, 4, 2, 1, 7, 6].**- Orders of Disjoint Cycles: [5, 2].
**Cable 2:[4, 5, 1, 2, 3, 7, 6].**- Orders of Disjoint Cycles: [5, 2].

**Conjugacy Class 20**of Group A5Z2 Graph20 - 10 examples without vertex overlap including Example20_2**Cable 1:[2, 4, 5, 3, 1, 6, 7].**- Orders of Disjoint Cycles: [5].
**Cable 2:[4, 5, 1, 2, 3, 7, 6].**- Orders of Disjoint Cycles: [5, 2].

**Conjugacy Class 21**of Group A5Z2 Graph21 - 7 examples without vertex overlap including Example21_2**Cable 1:[1, 5, 2, 4, 3, 6, 7].**- Orders of Disjoint Cycles: [3].
**Cable 2:[2, 4, 5, 3, 1, 7, 6].**- Orders of Disjoint Cycles: [5, 2].