- 2/12. d ls in A8 e ls in A8 f ls in A8. h is in A8 Coset Leaders: 1 I
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1 o Order Index135 No. of roots A0 ab a fill--ab a 01AA ab0000 c fill--00ab0000 c 0A10A ab000 d fill--000ab000 d 0A0100A0 cccddddd h fill--cccddddd h 00A0100A 0000a0b0 e*f fill--0000a0b0 e*f A a0b f*g fill a0b f*g A00A0A10 0a000b00 b*c*o*e fill--0a000b00 b*c*o*e 0000A001 ddcccddd b*a*c*o*h*c*b fill--ddcccddd b*a*c*o*h*c*b Coset leaders of A4 X A3 in A8 normalized initial vector is x0. 2*x0/d 1*v0: iv*r/lrl d/ a b c d e f g Roots of the group: A B C o E F G /12
2 H Roots of the subgroup: a b c d e f g h a ls in A8 b ls in A8 C ls in A8 d ls in A8 e ls in A8 f ls in A8 g ls in A8 h is in A8 Coset Leaders: 1 I 2 1*B 3 1*E 4 1*F 5 1*G 6 1*B*A 7 1*B*C 8 1*B*E 9 1*B*F 10 1*B*G 1*E*B *E*O 12 1*E*G 1*F*B 9 1*G*B 10 1*G*E *B*A*C 14 1*B*A*E 15 1*B*A*F 16 1*B*A*G 1*B*C*A *B*C*O 18 1*B*C*E - 2/12
3 19 I*B*C*F 20 I*B*C*G 21 I*B*C*H I*B*E*A 14 I*B*E*C I*B*E*D 23 I*B*E*G I*B*F*A 15 I*B*F*C 19 I*B*G*A 16 I*B*G*C 20 I*B*G*E 23 I*E*D*B I*E*D*C 25 I*E*D*G I*E*G*B 23 I*E*G*D I*E*G*F 27 I*B*A*C*B 28 I*B*A*C*D 29 I*B*A*C*E 30 I*B*A*C*F 31 I*B*A*C*G 32 I*B*A*C*H I*B*A*E*C I*B*A*E*D 34 I*B*A*E*G I*B*A*F*C 30 I*B*A*G*C 31 I*B*A*G*E 34 I*B*C*D*A I*B*C*D*F 36 I*B*C*D*G 37 I*B*C*D*H I*B*C*E*A I*B*C*E*G 39 I*B*C*E*H I*B*C*F*A 30 I*B*C*F*D I*B*C*F*H I*B*C*G*A 31 I*B*C*G*D 36-3/12 -
4 I*B*C*G*E :: I*B*C*G*H I*B*C*H*A :: 32 I*B*C*H*D 37 I*B*C*H*E 39 I*B*C*H*F 40 I*B*C*H*G :: 41 I*B*E*D*A I*B*E*D*G I*B*E*G*A 34 I*B*E*G*C 38 I*B*E*G*D :: I*B*E*G*F 44 I*E*D*C*G 45 I*E*D*C*H I*E*D*G*B 42 I*E*D*G*C :: I*E*D*G*F I*E*G*F*B 43 I*E*G*F*D 46 :: 47 I*B*A*C*B*D 48 I*B*A*C*B*E 49 I*B*A*C*B*F 50 I*B*A*C*B*G 51 I*B*A*C*B*H I*B*A*C*D*B 47 :: 52 I*B*A*C*D*F 53 I*B*A*C*D*G 54 I*B*A*C*D*H I*B*A*C*E*B I*B*A*C*E*G 56 I*B*A*C*E*H I*B*A*C*F*B 49 I*B*A*C*F*D I*B*A*C*F*H I*B*A*C*G*B 50 I*B*A*C*G*D 53 I*B*A*C*G*E I*B*A*C*G*H I*B*A*C*H*B 51 I*B*A*C*H*D 54 I*B*A*C*H*E 56-4/12
5 I*B*A*C*H*F 57 I*B*A*C*H*G I*B*A*E*D*G I*B*A*E*G*C 55 I*B*A*E*G*D I*B*A*E*G*F I*B*C*D*F*A I*B*C*D*F*E 62 I*B*C*D*F*H I*B*C*D*G*A I*B*C*D*G*H I*B*C*D*H*A I*B*C*D*H*C I*B*C*D*H*F 62 I*B*C*D*H*G 63 I*B*C*E*G*A I*B*C*E*G*F 66 I*B*C*E*G*H I*B*C*E*H*A 56 I*B*C*E*H*G 66 I*B*C*F*H*A 57 I*B*C*F*H*D 62 I*B*C*G*H*A 58 I*B*C*G*H*D 63 I*B*C*G*H*E 66 I*B*E*D*G*A I*B*E*D*G*F I*B*E*G*F*A 60 I*B*E*G*F*C 65 I*B*E*G*F*D I*E*D*C*G*F 69 I*E*D*C*G*H I*E*D*C*H*G 69 I*E*D*G*F*B 67 I*E*D*G*F*C I*E*D*G*F*E 71 I*B*A*C*B*D*C 72 I*B*A*C*B*D*F 73 I*B*A*C*B*D*G 74 I*B*A*C*B*D*H 75 I*B*A*C*B*E*G 76 I*B*A*C*B*E*H - 5/12
6 I*B*A*C*B*F*D I*B*A*C*B*F*H I*B*A*C*B*G*D 73 I*B*A*C*B*G*E I*B*A*C*B*G*H 79 I*B*A*C*B*H*C I*B*A*C*B*H*D 74 I*B*A*C*B*H*E 76 I*B*A*C*B*H*F 77 I*B*A*C*B*H*G 78 I*B*A*C*D*F*B I*B*A*C*D*F*E 81 I*B*A*C*D*F*H I*B*A*C*D*G*B I*B*A*C*D*G*H I*B*A*C*D*H*B I*B*A*C*D*H*C I*B*A*C*D*H*F 81 I*B*A*C*D*H*G 82 I*B*A*C*E*G*B I*B*A*C*E*G*F 85 I*B*A*C*E*G*H I*B*A*C*E*H*B 76 I*B*A*C*E*H*G 85 I*B*A*C*F*H*B 77 I*B*A*C*F*H*D 81 I*B*A*C*G*H*B 78 I*B*A*C*G*H*D 82 I*B*A*C*G*H*E I*B*A*E*D*G*F I*B*A*E*G*F*C 84 I*B*A*E*G*F*D 86 I*B*C*D*F*E*A I*B*C*D*F*E*H I*B*C*D*F*H*A I*B*C*D*F*H*C I*B*C*D*F*H*E 87 I*B*C*D*G*H*A I*B*C*D*G*H*C I*B*C*D*H*C*A I*B*C*D*H*C*B I*B*C*D*H*C*F 88-6/12
7 I*B*C*D*H*C*G 89 I*B*C*E*G*F*A I*B*C*E*G*F*H I*B*C*E*G*H*A 85 I*B*C*E*G*H*F 91 I*B*E*D*G*F*A I*B*E*D*G*F*E 93 I*E*D*C*G*F*E 94 I*E*D*C*G*F*H I*E*D*C*G*H*F 94 I*E*D*G*F*E*B 92 I*E*D*G*F*E*C I*B*A*C*B*D*C*F 96 I*B*A*C*B*D*C*G I*B*A*C*B*D*F*C I*B*A*C*B*D*F*E 98 I*B*A*C*B*D*F*H I*B*A*C*B*D*G*C I*B*A*C*B*D*G*H I*B*A*C*B*D*H*F 98 I*B*A*C*B*D*H*G I*B*A*C*B*E*G*F 101 I*B*A*C*B*E*G*H 102 I*B*A*C*B*E*H*C I*B*A*C*B*E*H*G I*B*A*C*B*F*H*C I*B*A*C*B*F*H*D I*B*A*C*B*G*H*C I*B*A*C*B*G*H*D 99 I*B*A*C*B*G*H*E 101 I*B*A*C*B*H*C*E 102 I*B*A*C*B*H*C*F 103 I*B*A*C*B*H*C*G 104 I*B*A*C*D*F*E*B I*B*A*C*D*F*E*H I*B*A*C*D*F*H*B I*B*A*C*D*F*H*C I*B*A*C*D*F*H*E 105 I*B*A*C*D*G*H*B I*B*A*C*D*G*H*C I*B*A*C*D*H*C*F 106 I*B*A*C*D*H*C*G 107-7/12
8 I*B*A*C*E*G*F*B 108 I*B*A*C*E*G*F*H I*B*A*C*E*G*H*B I*B*A*C*E*G*H*F 109 I*B*A*E*D*G*F*E I*B*C*D*F*E*H*A 110 I*B*C*D*F*E*H*C I*B*C*D*F*H*C*A 111 I*B*C*D*F*H*C*B I*B*C*D*F*H*C*E I*B*C*D*G*H*C*A 112 I*B*C*D*G*H*C*B I*B*C*D*H*C*B*F I*B*C*D*H*C*B*G I*B*C*E*G*F*H*A I*B*E*D*G*F*E*A 113 I*E*D*C*G*F*E*D 114 I*E*D*C*G*F*E*H I*E*D*C*G*F*H*E 115 I*B*A*C*B*D*C*F*E I*B*A*C*B*D*F*E*C 116 I*B*A*C*B*D*F*E*H I*B*A*C*B*D*F*H*E 117 I*B*A*C*B*E*G*F*H 118 I*B*A*C*B*E*G*H*C I*B*A*C*B*E*G*H*F 119 I*B*A*C*B*E*H*C*D I*B*A*C*B*E*H*C*G I*B*A*C*B*G*H*C*E I*B*A*C*D*F*E*H*B 120 I*B*A*C*D*F*E*H*C I*B*A*C*D*F*H*C*E I*B*A*C*E*G*F*H*B I*B*C*D*F*E*H*C*A 121 I*B*C*D*F*E*H*C*B 122 I*B*C*D*F*E*H*C*D I*B*C*D*F*H*C*B*E 123 I*E*D*C*G*F*E*D*H I*E*D*C*G*F*E*H*D 124 I*B*A*C*B*D*C*F*E*D 125 I*B*A*C*B*E*G*F*H*C 126 I*B*A*C*B*E*G*H*C*D /12
9 I*B*A*C*B*E*G*H*C*F 125 I*B*A*C*B*E*H*C*D*G I*B*A*C*D*F*E*H*C*D 128 I*B*C*D*F*E*H*C*B*D I*B*C*D*F*E*H*C*D*A 127 I*B*C*D*F*E*H*C*D*B I*E*D*C*G*F*E*D*H*C 130 I*B*A*C*B*E*G*F*H*C*D I*B*A*C*B*E*G*H*C*D*F I*B*C*D*F*E*H*C*B*D*C 132 I*E*D*C*G*F*E*D*H*C*B 133 I*B*A*C*B*E*G*F*H*C*D*E 134 I*B*C*D*F*E*H*C*B*D*C*H 135 I*E*D*C*G*F*E*D*H*C*B*A Finished 35--there are 135 cosets counting the subgroup--max length 11. I - L) I*B - 6 I*B*A ' f, I*B*A*C - '1 I*B*A*C*B I*B*A*C*B*D -", I*B*A*C*B*D*C -''is I*B*A*C*B*D*C*F - 'is I*B*A*C*B*D*C*F*E I*B*A*C*B*D*C*F*E*D -9 I*B*A*C*B*D*C*G - g I*B*A*C*B*D*F ~ ~ I*B*A*C*B*D*F*E -~ I*B*A*C*B*D*F*E*H I*B*A*C*B*D*F*H - 9 I*B*A*C*B*D*G -q I*B*A*C*B*D*G*H -9 I*B*A*C*B*D*H ~ Cf I*B*A*C*B*E - '7 I*B*A*C*B*E*G - 21 I*B*A*C*B*E*G*F _ - c I*B*A*C*B*E*G*F*H - I.:) I*B*A*C*B*E*G*F*H*C I*B*A*C*B*E*G*F*H*C*D- i1 I*B*A*C*B*E*G*F*H*C*D*E -!z I*B*A*C*B*E*G*H f'j I*B*A*C*B*E*G*H*C -II - 9/12
10 I*B*A*C*B*E*G*H*C*D -n I*B*A*C*B*E*H -'8 I*B*A*C*B*E*H*C - c I*B*A*C*B*E*H*C*D-9 I*B*A*C*B*F -9 I*B*A*C*B*F*H _Lj I*B*A*C*B*F*H*C -1 I*B*A*C*B*G _Cj I*B*A*C*B*G*H -IV I*B*A*C*B*G*H*C -II I*B*A*C*B*H - to I*B*A*C*B*H*C - II I*B*A*C*D - 8 I*B*A*C*D*F -10 I*B*A*C*D*F*E r 10 I*B*A*C*D*F*E*H II -_. I*B*A*C*D*F*E*H*C -\ z. I*B*A*C*D*F*E*H*C*D -13 I*B*A*C*D*F*H _c I*B*A*C*D*F*H*C ~ I*B*A*C*D*G - '?f I*B*A*C*D*G*H I*B*A*C*D*G*H*C -~ I*B*A*C*D*H -'1 I*B*A*C*D*H*C I*B*A*C*E - '7 I*B*A*C*E*G - c, I*B*A*C*E*G*F -10 I*B*A*C*E*G*F*H ~IO I*B*A*C*E*G*H - "t I*B*A*C*E*H - 7 I*B*A*C*F -Cf I*B*A*C*F*H -CJ I*B*A*C*G -<1 I*B*A*C*G*H - 9 I*B*A*C*H - Cj' I*B*A*E -6 I*B*A*E*D {p I*B*A*E*D*G -! I*B*A*E*D*G*F -"3 I*B*A*E*D*G*F*E - ~ I*B*A*E*G - '7-10/12
11 1*B*A*E*G*F-7 1*B*A*F - s 1*B*A*G - G 1*B*C -"'l> 1*B*C*O - 7 1*B*C*D*F - ry 1*B*C*D*F*E - 7 1*B*C*D*F*E*H - '6 1* B*C*O* F* E*H*C -10 1*B*C*O*F*E*H*C*B -10 1*B*C*D*F*E*H*C*B*O -II 1*B*C*D*F*E*H*C*B*O*C -12 1*B*C*D*F*E*H*C*B*O*C*H -IL. 1*B*C*O*F*E*H*C*O -10 1*B*C*O*F*H -~ 1*B*C*O*F*H*C - G 1*B*C*D*F*H*C*B-4 1*B*C*O*G -'7 1*B*C*D*G*H _c; 1*B*C*D*G*H*C - q 1*B*C*D*G*H*C*B -9 1*B*C*D*H - g 1*B*C*O*H*C 1*B*C*D*H*C*B -9 1*B*C*E - '7 1*B*C*E*G - "g 1*B*C*E*G*F --'is 1*B*C*E*G*F*H -~ 1*B*C*E*G*H - '3 1*B*C*E*H - 7 1*B*C*F - '7 1*B*C*F*H -7 1*B*C*G - 'b 1*B*C*G*H -~ 1*B*C*H - ~ 1*B*E - f., 1*B*E*D ~ & 1*B*E*D*G - 7 1*B*E*D*G*F - ~ 1*B*E*D*G*F*E - <6 1*B*E*G ~ 1*B*E*G*F /12
12 I*B*F - 5"" I*B*G I*E-Li I*E*D-1.. I*E*D*C-s I*E*D*C*G-r"J I*E*D*C*G*F-~ I*E*D*C*G*F*E-~ I*E*D*C*G*F*E*D I*E*D*C*G*F*E*D*H - :J I*E*D*C*G*F*E*D*H*C I*E*D*C*G*F*E*D*H*C*B- ~ I*E*D*C*G*F*E*D*H*C*B*A - I*E*D*C*G*F*E*H I*E*D*C*G*F*H - ~ I*E*D*C*G*H -1 I*E*D*C*H - /,p I*E*D*G - ~ I*E*D*G*F -'"7 I*E*D*G*F*E -7 I*E*G /..f I*E*G*F -lj I*F-3 I*G,'-l lot Ie, 7-12/12
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