metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C68.4C4, C4.Dic17, C4.15D34, C17⋊4M4(2), C22.Dic17, C68.15C22, C17⋊3C8⋊5C2, (C2×C34).5C4, (C2×C68).5C2, (C2×C4).2D17, C34.14(C2×C4), C2.3(C2×Dic17), SmallGroup(272,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C68.4C4
G = < a,b | a68=1, b4=a34, bab-1=a-1 >
Smallest permutation representation of C68.4C4
►On 136 points
Generators in S136
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68)(69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136)
(1 90 18 73 35 124 52 107)(2 89 19 72 36 123 53 106)(3 88 20 71 37 122 54 105)(4 87 21 70 38 121 55 104)(5 86 22 69 39 120 56 103)(6 85 23 136 40 119 57 102)(7 84 24 135 41 118 58 101)(8 83 25 134 42 117 59 100)(9 82 26 133 43 116 60 99)(10 81 27 132 44 115 61 98)(11 80 28 131 45 114 62 97)(12 79 29 130 46 113 63 96)(13 78 30 129 47 112 64 95)(14 77 31 128 48 111 65 94)(15 76 32 127 49 110 66 93)(16 75 33 126 50 109 67 92)(17 74 34 125 51 108 68 91)
G:=sub<Sym(136)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,90,18,73,35,124,52,107)(2,89,19,72,36,123,53,106)(3,88,20,71,37,122,54,105)(4,87,21,70,38,121,55,104)(5,86,22,69,39,120,56,103)(6,85,23,136,40,119,57,102)(7,84,24,135,41,118,58,101)(8,83,25,134,42,117,59,100)(9,82,26,133,43,116,60,99)(10,81,27,132,44,115,61,98)(11,80,28,131,45,114,62,97)(12,79,29,130,46,113,63,96)(13,78,30,129,47,112,64,95)(14,77,31,128,48,111,65,94)(15,76,32,127,49,110,66,93)(16,75,33,126,50,109,67,92)(17,74,34,125,51,108,68,91)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,90,18,73,35,124,52,107)(2,89,19,72,36,123,53,106)(3,88,20,71,37,122,54,105)(4,87,21,70,38,121,55,104)(5,86,22,69,39,120,56,103)(6,85,23,136,40,119,57,102)(7,84,24,135,41,118,58,101)(8,83,25,134,42,117,59,100)(9,82,26,133,43,116,60,99)(10,81,27,132,44,115,61,98)(11,80,28,131,45,114,62,97)(12,79,29,130,46,113,63,96)(13,78,30,129,47,112,64,95)(14,77,31,128,48,111,65,94)(15,76,32,127,49,110,66,93)(16,75,33,126,50,109,67,92)(17,74,34,125,51,108,68,91) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68),(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)], [(1,90,18,73,35,124,52,107),(2,89,19,72,36,123,53,106),(3,88,20,71,37,122,54,105),(4,87,21,70,38,121,55,104),(5,86,22,69,39,120,56,103),(6,85,23,136,40,119,57,102),(7,84,24,135,41,118,58,101),(8,83,25,134,42,117,59,100),(9,82,26,133,43,116,60,99),(10,81,27,132,44,115,61,98),(11,80,28,131,45,114,62,97),(12,79,29,130,46,113,63,96),(13,78,30,129,47,112,64,95),(14,77,31,128,48,111,65,94),(15,76,32,127,49,110,66,93),(16,75,33,126,50,109,67,92),(17,74,34,125,51,108,68,91)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 17A | ··· | 17H | 34A | ··· | 34X | 68A | ··· | 68AF |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 34 | 34 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | M4(2) | D17 | Dic17 | D34 | Dic17 | C68.4C4 |
| kernel | C68.4C4 | C17⋊3C8 | C2×C68 | C68 | C2×C34 | C17 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 32 |
Matrix representation of C68.4C4 ►in GL2(𝔽137) generated by
| 44 | 0 |
| 0 | 109 |
| 0 | 1 |
| 37 | 0 |
G:=sub<GL(2,GF(137))| [44,0,0,109],[0,37,1,0] >;
C68.4C4 in GAP, Magma, Sage, TeX
C_{68}._4C_4 % in TeX
G:=Group("C68.4C4"); // GroupNames label
G:=SmallGroup(272,10);
// by ID
G=gap.SmallGroup(272,10);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,101,42,6404]);
// Polycyclic
G:=Group<a,b|a^68=1,b^4=a^34,b*a*b^-1=a^-1>;
// generators/relations
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