metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic34, C17⋊Q8, C4.D17, C68.1C2, C2.3D34, C34.1C22, Dic17.1C2, SmallGroup(136,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic34
G = < a,b | a68=1, b2=a34, bab-1=a-1 >
Smallest permutation representation of Dic34
►Regular action 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 97 35 131)(2 96 36 130)(3 95 37 129)(4 94 38 128)(5 93 39 127)(6 92 40 126)(7 91 41 125)(8 90 42 124)(9 89 43 123)(10 88 44 122)(11 87 45 121)(12 86 46 120)(13 85 47 119)(14 84 48 118)(15 83 49 117)(16 82 50 116)(17 81 51 115)(18 80 52 114)(19 79 53 113)(20 78 54 112)(21 77 55 111)(22 76 56 110)(23 75 57 109)(24 74 58 108)(25 73 59 107)(26 72 60 106)(27 71 61 105)(28 70 62 104)(29 69 63 103)(30 136 64 102)(31 135 65 101)(32 134 66 100)(33 133 67 99)(34 132 68 98)
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), (1,97,35,131)(2,96,36,130)(3,95,37,129)(4,94,38,128)(5,93,39,127)(6,92,40,126)(7,91,41,125)(8,90,42,124)(9,89,43,123)(10,88,44,122)(11,87,45,121)(12,86,46,120)(13,85,47,119)(14,84,48,118)(15,83,49,117)(16,82,50,116)(17,81,51,115)(18,80,52,114)(19,79,53,113)(20,78,54,112)(21,77,55,111)(22,76,56,110)(23,75,57,109)(24,74,58,108)(25,73,59,107)(26,72,60,106)(27,71,61,105)(28,70,62,104)(29,69,63,103)(30,136,64,102)(31,135,65,101)(32,134,66,100)(33,133,67,99)(34,132,68,98)>;
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), (1,97,35,131)(2,96,36,130)(3,95,37,129)(4,94,38,128)(5,93,39,127)(6,92,40,126)(7,91,41,125)(8,90,42,124)(9,89,43,123)(10,88,44,122)(11,87,45,121)(12,86,46,120)(13,85,47,119)(14,84,48,118)(15,83,49,117)(16,82,50,116)(17,81,51,115)(18,80,52,114)(19,79,53,113)(20,78,54,112)(21,77,55,111)(22,76,56,110)(23,75,57,109)(24,74,58,108)(25,73,59,107)(26,72,60,106)(27,71,61,105)(28,70,62,104)(29,69,63,103)(30,136,64,102)(31,135,65,101)(32,134,66,100)(33,133,67,99)(34,132,68,98) );
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)], [(1,97,35,131),(2,96,36,130),(3,95,37,129),(4,94,38,128),(5,93,39,127),(6,92,40,126),(7,91,41,125),(8,90,42,124),(9,89,43,123),(10,88,44,122),(11,87,45,121),(12,86,46,120),(13,85,47,119),(14,84,48,118),(15,83,49,117),(16,82,50,116),(17,81,51,115),(18,80,52,114),(19,79,53,113),(20,78,54,112),(21,77,55,111),(22,76,56,110),(23,75,57,109),(24,74,58,108),(25,73,59,107),(26,72,60,106),(27,71,61,105),(28,70,62,104),(29,69,63,103),(30,136,64,102),(31,135,65,101),(32,134,66,100),(33,133,67,99),(34,132,68,98)]])
Dic34 is a maximal subgroup of
C136⋊C2 Dic68 D4.D17 C17⋊Q16 D68⋊5C2 D4⋊2D17 Q8×D17 C51⋊Q8 Dic102
Dic34 is a maximal quotient of
C34.D4 C68⋊3C4 C51⋊Q8 Dic102
37 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68P |
| order | 1 | 2 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 2 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
37 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D17 | D34 | Dic34 |
| kernel | Dic34 | Dic17 | C68 | C17 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 8 | 8 | 16 |
Matrix representation of Dic34 ►in GL2(𝔽137) generated by
| 104 | 23 |
| 114 | 16 |
| 64 | 87 |
| 134 | 73 |
G:=sub<GL(2,GF(137))| [104,114,23,16],[64,134,87,73] >;
Dic34 in GAP, Magma, Sage, TeX
{\rm Dic}_{34} % in TeX
G:=Group("Dic34"); // GroupNames label
G:=SmallGroup(136,4);
// by ID
G=gap.SmallGroup(136,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-17,16,49,21,2051]);
// Polycyclic
G:=Group<a,b|a^68=1,b^2=a^34,b*a*b^-1=a^-1>;
// generators/relations
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