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G = Dic34order 136 = 23·17

Dicyclic group

Aliases: Dic34, C17⋊Q8, C4.D17, C68.1C2, C2.3D34, C34.1C22, Dic17.1C2, SmallGroup(136,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C34 — Dic34
 Chief series C1 — C17 — C34 — Dic17 — Dic34
 Lower central C17 — C34 — Dic34
 Upper central C1 — C2 — C4

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  D685C2  D42D17  Q8×D17  C51⋊Q8  Dic102
Dic34 is a maximal quotient of
C34.D4  C683C4  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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