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G = Dic34order 136 = 23·17

Dicyclic group

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

C1C34 — Dic34
C1C17C34Dic17 — Dic34
C17C34 — Dic34
C1C2C4

Generators and relations for Dic34
 G = < a,b | a68=1, b2=a34, bab-1=a-1 >

Subgroups: 60 in 12 conjugacy classes, 9 normal (7 characteristic)
Quotients: C1, C2, C22, Q8, D17, D34, Dic34
17C4
17C4
17Q8

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  Q8xD17  C51:Q8  Dic102
Dic34 is a maximal quotient of
C34.D4  C68:3C4  C51:Q8  Dic102

37 conjugacy classes

class 1  2 4A4B4C17A···17H34A···34H68A···68P
order1244417···1734···3468···68
size11234342···22···22···2

37 irreducible representations

dim1112222
type+++-++-
imageC1C2C2Q8D17D34Dic34
kernelDic34Dic17C68C17C4C2C1
# reps12118816

Matrix representation of Dic34 in GL2(F137) generated by

10423
11416
,
6487
13473
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

Export

Subgroup lattice of Dic34 in TeX

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