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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 >

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)
(1 134 35 100)(2 133 36 99)(3 132 37 98)(4 131 38 97)(5 130 39 96)(6 129 40 95)(7 128 41 94)(8 127 42 93)(9 126 43 92)(10 125 44 91)(11 124 45 90)(12 123 46 89)(13 122 47 88)(14 121 48 87)(15 120 49 86)(16 119 50 85)(17 118 51 84)(18 117 52 83)(19 116 53 82)(20 115 54 81)(21 114 55 80)(22 113 56 79)(23 112 57 78)(24 111 58 77)(25 110 59 76)(26 109 60 75)(27 108 61 74)(28 107 62 73)(29 106 63 72)(30 105 64 71)(31 104 65 70)(32 103 66 69)(33 102 67 136)(34 101 68 135)

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), (1,134,35,100)(2,133,36,99)(3,132,37,98)(4,131,38,97)(5,130,39,96)(6,129,40,95)(7,128,41,94)(8,127,42,93)(9,126,43,92)(10,125,44,91)(11,124,45,90)(12,123,46,89)(13,122,47,88)(14,121,48,87)(15,120,49,86)(16,119,50,85)(17,118,51,84)(18,117,52,83)(19,116,53,82)(20,115,54,81)(21,114,55,80)(22,113,56,79)(23,112,57,78)(24,111,58,77)(25,110,59,76)(26,109,60,75)(27,108,61,74)(28,107,62,73)(29,106,63,72)(30,105,64,71)(31,104,65,70)(32,103,66,69)(33,102,67,136)(34,101,68,135)>;

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), (1,134,35,100)(2,133,36,99)(3,132,37,98)(4,131,38,97)(5,130,39,96)(6,129,40,95)(7,128,41,94)(8,127,42,93)(9,126,43,92)(10,125,44,91)(11,124,45,90)(12,123,46,89)(13,122,47,88)(14,121,48,87)(15,120,49,86)(16,119,50,85)(17,118,51,84)(18,117,52,83)(19,116,53,82)(20,115,54,81)(21,114,55,80)(22,113,56,79)(23,112,57,78)(24,111,58,77)(25,110,59,76)(26,109,60,75)(27,108,61,74)(28,107,62,73)(29,106,63,72)(30,105,64,71)(31,104,65,70)(32,103,66,69)(33,102,67,136)(34,101,68,135) );

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)], [(1,134,35,100),(2,133,36,99),(3,132,37,98),(4,131,38,97),(5,130,39,96),(6,129,40,95),(7,128,41,94),(8,127,42,93),(9,126,43,92),(10,125,44,91),(11,124,45,90),(12,123,46,89),(13,122,47,88),(14,121,48,87),(15,120,49,86),(16,119,50,85),(17,118,51,84),(18,117,52,83),(19,116,53,82),(20,115,54,81),(21,114,55,80),(22,113,56,79),(23,112,57,78),(24,111,58,77),(25,110,59,76),(26,109,60,75),(27,108,61,74),(28,107,62,73),(29,106,63,72),(30,105,64,71),(31,104,65,70),(32,103,66,69),(33,102,67,136),(34,101,68,135)])

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 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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