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G = Dic39order 156 = 22·3·13

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic39, C393C4, C26.S3, C6.D13, C2.D39, C3⋊Dic13, C78.1C2, C132Dic3, SmallGroup(156,5)

Series: Derived Chief Lower central Upper central

C1C39 — Dic39
C1C13C39C78 — Dic39
C39 — Dic39
C1C2

Generators and relations for Dic39
 G = < a,b | a78=1, b2=a39, bab-1=a-1 >

39C4
13Dic3
3Dic13

Smallest permutation representation of Dic39
Regular action on 156 points
Generators in S156
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156)
(1 137 40 98)(2 136 41 97)(3 135 42 96)(4 134 43 95)(5 133 44 94)(6 132 45 93)(7 131 46 92)(8 130 47 91)(9 129 48 90)(10 128 49 89)(11 127 50 88)(12 126 51 87)(13 125 52 86)(14 124 53 85)(15 123 54 84)(16 122 55 83)(17 121 56 82)(18 120 57 81)(19 119 58 80)(20 118 59 79)(21 117 60 156)(22 116 61 155)(23 115 62 154)(24 114 63 153)(25 113 64 152)(26 112 65 151)(27 111 66 150)(28 110 67 149)(29 109 68 148)(30 108 69 147)(31 107 70 146)(32 106 71 145)(33 105 72 144)(34 104 73 143)(35 103 74 142)(36 102 75 141)(37 101 76 140)(38 100 77 139)(39 99 78 138)

G:=sub<Sym(156)| (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)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,137,40,98)(2,136,41,97)(3,135,42,96)(4,134,43,95)(5,133,44,94)(6,132,45,93)(7,131,46,92)(8,130,47,91)(9,129,48,90)(10,128,49,89)(11,127,50,88)(12,126,51,87)(13,125,52,86)(14,124,53,85)(15,123,54,84)(16,122,55,83)(17,121,56,82)(18,120,57,81)(19,119,58,80)(20,118,59,79)(21,117,60,156)(22,116,61,155)(23,115,62,154)(24,114,63,153)(25,113,64,152)(26,112,65,151)(27,111,66,150)(28,110,67,149)(29,109,68,148)(30,108,69,147)(31,107,70,146)(32,106,71,145)(33,105,72,144)(34,104,73,143)(35,103,74,142)(36,102,75,141)(37,101,76,140)(38,100,77,139)(39,99,78,138)>;

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)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,137,40,98)(2,136,41,97)(3,135,42,96)(4,134,43,95)(5,133,44,94)(6,132,45,93)(7,131,46,92)(8,130,47,91)(9,129,48,90)(10,128,49,89)(11,127,50,88)(12,126,51,87)(13,125,52,86)(14,124,53,85)(15,123,54,84)(16,122,55,83)(17,121,56,82)(18,120,57,81)(19,119,58,80)(20,118,59,79)(21,117,60,156)(22,116,61,155)(23,115,62,154)(24,114,63,153)(25,113,64,152)(26,112,65,151)(27,111,66,150)(28,110,67,149)(29,109,68,148)(30,108,69,147)(31,107,70,146)(32,106,71,145)(33,105,72,144)(34,104,73,143)(35,103,74,142)(36,102,75,141)(37,101,76,140)(38,100,77,139)(39,99,78,138) );

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),(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)], [(1,137,40,98),(2,136,41,97),(3,135,42,96),(4,134,43,95),(5,133,44,94),(6,132,45,93),(7,131,46,92),(8,130,47,91),(9,129,48,90),(10,128,49,89),(11,127,50,88),(12,126,51,87),(13,125,52,86),(14,124,53,85),(15,123,54,84),(16,122,55,83),(17,121,56,82),(18,120,57,81),(19,119,58,80),(20,118,59,79),(21,117,60,156),(22,116,61,155),(23,115,62,154),(24,114,63,153),(25,113,64,152),(26,112,65,151),(27,111,66,150),(28,110,67,149),(29,109,68,148),(30,108,69,147),(31,107,70,146),(32,106,71,145),(33,105,72,144),(34,104,73,143),(35,103,74,142),(36,102,75,141),(37,101,76,140),(38,100,77,139),(39,99,78,138)])

Dic39 is a maximal subgroup of
Dic3×D13  S3×Dic13  C39⋊D4  C39⋊Q8  Dic78  C4×D39  C397D4  Dic117  C393C12  C3⋊Dic39
Dic39 is a maximal quotient of
C393C8  Dic117  C3⋊Dic39

42 conjugacy classes

class 1  2  3 4A4B 6 13A···13F26A···26F39A···39L78A···78L
order12344613···1326···2639···3978···78
size112393922···22···22···22···2

42 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4S3Dic3D13Dic13D39Dic39
kernelDic39C78C39C26C13C6C3C2C1
# reps11211661212

Matrix representation of Dic39 in GL2(𝔽157) generated by

47126
13866
,
2523
82132
G:=sub<GL(2,GF(157))| [47,138,126,66],[25,82,23,132] >;

Dic39 in GAP, Magma, Sage, TeX

{\rm Dic}_{39}
% in TeX

G:=Group("Dic39");
// GroupNames label

G:=SmallGroup(156,5);
// by ID

G=gap.SmallGroup(156,5);
# by ID

G:=PCGroup([4,-2,-2,-3,-13,8,98,2307]);
// Polycyclic

G:=Group<a,b|a^78=1,b^2=a^39,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic39 in TeX

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