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G = Dic39order 156 = 22·3·13

Dicyclic group

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C39 — Dic39
 Chief series C1 — C13 — C39 — C78 — Dic39
 Lower central C39 — Dic39
 Upper central C1 — C2

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

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

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

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

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

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 4A 4B 6 13A ··· 13F 26A ··· 26F 39A ··· 39L 78A ··· 78L order 1 2 3 4 4 6 13 ··· 13 26 ··· 26 39 ··· 39 78 ··· 78 size 1 1 2 39 39 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + - + - + - image C1 C2 C4 S3 Dic3 D13 Dic13 D39 Dic39 kernel Dic39 C78 C39 C26 C13 C6 C3 C2 C1 # reps 1 1 2 1 1 6 6 12 12

Matrix representation of Dic39 in GL2(𝔽157) generated by

 47 126 138 66
,
 25 23 82 132
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

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