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G = Dic43order 172 = 22·43

Dicyclic group

Aliases: Dic43, C43⋊C4, C86.C2, C2.D43, SmallGroup(172,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C43 — Dic43
 Chief series C1 — C43 — C86 — Dic43
 Lower central C43 — Dic43
 Upper central C1 — C2

Generators and relations for Dic43
G = < a,b | a86=1, b2=a43, bab-1=a-1 >

Smallest permutation representation of Dic43
Regular action on 172 points
Generators in S172
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172)
(1 142 44 99)(2 141 45 98)(3 140 46 97)(4 139 47 96)(5 138 48 95)(6 137 49 94)(7 136 50 93)(8 135 51 92)(9 134 52 91)(10 133 53 90)(11 132 54 89)(12 131 55 88)(13 130 56 87)(14 129 57 172)(15 128 58 171)(16 127 59 170)(17 126 60 169)(18 125 61 168)(19 124 62 167)(20 123 63 166)(21 122 64 165)(22 121 65 164)(23 120 66 163)(24 119 67 162)(25 118 68 161)(26 117 69 160)(27 116 70 159)(28 115 71 158)(29 114 72 157)(30 113 73 156)(31 112 74 155)(32 111 75 154)(33 110 76 153)(34 109 77 152)(35 108 78 151)(36 107 79 150)(37 106 80 149)(38 105 81 148)(39 104 82 147)(40 103 83 146)(41 102 84 145)(42 101 85 144)(43 100 86 143)

G:=sub<Sym(172)| (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)(87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172), (1,142,44,99)(2,141,45,98)(3,140,46,97)(4,139,47,96)(5,138,48,95)(6,137,49,94)(7,136,50,93)(8,135,51,92)(9,134,52,91)(10,133,53,90)(11,132,54,89)(12,131,55,88)(13,130,56,87)(14,129,57,172)(15,128,58,171)(16,127,59,170)(17,126,60,169)(18,125,61,168)(19,124,62,167)(20,123,63,166)(21,122,64,165)(22,121,65,164)(23,120,66,163)(24,119,67,162)(25,118,68,161)(26,117,69,160)(27,116,70,159)(28,115,71,158)(29,114,72,157)(30,113,73,156)(31,112,74,155)(32,111,75,154)(33,110,76,153)(34,109,77,152)(35,108,78,151)(36,107,79,150)(37,106,80,149)(38,105,81,148)(39,104,82,147)(40,103,83,146)(41,102,84,145)(42,101,85,144)(43,100,86,143)>;

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)(87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172), (1,142,44,99)(2,141,45,98)(3,140,46,97)(4,139,47,96)(5,138,48,95)(6,137,49,94)(7,136,50,93)(8,135,51,92)(9,134,52,91)(10,133,53,90)(11,132,54,89)(12,131,55,88)(13,130,56,87)(14,129,57,172)(15,128,58,171)(16,127,59,170)(17,126,60,169)(18,125,61,168)(19,124,62,167)(20,123,63,166)(21,122,64,165)(22,121,65,164)(23,120,66,163)(24,119,67,162)(25,118,68,161)(26,117,69,160)(27,116,70,159)(28,115,71,158)(29,114,72,157)(30,113,73,156)(31,112,74,155)(32,111,75,154)(33,110,76,153)(34,109,77,152)(35,108,78,151)(36,107,79,150)(37,106,80,149)(38,105,81,148)(39,104,82,147)(40,103,83,146)(41,102,84,145)(42,101,85,144)(43,100,86,143) );

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),(87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172)], [(1,142,44,99),(2,141,45,98),(3,140,46,97),(4,139,47,96),(5,138,48,95),(6,137,49,94),(7,136,50,93),(8,135,51,92),(9,134,52,91),(10,133,53,90),(11,132,54,89),(12,131,55,88),(13,130,56,87),(14,129,57,172),(15,128,58,171),(16,127,59,170),(17,126,60,169),(18,125,61,168),(19,124,62,167),(20,123,63,166),(21,122,64,165),(22,121,65,164),(23,120,66,163),(24,119,67,162),(25,118,68,161),(26,117,69,160),(27,116,70,159),(28,115,71,158),(29,114,72,157),(30,113,73,156),(31,112,74,155),(32,111,75,154),(33,110,76,153),(34,109,77,152),(35,108,78,151),(36,107,79,150),(37,106,80,149),(38,105,81,148),(39,104,82,147),(40,103,83,146),(41,102,84,145),(42,101,85,144),(43,100,86,143)]])

Dic43 is a maximal subgroup of   Dic86  C4×D43  C43⋊D4
Dic43 is a maximal quotient of   C43⋊C8

46 conjugacy classes

 class 1 2 4A 4B 43A ··· 43U 86A ··· 86U order 1 2 4 4 43 ··· 43 86 ··· 86 size 1 1 43 43 2 ··· 2 2 ··· 2

46 irreducible representations

 dim 1 1 1 2 2 type + + + - image C1 C2 C4 D43 Dic43 kernel Dic43 C86 C43 C2 C1 # reps 1 1 2 21 21

Matrix representation of Dic43 in GL3(𝔽173) generated by

 172 0 0 0 128 172 0 1 0
,
 93 0 0 0 39 8 0 156 134
G:=sub<GL(3,GF(173))| [172,0,0,0,128,1,0,172,0],[93,0,0,0,39,156,0,8,134] >;

Dic43 in GAP, Magma, Sage, TeX

{\rm Dic}_{43}
% in TeX

G:=Group("Dic43");
// GroupNames label

G:=SmallGroup(172,1);
// by ID

G=gap.SmallGroup(172,1);
# by ID

G:=PCGroup([3,-2,-2,-43,6,1514]);
// Polycyclic

G:=Group<a,b|a^86=1,b^2=a^43,b*a*b^-1=a^-1>;
// generators/relations

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