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G = Dic43order 172 = 22·43

Dicyclic group

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

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

Series: Derived Chief Lower central Upper central

C1C43 — Dic43
C1C43C86 — Dic43
C43 — Dic43
C1C2

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

43C4

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

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

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

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

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 4A4B43A···43U86A···86U
order124443···4386···86
size1143432···22···2

46 irreducible representations

dim11122
type+++-
imageC1C2C4D43Dic43
kernelDic43C86C43C2C1
# reps1122121

Matrix representation of Dic43 in GL3(𝔽173) generated by

17200
0128172
010
,
9300
0398
0156134
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

Export

Subgroup lattice of Dic43 in TeX

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