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G = C2xC86order 172 = 22·43

Abelian group of type [2,86]

direct product, abelian, monomial, 2-elementary

Aliases: C2xC86, SmallGroup(172,4)

Series: Derived Chief Lower central Upper central

C1 — C2xC86
C1C43C86 — C2xC86
C1 — C2xC86
C1 — C2xC86

Generators and relations for C2xC86
 G = < a,b | a2=b86=1, ab=ba >

Subgroups: 10, all normal (4 characteristic)
Quotients: C1, C2, C22, C43, C86, C2xC86

Smallest permutation representation of C2xC86
Regular action on 172 points
Generators in S172
(1 131)(2 132)(3 133)(4 134)(5 135)(6 136)(7 137)(8 138)(9 139)(10 140)(11 141)(12 142)(13 143)(14 144)(15 145)(16 146)(17 147)(18 148)(19 149)(20 150)(21 151)(22 152)(23 153)(24 154)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 161)(32 162)(33 163)(34 164)(35 165)(36 166)(37 167)(38 168)(39 169)(40 170)(41 171)(42 172)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)(49 93)(50 94)(51 95)(52 96)(53 97)(54 98)(55 99)(56 100)(57 101)(58 102)(59 103)(60 104)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 121)(78 122)(79 123)(80 124)(81 125)(82 126)(83 127)(84 128)(85 129)(86 130)
(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)

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

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

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

C2xC86 is a maximal subgroup of   C43:D4

172 conjugacy classes

class 1 2A2B2C43A···43AP86A···86DV
order122243···4386···86
size11111···11···1

172 irreducible representations

dim1111
type++
imageC1C2C43C86
kernelC2xC86C86C22C2
# reps1342126

Matrix representation of C2xC86 in GL2(F173) generated by

1720
01
,
560
040
G:=sub<GL(2,GF(173))| [172,0,0,1],[56,0,0,40] >;

C2xC86 in GAP, Magma, Sage, TeX

C_2\times C_{86}
% in TeX

G:=Group("C2xC86");
// GroupNames label

G:=SmallGroup(172,4);
// by ID

G=gap.SmallGroup(172,4);
# by ID

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

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

Export

Subgroup lattice of C2xC86 in TeX

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