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G = C2×C86order 172 = 22·43

Abelian group of type [2,86]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C86, SmallGroup(172,4)

Series: Derived Chief Lower central Upper central

C1 — C2×C86
C1C43C86 — C2×C86
C1 — C2×C86
C1 — C2×C86

Generators and relations for C2×C86
 G = < a,b | a2=b86=1, ab=ba >


Smallest permutation representation of C2×C86
Regular action on 172 points
Generators in S172
(1 164)(2 165)(3 166)(4 167)(5 168)(6 169)(7 170)(8 171)(9 172)(10 87)(11 88)(12 89)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 98)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 121)(45 122)(46 123)(47 124)(48 125)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 132)(56 133)(57 134)(58 135)(59 136)(60 137)(61 138)(62 139)(63 140)(64 141)(65 142)(66 143)(67 144)(68 145)(69 146)(70 147)(71 148)(72 149)(73 150)(74 151)(75 152)(76 153)(77 154)(78 155)(79 156)(80 157)(81 158)(82 159)(83 160)(84 161)(85 162)(86 163)
(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)| (1,164)(2,165)(3,166)(4,167)(5,168)(6,169)(7,170)(8,171)(9,172)(10,87)(11,88)(12,89)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,97)(21,98)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,118)(42,119)(43,120)(44,121)(45,122)(46,123)(47,124)(48,125)(49,126)(50,127)(51,128)(52,129)(53,130)(54,131)(55,132)(56,133)(57,134)(58,135)(59,136)(60,137)(61,138)(62,139)(63,140)(64,141)(65,142)(66,143)(67,144)(68,145)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,155)(79,156)(80,157)(81,158)(82,159)(83,160)(84,161)(85,162)(86,163), (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( (1,164)(2,165)(3,166)(4,167)(5,168)(6,169)(7,170)(8,171)(9,172)(10,87)(11,88)(12,89)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,97)(21,98)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,118)(42,119)(43,120)(44,121)(45,122)(46,123)(47,124)(48,125)(49,126)(50,127)(51,128)(52,129)(53,130)(54,131)(55,132)(56,133)(57,134)(58,135)(59,136)(60,137)(61,138)(62,139)(63,140)(64,141)(65,142)(66,143)(67,144)(68,145)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,155)(79,156)(80,157)(81,158)(82,159)(83,160)(84,161)(85,162)(86,163), (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([(1,164),(2,165),(3,166),(4,167),(5,168),(6,169),(7,170),(8,171),(9,172),(10,87),(11,88),(12,89),(13,90),(14,91),(15,92),(16,93),(17,94),(18,95),(19,96),(20,97),(21,98),(22,99),(23,100),(24,101),(25,102),(26,103),(27,104),(28,105),(29,106),(30,107),(31,108),(32,109),(33,110),(34,111),(35,112),(36,113),(37,114),(38,115),(39,116),(40,117),(41,118),(42,119),(43,120),(44,121),(45,122),(46,123),(47,124),(48,125),(49,126),(50,127),(51,128),(52,129),(53,130),(54,131),(55,132),(56,133),(57,134),(58,135),(59,136),(60,137),(61,138),(62,139),(63,140),(64,141),(65,142),(66,143),(67,144),(68,145),(69,146),(70,147),(71,148),(72,149),(73,150),(74,151),(75,152),(76,153),(77,154),(78,155),(79,156),(80,157),(81,158),(82,159),(83,160),(84,161),(85,162),(86,163)], [(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)])

C2×C86 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
kernelC2×C86C86C22C2
# reps1342126

Matrix representation of C2×C86 in GL2(𝔽173) generated by

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

C2×C86 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 C2×C86 in TeX

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