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G = C2×C102order 204 = 22·3·17

Abelian group of type [2,102]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C102, SmallGroup(204,12)

Series: Derived Chief Lower central Upper central

C1 — C2×C102
C1C17C51C102 — C2×C102
C1 — C2×C102
C1 — C2×C102

Generators and relations for C2×C102
 G = < a,b | a2=b102=1, ab=ba >


Smallest permutation representation of C2×C102
Regular action on 204 points
Generators in S204
(1 176)(2 177)(3 178)(4 179)(5 180)(6 181)(7 182)(8 183)(9 184)(10 185)(11 186)(12 187)(13 188)(14 189)(15 190)(16 191)(17 192)(18 193)(19 194)(20 195)(21 196)(22 197)(23 198)(24 199)(25 200)(26 201)(27 202)(28 203)(29 204)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 113)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 141)(69 142)(70 143)(71 144)(72 145)(73 146)(74 147)(75 148)(76 149)(77 150)(78 151)(79 152)(80 153)(81 154)(82 155)(83 156)(84 157)(85 158)(86 159)(87 160)(88 161)(89 162)(90 163)(91 164)(92 165)(93 166)(94 167)(95 168)(96 169)(97 170)(98 171)(99 172)(100 173)(101 174)(102 175)
(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)(103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204)

G:=sub<Sym(204)| (1,176)(2,177)(3,178)(4,179)(5,180)(6,181)(7,182)(8,183)(9,184)(10,185)(11,186)(12,187)(13,188)(14,189)(15,190)(16,191)(17,192)(18,193)(19,194)(20,195)(21,196)(22,197)(23,198)(24,199)(25,200)(26,201)(27,202)(28,203)(29,204)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,145)(73,146)(74,147)(75,148)(76,149)(77,150)(78,151)(79,152)(80,153)(81,154)(82,155)(83,156)(84,157)(85,158)(86,159)(87,160)(88,161)(89,162)(90,163)(91,164)(92,165)(93,166)(94,167)(95,168)(96,169)(97,170)(98,171)(99,172)(100,173)(101,174)(102,175), (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)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204)>;

G:=Group( (1,176)(2,177)(3,178)(4,179)(5,180)(6,181)(7,182)(8,183)(9,184)(10,185)(11,186)(12,187)(13,188)(14,189)(15,190)(16,191)(17,192)(18,193)(19,194)(20,195)(21,196)(22,197)(23,198)(24,199)(25,200)(26,201)(27,202)(28,203)(29,204)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,145)(73,146)(74,147)(75,148)(76,149)(77,150)(78,151)(79,152)(80,153)(81,154)(82,155)(83,156)(84,157)(85,158)(86,159)(87,160)(88,161)(89,162)(90,163)(91,164)(92,165)(93,166)(94,167)(95,168)(96,169)(97,170)(98,171)(99,172)(100,173)(101,174)(102,175), (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)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204) );

G=PermutationGroup([(1,176),(2,177),(3,178),(4,179),(5,180),(6,181),(7,182),(8,183),(9,184),(10,185),(11,186),(12,187),(13,188),(14,189),(15,190),(16,191),(17,192),(18,193),(19,194),(20,195),(21,196),(22,197),(23,198),(24,199),(25,200),(26,201),(27,202),(28,203),(29,204),(30,103),(31,104),(32,105),(33,106),(34,107),(35,108),(36,109),(37,110),(38,111),(39,112),(40,113),(41,114),(42,115),(43,116),(44,117),(45,118),(46,119),(47,120),(48,121),(49,122),(50,123),(51,124),(52,125),(53,126),(54,127),(55,128),(56,129),(57,130),(58,131),(59,132),(60,133),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,141),(69,142),(70,143),(71,144),(72,145),(73,146),(74,147),(75,148),(76,149),(77,150),(78,151),(79,152),(80,153),(81,154),(82,155),(83,156),(84,157),(85,158),(86,159),(87,160),(88,161),(89,162),(90,163),(91,164),(92,165),(93,166),(94,167),(95,168),(96,169),(97,170),(98,171),(99,172),(100,173),(101,174),(102,175)], [(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),(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204)])

C2×C102 is a maximal subgroup of   C517D4

204 conjugacy classes

class 1 2A2B2C3A3B6A···6F17A···17P34A···34AV51A···51AF102A···102CR
order1222336···617···1734···3451···51102···102
size1111111···11···11···11···11···1

204 irreducible representations

dim11111111
type++
imageC1C2C3C6C17C34C51C102
kernelC2×C102C102C2×C34C34C2×C6C6C22C2
# reps132616483296

Matrix representation of C2×C102 in GL2(𝔽103) generated by

1020
0102
,
450
076
G:=sub<GL(2,GF(103))| [102,0,0,102],[45,0,0,76] >;

C2×C102 in GAP, Magma, Sage, TeX

C_2\times C_{102}
% in TeX

G:=Group("C2xC102");
// GroupNames label

G:=SmallGroup(204,12);
// by ID

G=gap.SmallGroup(204,12);
# by ID

G:=PCGroup([4,-2,-2,-3,-17]);
// Polycyclic

G:=Group<a,b|a^2=b^102=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C102 in TeX

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