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G = C2×C130order 260 = 22·5·13

Abelian group of type [2,130]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C130, SmallGroup(260,15)

Series: Derived Chief Lower central Upper central

C1 — C2×C130
C1C13C65C130 — C2×C130
C1 — C2×C130
C1 — C2×C130

Generators and relations for C2×C130
 G = < a,b | a2=b130=1, ab=ba >


Smallest permutation representation of C2×C130
Regular action on 260 points
Generators in S260
(1 252)(2 253)(3 254)(4 255)(5 256)(6 257)(7 258)(8 259)(9 260)(10 131)(11 132)(12 133)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 141)(21 142)(22 143)(23 144)(24 145)(25 146)(26 147)(27 148)(28 149)(29 150)(30 151)(31 152)(32 153)(33 154)(34 155)(35 156)(36 157)(37 158)(38 159)(39 160)(40 161)(41 162)(42 163)(43 164)(44 165)(45 166)(46 167)(47 168)(48 169)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 177)(57 178)(58 179)(59 180)(60 181)(61 182)(62 183)(63 184)(64 185)(65 186)(66 187)(67 188)(68 189)(69 190)(70 191)(71 192)(72 193)(73 194)(74 195)(75 196)(76 197)(77 198)(78 199)(79 200)(80 201)(81 202)(82 203)(83 204)(84 205)(85 206)(86 207)(87 208)(88 209)(89 210)(90 211)(91 212)(92 213)(93 214)(94 215)(95 216)(96 217)(97 218)(98 219)(99 220)(100 221)(101 222)(102 223)(103 224)(104 225)(105 226)(106 227)(107 228)(108 229)(109 230)(110 231)(111 232)(112 233)(113 234)(114 235)(115 236)(116 237)(117 238)(118 239)(119 240)(120 241)(121 242)(122 243)(123 244)(124 245)(125 246)(126 247)(127 248)(128 249)(129 250)(130 251)
(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)(131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260)

G:=sub<Sym(260)| (1,252)(2,253)(3,254)(4,255)(5,256)(6,257)(7,258)(8,259)(9,260)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,141)(21,142)(22,143)(23,144)(24,145)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,157)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,178)(58,179)(59,180)(60,181)(61,182)(62,183)(63,184)(64,185)(65,186)(66,187)(67,188)(68,189)(69,190)(70,191)(71,192)(72,193)(73,194)(74,195)(75,196)(76,197)(77,198)(78,199)(79,200)(80,201)(81,202)(82,203)(83,204)(84,205)(85,206)(86,207)(87,208)(88,209)(89,210)(90,211)(91,212)(92,213)(93,214)(94,215)(95,216)(96,217)(97,218)(98,219)(99,220)(100,221)(101,222)(102,223)(103,224)(104,225)(105,226)(106,227)(107,228)(108,229)(109,230)(110,231)(111,232)(112,233)(113,234)(114,235)(115,236)(116,237)(117,238)(118,239)(119,240)(120,241)(121,242)(122,243)(123,244)(124,245)(125,246)(126,247)(127,248)(128,249)(129,250)(130,251), (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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)>;

G:=Group( (1,252)(2,253)(3,254)(4,255)(5,256)(6,257)(7,258)(8,259)(9,260)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,141)(21,142)(22,143)(23,144)(24,145)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,157)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,178)(58,179)(59,180)(60,181)(61,182)(62,183)(63,184)(64,185)(65,186)(66,187)(67,188)(68,189)(69,190)(70,191)(71,192)(72,193)(73,194)(74,195)(75,196)(76,197)(77,198)(78,199)(79,200)(80,201)(81,202)(82,203)(83,204)(84,205)(85,206)(86,207)(87,208)(88,209)(89,210)(90,211)(91,212)(92,213)(93,214)(94,215)(95,216)(96,217)(97,218)(98,219)(99,220)(100,221)(101,222)(102,223)(103,224)(104,225)(105,226)(106,227)(107,228)(108,229)(109,230)(110,231)(111,232)(112,233)(113,234)(114,235)(115,236)(116,237)(117,238)(118,239)(119,240)(120,241)(121,242)(122,243)(123,244)(124,245)(125,246)(126,247)(127,248)(128,249)(129,250)(130,251), (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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260) );

G=PermutationGroup([(1,252),(2,253),(3,254),(4,255),(5,256),(6,257),(7,258),(8,259),(9,260),(10,131),(11,132),(12,133),(13,134),(14,135),(15,136),(16,137),(17,138),(18,139),(19,140),(20,141),(21,142),(22,143),(23,144),(24,145),(25,146),(26,147),(27,148),(28,149),(29,150),(30,151),(31,152),(32,153),(33,154),(34,155),(35,156),(36,157),(37,158),(38,159),(39,160),(40,161),(41,162),(42,163),(43,164),(44,165),(45,166),(46,167),(47,168),(48,169),(49,170),(50,171),(51,172),(52,173),(53,174),(54,175),(55,176),(56,177),(57,178),(58,179),(59,180),(60,181),(61,182),(62,183),(63,184),(64,185),(65,186),(66,187),(67,188),(68,189),(69,190),(70,191),(71,192),(72,193),(73,194),(74,195),(75,196),(76,197),(77,198),(78,199),(79,200),(80,201),(81,202),(82,203),(83,204),(84,205),(85,206),(86,207),(87,208),(88,209),(89,210),(90,211),(91,212),(92,213),(93,214),(94,215),(95,216),(96,217),(97,218),(98,219),(99,220),(100,221),(101,222),(102,223),(103,224),(104,225),(105,226),(106,227),(107,228),(108,229),(109,230),(110,231),(111,232),(112,233),(113,234),(114,235),(115,236),(116,237),(117,238),(118,239),(119,240),(120,241),(121,242),(122,243),(123,244),(124,245),(125,246),(126,247),(127,248),(128,249),(129,250),(130,251)], [(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),(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)])

260 conjugacy classes

class 1 2A2B2C5A5B5C5D10A···10L13A···13L26A···26AJ65A···65AV130A···130EN
order1222555510···1013···1326···2665···65130···130
size111111111···11···11···11···11···1

260 irreducible representations

dim11111111
type++
imageC1C2C5C10C13C26C65C130
kernelC2×C130C130C2×C26C26C2×C10C10C22C2
# reps13412123648144

Matrix representation of C2×C130 in GL2(𝔽131) generated by

1300
01
,
260
051
G:=sub<GL(2,GF(131))| [130,0,0,1],[26,0,0,51] >;

C2×C130 in GAP, Magma, Sage, TeX

C_2\times C_{130}
% in TeX

G:=Group("C2xC130");
// GroupNames label

G:=SmallGroup(260,15);
// by ID

G=gap.SmallGroup(260,15);
# by ID

G:=PCGroup([4,-2,-2,-5,-13]);
// Polycyclic

G:=Group<a,b|a^2=b^130=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C130 in TeX

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