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G = C2×C170order 340 = 22·5·17

Abelian group of type [2,170]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C170, SmallGroup(340,15)

Series: Derived Chief Lower central Upper central

C1 — C2×C170
C1C17C85C170 — C2×C170
C1 — C2×C170
C1 — C2×C170

Generators and relations for C2×C170
 G = < a,b | a2=b170=1, ab=ba >


Smallest permutation representation of C2×C170
Regular action on 340 points
Generators in S340
(1 206)(2 207)(3 208)(4 209)(5 210)(6 211)(7 212)(8 213)(9 214)(10 215)(11 216)(12 217)(13 218)(14 219)(15 220)(16 221)(17 222)(18 223)(19 224)(20 225)(21 226)(22 227)(23 228)(24 229)(25 230)(26 231)(27 232)(28 233)(29 234)(30 235)(31 236)(32 237)(33 238)(34 239)(35 240)(36 241)(37 242)(38 243)(39 244)(40 245)(41 246)(42 247)(43 248)(44 249)(45 250)(46 251)(47 252)(48 253)(49 254)(50 255)(51 256)(52 257)(53 258)(54 259)(55 260)(56 261)(57 262)(58 263)(59 264)(60 265)(61 266)(62 267)(63 268)(64 269)(65 270)(66 271)(67 272)(68 273)(69 274)(70 275)(71 276)(72 277)(73 278)(74 279)(75 280)(76 281)(77 282)(78 283)(79 284)(80 285)(81 286)(82 287)(83 288)(84 289)(85 290)(86 291)(87 292)(88 293)(89 294)(90 295)(91 296)(92 297)(93 298)(94 299)(95 300)(96 301)(97 302)(98 303)(99 304)(100 305)(101 306)(102 307)(103 308)(104 309)(105 310)(106 311)(107 312)(108 313)(109 314)(110 315)(111 316)(112 317)(113 318)(114 319)(115 320)(116 321)(117 322)(118 323)(119 324)(120 325)(121 326)(122 327)(123 328)(124 329)(125 330)(126 331)(127 332)(128 333)(129 334)(130 335)(131 336)(132 337)(133 338)(134 339)(135 340)(136 171)(137 172)(138 173)(139 174)(140 175)(141 176)(142 177)(143 178)(144 179)(145 180)(146 181)(147 182)(148 183)(149 184)(150 185)(151 186)(152 187)(153 188)(154 189)(155 190)(156 191)(157 192)(158 193)(159 194)(160 195)(161 196)(162 197)(163 198)(164 199)(165 200)(166 201)(167 202)(168 203)(169 204)(170 205)
(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)(171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340)

G:=sub<Sym(340)| (1,206)(2,207)(3,208)(4,209)(5,210)(6,211)(7,212)(8,213)(9,214)(10,215)(11,216)(12,217)(13,218)(14,219)(15,220)(16,221)(17,222)(18,223)(19,224)(20,225)(21,226)(22,227)(23,228)(24,229)(25,230)(26,231)(27,232)(28,233)(29,234)(30,235)(31,236)(32,237)(33,238)(34,239)(35,240)(36,241)(37,242)(38,243)(39,244)(40,245)(41,246)(42,247)(43,248)(44,249)(45,250)(46,251)(47,252)(48,253)(49,254)(50,255)(51,256)(52,257)(53,258)(54,259)(55,260)(56,261)(57,262)(58,263)(59,264)(60,265)(61,266)(62,267)(63,268)(64,269)(65,270)(66,271)(67,272)(68,273)(69,274)(70,275)(71,276)(72,277)(73,278)(74,279)(75,280)(76,281)(77,282)(78,283)(79,284)(80,285)(81,286)(82,287)(83,288)(84,289)(85,290)(86,291)(87,292)(88,293)(89,294)(90,295)(91,296)(92,297)(93,298)(94,299)(95,300)(96,301)(97,302)(98,303)(99,304)(100,305)(101,306)(102,307)(103,308)(104,309)(105,310)(106,311)(107,312)(108,313)(109,314)(110,315)(111,316)(112,317)(113,318)(114,319)(115,320)(116,321)(117,322)(118,323)(119,324)(120,325)(121,326)(122,327)(123,328)(124,329)(125,330)(126,331)(127,332)(128,333)(129,334)(130,335)(131,336)(132,337)(133,338)(134,339)(135,340)(136,171)(137,172)(138,173)(139,174)(140,175)(141,176)(142,177)(143,178)(144,179)(145,180)(146,181)(147,182)(148,183)(149,184)(150,185)(151,186)(152,187)(153,188)(154,189)(155,190)(156,191)(157,192)(158,193)(159,194)(160,195)(161,196)(162,197)(163,198)(164,199)(165,200)(166,201)(167,202)(168,203)(169,204)(170,205), (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)(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340)>;

G:=Group( (1,206)(2,207)(3,208)(4,209)(5,210)(6,211)(7,212)(8,213)(9,214)(10,215)(11,216)(12,217)(13,218)(14,219)(15,220)(16,221)(17,222)(18,223)(19,224)(20,225)(21,226)(22,227)(23,228)(24,229)(25,230)(26,231)(27,232)(28,233)(29,234)(30,235)(31,236)(32,237)(33,238)(34,239)(35,240)(36,241)(37,242)(38,243)(39,244)(40,245)(41,246)(42,247)(43,248)(44,249)(45,250)(46,251)(47,252)(48,253)(49,254)(50,255)(51,256)(52,257)(53,258)(54,259)(55,260)(56,261)(57,262)(58,263)(59,264)(60,265)(61,266)(62,267)(63,268)(64,269)(65,270)(66,271)(67,272)(68,273)(69,274)(70,275)(71,276)(72,277)(73,278)(74,279)(75,280)(76,281)(77,282)(78,283)(79,284)(80,285)(81,286)(82,287)(83,288)(84,289)(85,290)(86,291)(87,292)(88,293)(89,294)(90,295)(91,296)(92,297)(93,298)(94,299)(95,300)(96,301)(97,302)(98,303)(99,304)(100,305)(101,306)(102,307)(103,308)(104,309)(105,310)(106,311)(107,312)(108,313)(109,314)(110,315)(111,316)(112,317)(113,318)(114,319)(115,320)(116,321)(117,322)(118,323)(119,324)(120,325)(121,326)(122,327)(123,328)(124,329)(125,330)(126,331)(127,332)(128,333)(129,334)(130,335)(131,336)(132,337)(133,338)(134,339)(135,340)(136,171)(137,172)(138,173)(139,174)(140,175)(141,176)(142,177)(143,178)(144,179)(145,180)(146,181)(147,182)(148,183)(149,184)(150,185)(151,186)(152,187)(153,188)(154,189)(155,190)(156,191)(157,192)(158,193)(159,194)(160,195)(161,196)(162,197)(163,198)(164,199)(165,200)(166,201)(167,202)(168,203)(169,204)(170,205), (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)(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340) );

G=PermutationGroup([(1,206),(2,207),(3,208),(4,209),(5,210),(6,211),(7,212),(8,213),(9,214),(10,215),(11,216),(12,217),(13,218),(14,219),(15,220),(16,221),(17,222),(18,223),(19,224),(20,225),(21,226),(22,227),(23,228),(24,229),(25,230),(26,231),(27,232),(28,233),(29,234),(30,235),(31,236),(32,237),(33,238),(34,239),(35,240),(36,241),(37,242),(38,243),(39,244),(40,245),(41,246),(42,247),(43,248),(44,249),(45,250),(46,251),(47,252),(48,253),(49,254),(50,255),(51,256),(52,257),(53,258),(54,259),(55,260),(56,261),(57,262),(58,263),(59,264),(60,265),(61,266),(62,267),(63,268),(64,269),(65,270),(66,271),(67,272),(68,273),(69,274),(70,275),(71,276),(72,277),(73,278),(74,279),(75,280),(76,281),(77,282),(78,283),(79,284),(80,285),(81,286),(82,287),(83,288),(84,289),(85,290),(86,291),(87,292),(88,293),(89,294),(90,295),(91,296),(92,297),(93,298),(94,299),(95,300),(96,301),(97,302),(98,303),(99,304),(100,305),(101,306),(102,307),(103,308),(104,309),(105,310),(106,311),(107,312),(108,313),(109,314),(110,315),(111,316),(112,317),(113,318),(114,319),(115,320),(116,321),(117,322),(118,323),(119,324),(120,325),(121,326),(122,327),(123,328),(124,329),(125,330),(126,331),(127,332),(128,333),(129,334),(130,335),(131,336),(132,337),(133,338),(134,339),(135,340),(136,171),(137,172),(138,173),(139,174),(140,175),(141,176),(142,177),(143,178),(144,179),(145,180),(146,181),(147,182),(148,183),(149,184),(150,185),(151,186),(152,187),(153,188),(154,189),(155,190),(156,191),(157,192),(158,193),(159,194),(160,195),(161,196),(162,197),(163,198),(164,199),(165,200),(166,201),(167,202),(168,203),(169,204),(170,205)], [(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),(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340)])

340 conjugacy classes

class 1 2A2B2C5A5B5C5D10A···10L17A···17P34A···34AV85A···85BL170A···170GJ
order1222555510···1017···1734···3485···85170···170
size111111111···11···11···11···11···1

340 irreducible representations

dim11111111
type++
imageC1C2C5C10C17C34C85C170
kernelC2×C170C170C2×C34C34C2×C10C10C22C2
# reps13412164864192

Matrix representation of C2×C170 in GL2(𝔽1021) generated by

10200
01
,
8530
0234
G:=sub<GL(2,GF(1021))| [1020,0,0,1],[853,0,0,234] >;

C2×C170 in GAP, Magma, Sage, TeX

C_2\times C_{170}
% in TeX

G:=Group("C2xC170");
// GroupNames label

G:=SmallGroup(340,15);
// by ID

G=gap.SmallGroup(340,15);
# by ID

G:=PCGroup([4,-2,-2,-5,-17]);
// Polycyclic

G:=Group<a,b|a^2=b^170=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C170 in TeX

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