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G = C2×C190order 380 = 22·5·19

Abelian group of type [2,190]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C190, SmallGroup(380,11)

Series: Derived Chief Lower central Upper central

C1 — C2×C190
C1C19C95C190 — C2×C190
C1 — C2×C190
C1 — C2×C190

Generators and relations for C2×C190
 G = < a,b | a2=b190=1, ab=ba >


Smallest permutation representation of C2×C190
Regular action on 380 points
Generators in S380
(1 360)(2 361)(3 362)(4 363)(5 364)(6 365)(7 366)(8 367)(9 368)(10 369)(11 370)(12 371)(13 372)(14 373)(15 374)(16 375)(17 376)(18 377)(19 378)(20 379)(21 380)(22 191)(23 192)(24 193)(25 194)(26 195)(27 196)(28 197)(29 198)(30 199)(31 200)(32 201)(33 202)(34 203)(35 204)(36 205)(37 206)(38 207)(39 208)(40 209)(41 210)(42 211)(43 212)(44 213)(45 214)(46 215)(47 216)(48 217)(49 218)(50 219)(51 220)(52 221)(53 222)(54 223)(55 224)(56 225)(57 226)(58 227)(59 228)(60 229)(61 230)(62 231)(63 232)(64 233)(65 234)(66 235)(67 236)(68 237)(69 238)(70 239)(71 240)(72 241)(73 242)(74 243)(75 244)(76 245)(77 246)(78 247)(79 248)(80 249)(81 250)(82 251)(83 252)(84 253)(85 254)(86 255)(87 256)(88 257)(89 258)(90 259)(91 260)(92 261)(93 262)(94 263)(95 264)(96 265)(97 266)(98 267)(99 268)(100 269)(101 270)(102 271)(103 272)(104 273)(105 274)(106 275)(107 276)(108 277)(109 278)(110 279)(111 280)(112 281)(113 282)(114 283)(115 284)(116 285)(117 286)(118 287)(119 288)(120 289)(121 290)(122 291)(123 292)(124 293)(125 294)(126 295)(127 296)(128 297)(129 298)(130 299)(131 300)(132 301)(133 302)(134 303)(135 304)(136 305)(137 306)(138 307)(139 308)(140 309)(141 310)(142 311)(143 312)(144 313)(145 314)(146 315)(147 316)(148 317)(149 318)(150 319)(151 320)(152 321)(153 322)(154 323)(155 324)(156 325)(157 326)(158 327)(159 328)(160 329)(161 330)(162 331)(163 332)(164 333)(165 334)(166 335)(167 336)(168 337)(169 338)(170 339)(171 340)(172 341)(173 342)(174 343)(175 344)(176 345)(177 346)(178 347)(179 348)(180 349)(181 350)(182 351)(183 352)(184 353)(185 354)(186 355)(187 356)(188 357)(189 358)(190 359)
(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)(191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380)

G:=sub<Sym(380)| (1,360)(2,361)(3,362)(4,363)(5,364)(6,365)(7,366)(8,367)(9,368)(10,369)(11,370)(12,371)(13,372)(14,373)(15,374)(16,375)(17,376)(18,377)(19,378)(20,379)(21,380)(22,191)(23,192)(24,193)(25,194)(26,195)(27,196)(28,197)(29,198)(30,199)(31,200)(32,201)(33,202)(34,203)(35,204)(36,205)(37,206)(38,207)(39,208)(40,209)(41,210)(42,211)(43,212)(44,213)(45,214)(46,215)(47,216)(48,217)(49,218)(50,219)(51,220)(52,221)(53,222)(54,223)(55,224)(56,225)(57,226)(58,227)(59,228)(60,229)(61,230)(62,231)(63,232)(64,233)(65,234)(66,235)(67,236)(68,237)(69,238)(70,239)(71,240)(72,241)(73,242)(74,243)(75,244)(76,245)(77,246)(78,247)(79,248)(80,249)(81,250)(82,251)(83,252)(84,253)(85,254)(86,255)(87,256)(88,257)(89,258)(90,259)(91,260)(92,261)(93,262)(94,263)(95,264)(96,265)(97,266)(98,267)(99,268)(100,269)(101,270)(102,271)(103,272)(104,273)(105,274)(106,275)(107,276)(108,277)(109,278)(110,279)(111,280)(112,281)(113,282)(114,283)(115,284)(116,285)(117,286)(118,287)(119,288)(120,289)(121,290)(122,291)(123,292)(124,293)(125,294)(126,295)(127,296)(128,297)(129,298)(130,299)(131,300)(132,301)(133,302)(134,303)(135,304)(136,305)(137,306)(138,307)(139,308)(140,309)(141,310)(142,311)(143,312)(144,313)(145,314)(146,315)(147,316)(148,317)(149,318)(150,319)(151,320)(152,321)(153,322)(154,323)(155,324)(156,325)(157,326)(158,327)(159,328)(160,329)(161,330)(162,331)(163,332)(164,333)(165,334)(166,335)(167,336)(168,337)(169,338)(170,339)(171,340)(172,341)(173,342)(174,343)(175,344)(176,345)(177,346)(178,347)(179,348)(180,349)(181,350)(182,351)(183,352)(184,353)(185,354)(186,355)(187,356)(188,357)(189,358)(190,359), (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)(191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380)>;

G:=Group( (1,360)(2,361)(3,362)(4,363)(5,364)(6,365)(7,366)(8,367)(9,368)(10,369)(11,370)(12,371)(13,372)(14,373)(15,374)(16,375)(17,376)(18,377)(19,378)(20,379)(21,380)(22,191)(23,192)(24,193)(25,194)(26,195)(27,196)(28,197)(29,198)(30,199)(31,200)(32,201)(33,202)(34,203)(35,204)(36,205)(37,206)(38,207)(39,208)(40,209)(41,210)(42,211)(43,212)(44,213)(45,214)(46,215)(47,216)(48,217)(49,218)(50,219)(51,220)(52,221)(53,222)(54,223)(55,224)(56,225)(57,226)(58,227)(59,228)(60,229)(61,230)(62,231)(63,232)(64,233)(65,234)(66,235)(67,236)(68,237)(69,238)(70,239)(71,240)(72,241)(73,242)(74,243)(75,244)(76,245)(77,246)(78,247)(79,248)(80,249)(81,250)(82,251)(83,252)(84,253)(85,254)(86,255)(87,256)(88,257)(89,258)(90,259)(91,260)(92,261)(93,262)(94,263)(95,264)(96,265)(97,266)(98,267)(99,268)(100,269)(101,270)(102,271)(103,272)(104,273)(105,274)(106,275)(107,276)(108,277)(109,278)(110,279)(111,280)(112,281)(113,282)(114,283)(115,284)(116,285)(117,286)(118,287)(119,288)(120,289)(121,290)(122,291)(123,292)(124,293)(125,294)(126,295)(127,296)(128,297)(129,298)(130,299)(131,300)(132,301)(133,302)(134,303)(135,304)(136,305)(137,306)(138,307)(139,308)(140,309)(141,310)(142,311)(143,312)(144,313)(145,314)(146,315)(147,316)(148,317)(149,318)(150,319)(151,320)(152,321)(153,322)(154,323)(155,324)(156,325)(157,326)(158,327)(159,328)(160,329)(161,330)(162,331)(163,332)(164,333)(165,334)(166,335)(167,336)(168,337)(169,338)(170,339)(171,340)(172,341)(173,342)(174,343)(175,344)(176,345)(177,346)(178,347)(179,348)(180,349)(181,350)(182,351)(183,352)(184,353)(185,354)(186,355)(187,356)(188,357)(189,358)(190,359), (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)(191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380) );

G=PermutationGroup([(1,360),(2,361),(3,362),(4,363),(5,364),(6,365),(7,366),(8,367),(9,368),(10,369),(11,370),(12,371),(13,372),(14,373),(15,374),(16,375),(17,376),(18,377),(19,378),(20,379),(21,380),(22,191),(23,192),(24,193),(25,194),(26,195),(27,196),(28,197),(29,198),(30,199),(31,200),(32,201),(33,202),(34,203),(35,204),(36,205),(37,206),(38,207),(39,208),(40,209),(41,210),(42,211),(43,212),(44,213),(45,214),(46,215),(47,216),(48,217),(49,218),(50,219),(51,220),(52,221),(53,222),(54,223),(55,224),(56,225),(57,226),(58,227),(59,228),(60,229),(61,230),(62,231),(63,232),(64,233),(65,234),(66,235),(67,236),(68,237),(69,238),(70,239),(71,240),(72,241),(73,242),(74,243),(75,244),(76,245),(77,246),(78,247),(79,248),(80,249),(81,250),(82,251),(83,252),(84,253),(85,254),(86,255),(87,256),(88,257),(89,258),(90,259),(91,260),(92,261),(93,262),(94,263),(95,264),(96,265),(97,266),(98,267),(99,268),(100,269),(101,270),(102,271),(103,272),(104,273),(105,274),(106,275),(107,276),(108,277),(109,278),(110,279),(111,280),(112,281),(113,282),(114,283),(115,284),(116,285),(117,286),(118,287),(119,288),(120,289),(121,290),(122,291),(123,292),(124,293),(125,294),(126,295),(127,296),(128,297),(129,298),(130,299),(131,300),(132,301),(133,302),(134,303),(135,304),(136,305),(137,306),(138,307),(139,308),(140,309),(141,310),(142,311),(143,312),(144,313),(145,314),(146,315),(147,316),(148,317),(149,318),(150,319),(151,320),(152,321),(153,322),(154,323),(155,324),(156,325),(157,326),(158,327),(159,328),(160,329),(161,330),(162,331),(163,332),(164,333),(165,334),(166,335),(167,336),(168,337),(169,338),(170,339),(171,340),(172,341),(173,342),(174,343),(175,344),(176,345),(177,346),(178,347),(179,348),(180,349),(181,350),(182,351),(183,352),(184,353),(185,354),(186,355),(187,356),(188,357),(189,358),(190,359)], [(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),(191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380)])

380 conjugacy classes

class 1 2A2B2C5A5B5C5D10A···10L19A···19R38A···38BB95A···95BT190A···190HH
order1222555510···1019···1938···3895···95190···190
size111111111···11···11···11···11···1

380 irreducible representations

dim11111111
type++
imageC1C2C5C10C19C38C95C190
kernelC2×C190C190C2×C38C38C2×C10C10C22C2
# reps13412185472216

Matrix representation of C2×C190 in GL2(𝔽191) generated by

1900
01
,
400
0110
G:=sub<GL(2,GF(191))| [190,0,0,1],[40,0,0,110] >;

C2×C190 in GAP, Magma, Sage, TeX

C_2\times C_{190}
% in TeX

G:=Group("C2xC190");
// GroupNames label

G:=SmallGroup(380,11);
// by ID

G=gap.SmallGroup(380,11);
# by ID

G:=PCGroup([4,-2,-2,-5,-19]);
// Polycyclic

G:=Group<a,b|a^2=b^190=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C190 in TeX

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