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G = C2×C188order 376 = 23·47

Abelian group of type [2,188]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C188, SmallGroup(376,8)

Series: Derived Chief Lower central Upper central

C1 — C2×C188
C1C2C94C188 — C2×C188
C1 — C2×C188
C1 — C2×C188

Generators and relations for C2×C188
 G = < a,b | a2=b188=1, ab=ba >


Smallest permutation representation of C2×C188
Regular action on 376 points
Generators in S376
(1 321)(2 322)(3 323)(4 324)(5 325)(6 326)(7 327)(8 328)(9 329)(10 330)(11 331)(12 332)(13 333)(14 334)(15 335)(16 336)(17 337)(18 338)(19 339)(20 340)(21 341)(22 342)(23 343)(24 344)(25 345)(26 346)(27 347)(28 348)(29 349)(30 350)(31 351)(32 352)(33 353)(34 354)(35 355)(36 356)(37 357)(38 358)(39 359)(40 360)(41 361)(42 362)(43 363)(44 364)(45 365)(46 366)(47 367)(48 368)(49 369)(50 370)(51 371)(52 372)(53 373)(54 374)(55 375)(56 376)(57 189)(58 190)(59 191)(60 192)(61 193)(62 194)(63 195)(64 196)(65 197)(66 198)(67 199)(68 200)(69 201)(70 202)(71 203)(72 204)(73 205)(74 206)(75 207)(76 208)(77 209)(78 210)(79 211)(80 212)(81 213)(82 214)(83 215)(84 216)(85 217)(86 218)(87 219)(88 220)(89 221)(90 222)(91 223)(92 224)(93 225)(94 226)(95 227)(96 228)(97 229)(98 230)(99 231)(100 232)(101 233)(102 234)(103 235)(104 236)(105 237)(106 238)(107 239)(108 240)(109 241)(110 242)(111 243)(112 244)(113 245)(114 246)(115 247)(116 248)(117 249)(118 250)(119 251)(120 252)(121 253)(122 254)(123 255)(124 256)(125 257)(126 258)(127 259)(128 260)(129 261)(130 262)(131 263)(132 264)(133 265)(134 266)(135 267)(136 268)(137 269)(138 270)(139 271)(140 272)(141 273)(142 274)(143 275)(144 276)(145 277)(146 278)(147 279)(148 280)(149 281)(150 282)(151 283)(152 284)(153 285)(154 286)(155 287)(156 288)(157 289)(158 290)(159 291)(160 292)(161 293)(162 294)(163 295)(164 296)(165 297)(166 298)(167 299)(168 300)(169 301)(170 302)(171 303)(172 304)(173 305)(174 306)(175 307)(176 308)(177 309)(178 310)(179 311)(180 312)(181 313)(182 314)(183 315)(184 316)(185 317)(186 318)(187 319)(188 320)
(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)

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

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

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

376 conjugacy classes

class 1 2A2B2C4A4B4C4D47A···47AT94A···94EH188A···188GB
order1222444447···4794···94188···188
size111111111···11···11···1

376 irreducible representations

dim11111111
type+++
imageC1C2C2C4C47C94C94C188
kernelC2×C188C188C2×C94C94C2×C4C4C22C2
# reps1214469246184

Matrix representation of C2×C188 in GL2(𝔽941) generated by

10
0940
,
8270
0827
G:=sub<GL(2,GF(941))| [1,0,0,940],[827,0,0,827] >;

C2×C188 in GAP, Magma, Sage, TeX

C_2\times C_{188}
% in TeX

G:=Group("C2xC188");
// GroupNames label

G:=SmallGroup(376,8);
// by ID

G=gap.SmallGroup(376,8);
# by ID

G:=PCGroup([4,-2,-2,-47,-2,752]);
// Polycyclic

G:=Group<a,b|a^2=b^188=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C188 in TeX

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