direct product, abelian, monomial, 2-elementary
Aliases: C2×C194, SmallGroup(388,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C194 |
| C1 — C2×C194 |
| C1 — C2×C194 |
Generators and relations for C2×C194
G = < a,b | a2=b194=1, ab=ba >
Smallest permutation representation of C2×C194
►Regular action on 388 points
Generators in S388
(1 325)(2 326)(3 327)(4 328)(5 329)(6 330)(7 331)(8 332)(9 333)(10 334)(11 335)(12 336)(13 337)(14 338)(15 339)(16 340)(17 341)(18 342)(19 343)(20 344)(21 345)(22 346)(23 347)(24 348)(25 349)(26 350)(27 351)(28 352)(29 353)(30 354)(31 355)(32 356)(33 357)(34 358)(35 359)(36 360)(37 361)(38 362)(39 363)(40 364)(41 365)(42 366)(43 367)(44 368)(45 369)(46 370)(47 371)(48 372)(49 373)(50 374)(51 375)(52 376)(53 377)(54 378)(55 379)(56 380)(57 381)(58 382)(59 383)(60 384)(61 385)(62 386)(63 387)(64 388)(65 195)(66 196)(67 197)(68 198)(69 199)(70 200)(71 201)(72 202)(73 203)(74 204)(75 205)(76 206)(77 207)(78 208)(79 209)(80 210)(81 211)(82 212)(83 213)(84 214)(85 215)(86 216)(87 217)(88 218)(89 219)(90 220)(91 221)(92 222)(93 223)(94 224)(95 225)(96 226)(97 227)(98 228)(99 229)(100 230)(101 231)(102 232)(103 233)(104 234)(105 235)(106 236)(107 237)(108 238)(109 239)(110 240)(111 241)(112 242)(113 243)(114 244)(115 245)(116 246)(117 247)(118 248)(119 249)(120 250)(121 251)(122 252)(123 253)(124 254)(125 255)(126 256)(127 257)(128 258)(129 259)(130 260)(131 261)(132 262)(133 263)(134 264)(135 265)(136 266)(137 267)(138 268)(139 269)(140 270)(141 271)(142 272)(143 273)(144 274)(145 275)(146 276)(147 277)(148 278)(149 279)(150 280)(151 281)(152 282)(153 283)(154 284)(155 285)(156 286)(157 287)(158 288)(159 289)(160 290)(161 291)(162 292)(163 293)(164 294)(165 295)(166 296)(167 297)(168 298)(169 299)(170 300)(171 301)(172 302)(173 303)(174 304)(175 305)(176 306)(177 307)(178 308)(179 309)(180 310)(181 311)(182 312)(183 313)(184 314)(185 315)(186 316)(187 317)(188 318)(189 319)(190 320)(191 321)(192 322)(193 323)(194 324)
(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)(195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388)
G:=sub<Sym(388)| (1,325)(2,326)(3,327)(4,328)(5,329)(6,330)(7,331)(8,332)(9,333)(10,334)(11,335)(12,336)(13,337)(14,338)(15,339)(16,340)(17,341)(18,342)(19,343)(20,344)(21,345)(22,346)(23,347)(24,348)(25,349)(26,350)(27,351)(28,352)(29,353)(30,354)(31,355)(32,356)(33,357)(34,358)(35,359)(36,360)(37,361)(38,362)(39,363)(40,364)(41,365)(42,366)(43,367)(44,368)(45,369)(46,370)(47,371)(48,372)(49,373)(50,374)(51,375)(52,376)(53,377)(54,378)(55,379)(56,380)(57,381)(58,382)(59,383)(60,384)(61,385)(62,386)(63,387)(64,388)(65,195)(66,196)(67,197)(68,198)(69,199)(70,200)(71,201)(72,202)(73,203)(74,204)(75,205)(76,206)(77,207)(78,208)(79,209)(80,210)(81,211)(82,212)(83,213)(84,214)(85,215)(86,216)(87,217)(88,218)(89,219)(90,220)(91,221)(92,222)(93,223)(94,224)(95,225)(96,226)(97,227)(98,228)(99,229)(100,230)(101,231)(102,232)(103,233)(104,234)(105,235)(106,236)(107,237)(108,238)(109,239)(110,240)(111,241)(112,242)(113,243)(114,244)(115,245)(116,246)(117,247)(118,248)(119,249)(120,250)(121,251)(122,252)(123,253)(124,254)(125,255)(126,256)(127,257)(128,258)(129,259)(130,260)(131,261)(132,262)(133,263)(134,264)(135,265)(136,266)(137,267)(138,268)(139,269)(140,270)(141,271)(142,272)(143,273)(144,274)(145,275)(146,276)(147,277)(148,278)(149,279)(150,280)(151,281)(152,282)(153,283)(154,284)(155,285)(156,286)(157,287)(158,288)(159,289)(160,290)(161,291)(162,292)(163,293)(164,294)(165,295)(166,296)(167,297)(168,298)(169,299)(170,300)(171,301)(172,302)(173,303)(174,304)(175,305)(176,306)(177,307)(178,308)(179,309)(180,310)(181,311)(182,312)(183,313)(184,314)(185,315)(186,316)(187,317)(188,318)(189,319)(190,320)(191,321)(192,322)(193,323)(194,324), (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)(195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388)>;
G:=Group( (1,325)(2,326)(3,327)(4,328)(5,329)(6,330)(7,331)(8,332)(9,333)(10,334)(11,335)(12,336)(13,337)(14,338)(15,339)(16,340)(17,341)(18,342)(19,343)(20,344)(21,345)(22,346)(23,347)(24,348)(25,349)(26,350)(27,351)(28,352)(29,353)(30,354)(31,355)(32,356)(33,357)(34,358)(35,359)(36,360)(37,361)(38,362)(39,363)(40,364)(41,365)(42,366)(43,367)(44,368)(45,369)(46,370)(47,371)(48,372)(49,373)(50,374)(51,375)(52,376)(53,377)(54,378)(55,379)(56,380)(57,381)(58,382)(59,383)(60,384)(61,385)(62,386)(63,387)(64,388)(65,195)(66,196)(67,197)(68,198)(69,199)(70,200)(71,201)(72,202)(73,203)(74,204)(75,205)(76,206)(77,207)(78,208)(79,209)(80,210)(81,211)(82,212)(83,213)(84,214)(85,215)(86,216)(87,217)(88,218)(89,219)(90,220)(91,221)(92,222)(93,223)(94,224)(95,225)(96,226)(97,227)(98,228)(99,229)(100,230)(101,231)(102,232)(103,233)(104,234)(105,235)(106,236)(107,237)(108,238)(109,239)(110,240)(111,241)(112,242)(113,243)(114,244)(115,245)(116,246)(117,247)(118,248)(119,249)(120,250)(121,251)(122,252)(123,253)(124,254)(125,255)(126,256)(127,257)(128,258)(129,259)(130,260)(131,261)(132,262)(133,263)(134,264)(135,265)(136,266)(137,267)(138,268)(139,269)(140,270)(141,271)(142,272)(143,273)(144,274)(145,275)(146,276)(147,277)(148,278)(149,279)(150,280)(151,281)(152,282)(153,283)(154,284)(155,285)(156,286)(157,287)(158,288)(159,289)(160,290)(161,291)(162,292)(163,293)(164,294)(165,295)(166,296)(167,297)(168,298)(169,299)(170,300)(171,301)(172,302)(173,303)(174,304)(175,305)(176,306)(177,307)(178,308)(179,309)(180,310)(181,311)(182,312)(183,313)(184,314)(185,315)(186,316)(187,317)(188,318)(189,319)(190,320)(191,321)(192,322)(193,323)(194,324), (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)(195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388) );
G=PermutationGroup([[(1,325),(2,326),(3,327),(4,328),(5,329),(6,330),(7,331),(8,332),(9,333),(10,334),(11,335),(12,336),(13,337),(14,338),(15,339),(16,340),(17,341),(18,342),(19,343),(20,344),(21,345),(22,346),(23,347),(24,348),(25,349),(26,350),(27,351),(28,352),(29,353),(30,354),(31,355),(32,356),(33,357),(34,358),(35,359),(36,360),(37,361),(38,362),(39,363),(40,364),(41,365),(42,366),(43,367),(44,368),(45,369),(46,370),(47,371),(48,372),(49,373),(50,374),(51,375),(52,376),(53,377),(54,378),(55,379),(56,380),(57,381),(58,382),(59,383),(60,384),(61,385),(62,386),(63,387),(64,388),(65,195),(66,196),(67,197),(68,198),(69,199),(70,200),(71,201),(72,202),(73,203),(74,204),(75,205),(76,206),(77,207),(78,208),(79,209),(80,210),(81,211),(82,212),(83,213),(84,214),(85,215),(86,216),(87,217),(88,218),(89,219),(90,220),(91,221),(92,222),(93,223),(94,224),(95,225),(96,226),(97,227),(98,228),(99,229),(100,230),(101,231),(102,232),(103,233),(104,234),(105,235),(106,236),(107,237),(108,238),(109,239),(110,240),(111,241),(112,242),(113,243),(114,244),(115,245),(116,246),(117,247),(118,248),(119,249),(120,250),(121,251),(122,252),(123,253),(124,254),(125,255),(126,256),(127,257),(128,258),(129,259),(130,260),(131,261),(132,262),(133,263),(134,264),(135,265),(136,266),(137,267),(138,268),(139,269),(140,270),(141,271),(142,272),(143,273),(144,274),(145,275),(146,276),(147,277),(148,278),(149,279),(150,280),(151,281),(152,282),(153,283),(154,284),(155,285),(156,286),(157,287),(158,288),(159,289),(160,290),(161,291),(162,292),(163,293),(164,294),(165,295),(166,296),(167,297),(168,298),(169,299),(170,300),(171,301),(172,302),(173,303),(174,304),(175,305),(176,306),(177,307),(178,308),(179,309),(180,310),(181,311),(182,312),(183,313),(184,314),(185,315),(186,316),(187,317),(188,318),(189,319),(190,320),(191,321),(192,322),(193,323),(194,324)], [(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),(195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388)]])
388 conjugacy classes
| class | 1 | 2A | 2B | 2C | 97A | ··· | 97CR | 194A | ··· | 194KB |
| order | 1 | 2 | 2 | 2 | 97 | ··· | 97 | 194 | ··· | 194 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
388 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C97 | C194 |
| kernel | C2×C194 | C194 | C22 | C2 |
| # reps | 1 | 3 | 96 | 288 |
Matrix representation of C2×C194 ►in GL2(𝔽389) generated by
| 388 | 0 |
| 0 | 388 |
| 376 | 0 |
| 0 | 142 |
G:=sub<GL(2,GF(389))| [388,0,0,388],[376,0,0,142] >;
C2×C194 in GAP, Magma, Sage, TeX
C_2\times C_{194} % in TeX
G:=Group("C2xC194"); // GroupNames label
G:=SmallGroup(388,5);
// by ID
G=gap.SmallGroup(388,5);
# by ID
G:=PCGroup([3,-2,-2,-97]);
// Polycyclic
G:=Group<a,b|a^2=b^194=1,a*b=b*a>;
// generators/relations
Export