direct product, abelian, monomial, 2-elementary
Aliases: C2×C198, SmallGroup(396,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C198 |
| C1 — C2×C198 |
| C1 — C2×C198 |
Generators and relations for C2×C198
G = < a,b | a2=b198=1, ab=ba >
Smallest permutation representation of C2×C198
►Regular action on 396 points
Generators in S396
(1 292)(2 293)(3 294)(4 295)(5 296)(6 297)(7 298)(8 299)(9 300)(10 301)(11 302)(12 303)(13 304)(14 305)(15 306)(16 307)(17 308)(18 309)(19 310)(20 311)(21 312)(22 313)(23 314)(24 315)(25 316)(26 317)(27 318)(28 319)(29 320)(30 321)(31 322)(32 323)(33 324)(34 325)(35 326)(36 327)(37 328)(38 329)(39 330)(40 331)(41 332)(42 333)(43 334)(44 335)(45 336)(46 337)(47 338)(48 339)(49 340)(50 341)(51 342)(52 343)(53 344)(54 345)(55 346)(56 347)(57 348)(58 349)(59 350)(60 351)(61 352)(62 353)(63 354)(64 355)(65 356)(66 357)(67 358)(68 359)(69 360)(70 361)(71 362)(72 363)(73 364)(74 365)(75 366)(76 367)(77 368)(78 369)(79 370)(80 371)(81 372)(82 373)(83 374)(84 375)(85 376)(86 377)(87 378)(88 379)(89 380)(90 381)(91 382)(92 383)(93 384)(94 385)(95 386)(96 387)(97 388)(98 389)(99 390)(100 391)(101 392)(102 393)(103 394)(104 395)(105 396)(106 199)(107 200)(108 201)(109 202)(110 203)(111 204)(112 205)(113 206)(114 207)(115 208)(116 209)(117 210)(118 211)(119 212)(120 213)(121 214)(122 215)(123 216)(124 217)(125 218)(126 219)(127 220)(128 221)(129 222)(130 223)(131 224)(132 225)(133 226)(134 227)(135 228)(136 229)(137 230)(138 231)(139 232)(140 233)(141 234)(142 235)(143 236)(144 237)(145 238)(146 239)(147 240)(148 241)(149 242)(150 243)(151 244)(152 245)(153 246)(154 247)(155 248)(156 249)(157 250)(158 251)(159 252)(160 253)(161 254)(162 255)(163 256)(164 257)(165 258)(166 259)(167 260)(168 261)(169 262)(170 263)(171 264)(172 265)(173 266)(174 267)(175 268)(176 269)(177 270)(178 271)(179 272)(180 273)(181 274)(182 275)(183 276)(184 277)(185 278)(186 279)(187 280)(188 281)(189 282)(190 283)(191 284)(192 285)(193 286)(194 287)(195 288)(196 289)(197 290)(198 291)
(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)(199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396)
G:=sub<Sym(396)| (1,292)(2,293)(3,294)(4,295)(5,296)(6,297)(7,298)(8,299)(9,300)(10,301)(11,302)(12,303)(13,304)(14,305)(15,306)(16,307)(17,308)(18,309)(19,310)(20,311)(21,312)(22,313)(23,314)(24,315)(25,316)(26,317)(27,318)(28,319)(29,320)(30,321)(31,322)(32,323)(33,324)(34,325)(35,326)(36,327)(37,328)(38,329)(39,330)(40,331)(41,332)(42,333)(43,334)(44,335)(45,336)(46,337)(47,338)(48,339)(49,340)(50,341)(51,342)(52,343)(53,344)(54,345)(55,346)(56,347)(57,348)(58,349)(59,350)(60,351)(61,352)(62,353)(63,354)(64,355)(65,356)(66,357)(67,358)(68,359)(69,360)(70,361)(71,362)(72,363)(73,364)(74,365)(75,366)(76,367)(77,368)(78,369)(79,370)(80,371)(81,372)(82,373)(83,374)(84,375)(85,376)(86,377)(87,378)(88,379)(89,380)(90,381)(91,382)(92,383)(93,384)(94,385)(95,386)(96,387)(97,388)(98,389)(99,390)(100,391)(101,392)(102,393)(103,394)(104,395)(105,396)(106,199)(107,200)(108,201)(109,202)(110,203)(111,204)(112,205)(113,206)(114,207)(115,208)(116,209)(117,210)(118,211)(119,212)(120,213)(121,214)(122,215)(123,216)(124,217)(125,218)(126,219)(127,220)(128,221)(129,222)(130,223)(131,224)(132,225)(133,226)(134,227)(135,228)(136,229)(137,230)(138,231)(139,232)(140,233)(141,234)(142,235)(143,236)(144,237)(145,238)(146,239)(147,240)(148,241)(149,242)(150,243)(151,244)(152,245)(153,246)(154,247)(155,248)(156,249)(157,250)(158,251)(159,252)(160,253)(161,254)(162,255)(163,256)(164,257)(165,258)(166,259)(167,260)(168,261)(169,262)(170,263)(171,264)(172,265)(173,266)(174,267)(175,268)(176,269)(177,270)(178,271)(179,272)(180,273)(181,274)(182,275)(183,276)(184,277)(185,278)(186,279)(187,280)(188,281)(189,282)(190,283)(191,284)(192,285)(193,286)(194,287)(195,288)(196,289)(197,290)(198,291), (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)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)>;
G:=Group( (1,292)(2,293)(3,294)(4,295)(5,296)(6,297)(7,298)(8,299)(9,300)(10,301)(11,302)(12,303)(13,304)(14,305)(15,306)(16,307)(17,308)(18,309)(19,310)(20,311)(21,312)(22,313)(23,314)(24,315)(25,316)(26,317)(27,318)(28,319)(29,320)(30,321)(31,322)(32,323)(33,324)(34,325)(35,326)(36,327)(37,328)(38,329)(39,330)(40,331)(41,332)(42,333)(43,334)(44,335)(45,336)(46,337)(47,338)(48,339)(49,340)(50,341)(51,342)(52,343)(53,344)(54,345)(55,346)(56,347)(57,348)(58,349)(59,350)(60,351)(61,352)(62,353)(63,354)(64,355)(65,356)(66,357)(67,358)(68,359)(69,360)(70,361)(71,362)(72,363)(73,364)(74,365)(75,366)(76,367)(77,368)(78,369)(79,370)(80,371)(81,372)(82,373)(83,374)(84,375)(85,376)(86,377)(87,378)(88,379)(89,380)(90,381)(91,382)(92,383)(93,384)(94,385)(95,386)(96,387)(97,388)(98,389)(99,390)(100,391)(101,392)(102,393)(103,394)(104,395)(105,396)(106,199)(107,200)(108,201)(109,202)(110,203)(111,204)(112,205)(113,206)(114,207)(115,208)(116,209)(117,210)(118,211)(119,212)(120,213)(121,214)(122,215)(123,216)(124,217)(125,218)(126,219)(127,220)(128,221)(129,222)(130,223)(131,224)(132,225)(133,226)(134,227)(135,228)(136,229)(137,230)(138,231)(139,232)(140,233)(141,234)(142,235)(143,236)(144,237)(145,238)(146,239)(147,240)(148,241)(149,242)(150,243)(151,244)(152,245)(153,246)(154,247)(155,248)(156,249)(157,250)(158,251)(159,252)(160,253)(161,254)(162,255)(163,256)(164,257)(165,258)(166,259)(167,260)(168,261)(169,262)(170,263)(171,264)(172,265)(173,266)(174,267)(175,268)(176,269)(177,270)(178,271)(179,272)(180,273)(181,274)(182,275)(183,276)(184,277)(185,278)(186,279)(187,280)(188,281)(189,282)(190,283)(191,284)(192,285)(193,286)(194,287)(195,288)(196,289)(197,290)(198,291), (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)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396) );
G=PermutationGroup([[(1,292),(2,293),(3,294),(4,295),(5,296),(6,297),(7,298),(8,299),(9,300),(10,301),(11,302),(12,303),(13,304),(14,305),(15,306),(16,307),(17,308),(18,309),(19,310),(20,311),(21,312),(22,313),(23,314),(24,315),(25,316),(26,317),(27,318),(28,319),(29,320),(30,321),(31,322),(32,323),(33,324),(34,325),(35,326),(36,327),(37,328),(38,329),(39,330),(40,331),(41,332),(42,333),(43,334),(44,335),(45,336),(46,337),(47,338),(48,339),(49,340),(50,341),(51,342),(52,343),(53,344),(54,345),(55,346),(56,347),(57,348),(58,349),(59,350),(60,351),(61,352),(62,353),(63,354),(64,355),(65,356),(66,357),(67,358),(68,359),(69,360),(70,361),(71,362),(72,363),(73,364),(74,365),(75,366),(76,367),(77,368),(78,369),(79,370),(80,371),(81,372),(82,373),(83,374),(84,375),(85,376),(86,377),(87,378),(88,379),(89,380),(90,381),(91,382),(92,383),(93,384),(94,385),(95,386),(96,387),(97,388),(98,389),(99,390),(100,391),(101,392),(102,393),(103,394),(104,395),(105,396),(106,199),(107,200),(108,201),(109,202),(110,203),(111,204),(112,205),(113,206),(114,207),(115,208),(116,209),(117,210),(118,211),(119,212),(120,213),(121,214),(122,215),(123,216),(124,217),(125,218),(126,219),(127,220),(128,221),(129,222),(130,223),(131,224),(132,225),(133,226),(134,227),(135,228),(136,229),(137,230),(138,231),(139,232),(140,233),(141,234),(142,235),(143,236),(144,237),(145,238),(146,239),(147,240),(148,241),(149,242),(150,243),(151,244),(152,245),(153,246),(154,247),(155,248),(156,249),(157,250),(158,251),(159,252),(160,253),(161,254),(162,255),(163,256),(164,257),(165,258),(166,259),(167,260),(168,261),(169,262),(170,263),(171,264),(172,265),(173,266),(174,267),(175,268),(176,269),(177,270),(178,271),(179,272),(180,273),(181,274),(182,275),(183,276),(184,277),(185,278),(186,279),(187,280),(188,281),(189,282),(190,283),(191,284),(192,285),(193,286),(194,287),(195,288),(196,289),(197,290),(198,291)], [(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),(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)]])
396 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | ··· | 6F | 9A | ··· | 9F | 11A | ··· | 11J | 18A | ··· | 18R | 22A | ··· | 22AD | 33A | ··· | 33T | 66A | ··· | 66BH | 99A | ··· | 99BH | 198A | ··· | 198FX |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 11 | ··· | 11 | 18 | ··· | 18 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 | 99 | ··· | 99 | 198 | ··· | 198 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
396 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | C9 | C11 | C18 | C22 | C33 | C66 | C99 | C198 |
| kernel | C2×C198 | C198 | C2×C66 | C66 | C2×C22 | C2×C18 | C22 | C18 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 6 | 10 | 18 | 30 | 20 | 60 | 60 | 180 |
Matrix representation of C2×C198 ►in GL2(𝔽199) generated by
| 198 | 0 |
| 0 | 198 |
| 74 | 0 |
| 0 | 51 |
G:=sub<GL(2,GF(199))| [198,0,0,198],[74,0,0,51] >;
C2×C198 in GAP, Magma, Sage, TeX
C_2\times C_{198} % in TeX
G:=Group("C2xC198"); // GroupNames label
G:=SmallGroup(396,10);
// by ID
G=gap.SmallGroup(396,10);
# by ID
G:=PCGroup([5,-2,-2,-3,-11,-3,507]);
// Polycyclic
G:=Group<a,b|a^2=b^198=1,a*b=b*a>;
// generators/relations
Export