direct product, abelian, monomial, 2-elementary
Aliases: C2×C196, SmallGroup(392,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C196 |
| C1 — C2×C196 |
| C1 — C2×C196 |
Generators and relations for C2×C196
G = < a,b | a2=b196=1, ab=ba >
Smallest permutation representation of C2×C196
►Regular action on 392 points
Generators in S392
(1 199)(2 200)(3 201)(4 202)(5 203)(6 204)(7 205)(8 206)(9 207)(10 208)(11 209)(12 210)(13 211)(14 212)(15 213)(16 214)(17 215)(18 216)(19 217)(20 218)(21 219)(22 220)(23 221)(24 222)(25 223)(26 224)(27 225)(28 226)(29 227)(30 228)(31 229)(32 230)(33 231)(34 232)(35 233)(36 234)(37 235)(38 236)(39 237)(40 238)(41 239)(42 240)(43 241)(44 242)(45 243)(46 244)(47 245)(48 246)(49 247)(50 248)(51 249)(52 250)(53 251)(54 252)(55 253)(56 254)(57 255)(58 256)(59 257)(60 258)(61 259)(62 260)(63 261)(64 262)(65 263)(66 264)(67 265)(68 266)(69 267)(70 268)(71 269)(72 270)(73 271)(74 272)(75 273)(76 274)(77 275)(78 276)(79 277)(80 278)(81 279)(82 280)(83 281)(84 282)(85 283)(86 284)(87 285)(88 286)(89 287)(90 288)(91 289)(92 290)(93 291)(94 292)(95 293)(96 294)(97 295)(98 296)(99 297)(100 298)(101 299)(102 300)(103 301)(104 302)(105 303)(106 304)(107 305)(108 306)(109 307)(110 308)(111 309)(112 310)(113 311)(114 312)(115 313)(116 314)(117 315)(118 316)(119 317)(120 318)(121 319)(122 320)(123 321)(124 322)(125 323)(126 324)(127 325)(128 326)(129 327)(130 328)(131 329)(132 330)(133 331)(134 332)(135 333)(136 334)(137 335)(138 336)(139 337)(140 338)(141 339)(142 340)(143 341)(144 342)(145 343)(146 344)(147 345)(148 346)(149 347)(150 348)(151 349)(152 350)(153 351)(154 352)(155 353)(156 354)(157 355)(158 356)(159 357)(160 358)(161 359)(162 360)(163 361)(164 362)(165 363)(166 364)(167 365)(168 366)(169 367)(170 368)(171 369)(172 370)(173 371)(174 372)(175 373)(176 374)(177 375)(178 376)(179 377)(180 378)(181 379)(182 380)(183 381)(184 382)(185 383)(186 384)(187 385)(188 386)(189 387)(190 388)(191 389)(192 390)(193 391)(194 392)(195 197)(196 198)
(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)
G:=sub<Sym(392)| (1,199)(2,200)(3,201)(4,202)(5,203)(6,204)(7,205)(8,206)(9,207)(10,208)(11,209)(12,210)(13,211)(14,212)(15,213)(16,214)(17,215)(18,216)(19,217)(20,218)(21,219)(22,220)(23,221)(24,222)(25,223)(26,224)(27,225)(28,226)(29,227)(30,228)(31,229)(32,230)(33,231)(34,232)(35,233)(36,234)(37,235)(38,236)(39,237)(40,238)(41,239)(42,240)(43,241)(44,242)(45,243)(46,244)(47,245)(48,246)(49,247)(50,248)(51,249)(52,250)(53,251)(54,252)(55,253)(56,254)(57,255)(58,256)(59,257)(60,258)(61,259)(62,260)(63,261)(64,262)(65,263)(66,264)(67,265)(68,266)(69,267)(70,268)(71,269)(72,270)(73,271)(74,272)(75,273)(76,274)(77,275)(78,276)(79,277)(80,278)(81,279)(82,280)(83,281)(84,282)(85,283)(86,284)(87,285)(88,286)(89,287)(90,288)(91,289)(92,290)(93,291)(94,292)(95,293)(96,294)(97,295)(98,296)(99,297)(100,298)(101,299)(102,300)(103,301)(104,302)(105,303)(106,304)(107,305)(108,306)(109,307)(110,308)(111,309)(112,310)(113,311)(114,312)(115,313)(116,314)(117,315)(118,316)(119,317)(120,318)(121,319)(122,320)(123,321)(124,322)(125,323)(126,324)(127,325)(128,326)(129,327)(130,328)(131,329)(132,330)(133,331)(134,332)(135,333)(136,334)(137,335)(138,336)(139,337)(140,338)(141,339)(142,340)(143,341)(144,342)(145,343)(146,344)(147,345)(148,346)(149,347)(150,348)(151,349)(152,350)(153,351)(154,352)(155,353)(156,354)(157,355)(158,356)(159,357)(160,358)(161,359)(162,360)(163,361)(164,362)(165,363)(166,364)(167,365)(168,366)(169,367)(170,368)(171,369)(172,370)(173,371)(174,372)(175,373)(176,374)(177,375)(178,376)(179,377)(180,378)(181,379)(182,380)(183,381)(184,382)(185,383)(186,384)(187,385)(188,386)(189,387)(190,388)(191,389)(192,390)(193,391)(194,392)(195,197)(196,198), (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)>;
G:=Group( (1,199)(2,200)(3,201)(4,202)(5,203)(6,204)(7,205)(8,206)(9,207)(10,208)(11,209)(12,210)(13,211)(14,212)(15,213)(16,214)(17,215)(18,216)(19,217)(20,218)(21,219)(22,220)(23,221)(24,222)(25,223)(26,224)(27,225)(28,226)(29,227)(30,228)(31,229)(32,230)(33,231)(34,232)(35,233)(36,234)(37,235)(38,236)(39,237)(40,238)(41,239)(42,240)(43,241)(44,242)(45,243)(46,244)(47,245)(48,246)(49,247)(50,248)(51,249)(52,250)(53,251)(54,252)(55,253)(56,254)(57,255)(58,256)(59,257)(60,258)(61,259)(62,260)(63,261)(64,262)(65,263)(66,264)(67,265)(68,266)(69,267)(70,268)(71,269)(72,270)(73,271)(74,272)(75,273)(76,274)(77,275)(78,276)(79,277)(80,278)(81,279)(82,280)(83,281)(84,282)(85,283)(86,284)(87,285)(88,286)(89,287)(90,288)(91,289)(92,290)(93,291)(94,292)(95,293)(96,294)(97,295)(98,296)(99,297)(100,298)(101,299)(102,300)(103,301)(104,302)(105,303)(106,304)(107,305)(108,306)(109,307)(110,308)(111,309)(112,310)(113,311)(114,312)(115,313)(116,314)(117,315)(118,316)(119,317)(120,318)(121,319)(122,320)(123,321)(124,322)(125,323)(126,324)(127,325)(128,326)(129,327)(130,328)(131,329)(132,330)(133,331)(134,332)(135,333)(136,334)(137,335)(138,336)(139,337)(140,338)(141,339)(142,340)(143,341)(144,342)(145,343)(146,344)(147,345)(148,346)(149,347)(150,348)(151,349)(152,350)(153,351)(154,352)(155,353)(156,354)(157,355)(158,356)(159,357)(160,358)(161,359)(162,360)(163,361)(164,362)(165,363)(166,364)(167,365)(168,366)(169,367)(170,368)(171,369)(172,370)(173,371)(174,372)(175,373)(176,374)(177,375)(178,376)(179,377)(180,378)(181,379)(182,380)(183,381)(184,382)(185,383)(186,384)(187,385)(188,386)(189,387)(190,388)(191,389)(192,390)(193,391)(194,392)(195,197)(196,198), (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) );
G=PermutationGroup([[(1,199),(2,200),(3,201),(4,202),(5,203),(6,204),(7,205),(8,206),(9,207),(10,208),(11,209),(12,210),(13,211),(14,212),(15,213),(16,214),(17,215),(18,216),(19,217),(20,218),(21,219),(22,220),(23,221),(24,222),(25,223),(26,224),(27,225),(28,226),(29,227),(30,228),(31,229),(32,230),(33,231),(34,232),(35,233),(36,234),(37,235),(38,236),(39,237),(40,238),(41,239),(42,240),(43,241),(44,242),(45,243),(46,244),(47,245),(48,246),(49,247),(50,248),(51,249),(52,250),(53,251),(54,252),(55,253),(56,254),(57,255),(58,256),(59,257),(60,258),(61,259),(62,260),(63,261),(64,262),(65,263),(66,264),(67,265),(68,266),(69,267),(70,268),(71,269),(72,270),(73,271),(74,272),(75,273),(76,274),(77,275),(78,276),(79,277),(80,278),(81,279),(82,280),(83,281),(84,282),(85,283),(86,284),(87,285),(88,286),(89,287),(90,288),(91,289),(92,290),(93,291),(94,292),(95,293),(96,294),(97,295),(98,296),(99,297),(100,298),(101,299),(102,300),(103,301),(104,302),(105,303),(106,304),(107,305),(108,306),(109,307),(110,308),(111,309),(112,310),(113,311),(114,312),(115,313),(116,314),(117,315),(118,316),(119,317),(120,318),(121,319),(122,320),(123,321),(124,322),(125,323),(126,324),(127,325),(128,326),(129,327),(130,328),(131,329),(132,330),(133,331),(134,332),(135,333),(136,334),(137,335),(138,336),(139,337),(140,338),(141,339),(142,340),(143,341),(144,342),(145,343),(146,344),(147,345),(148,346),(149,347),(150,348),(151,349),(152,350),(153,351),(154,352),(155,353),(156,354),(157,355),(158,356),(159,357),(160,358),(161,359),(162,360),(163,361),(164,362),(165,363),(166,364),(167,365),(168,366),(169,367),(170,368),(171,369),(172,370),(173,371),(174,372),(175,373),(176,374),(177,375),(178,376),(179,377),(180,378),(181,379),(182,380),(183,381),(184,382),(185,383),(186,384),(187,385),(188,386),(189,387),(190,388),(191,389),(192,390),(193,391),(194,392),(195,197),(196,198)], [(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)]])
392 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 14A | ··· | 14R | 28A | ··· | 28X | 49A | ··· | 49AP | 98A | ··· | 98DV | 196A | ··· | 196FL |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 49 | ··· | 49 | 98 | ··· | 98 | 196 | ··· | 196 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
392 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | C49 | C98 | C98 | C196 |
| kernel | C2×C196 | C196 | C2×C98 | C98 | C2×C28 | C28 | C2×C14 | C14 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 42 | 84 | 42 | 168 |
Matrix representation of C2×C196 ►in GL2(𝔽197) generated by
| 196 | 0 |
| 0 | 1 |
| 157 | 0 |
| 0 | 103 |
G:=sub<GL(2,GF(197))| [196,0,0,1],[157,0,0,103] >;
C2×C196 in GAP, Magma, Sage, TeX
C_2\times C_{196} % in TeX
G:=Group("C2xC196"); // GroupNames label
G:=SmallGroup(392,8);
// by ID
G=gap.SmallGroup(392,8);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-7,140,222]);
// Polycyclic
G:=Group<a,b|a^2=b^196=1,a*b=b*a>;
// generators/relations
Export