direct product, abelian, monomial, 2-elementary
Aliases: C22×C102, SmallGroup(408,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C102 |
| C1 — C22×C102 |
| C1 — C22×C102 |
Generators and relations for C22×C102
G = < a,b,c | a2=b2=c102=1, ab=ba, ac=ca, bc=cb >
Subgroups: 64, all normal (8 characteristic)
C1, C2, C3, C22, C6, C23, C2×C6, C17, C22×C6, C34, C51, C2×C34, C102, C22×C34, C2×C102, C22×C102
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C17, C22×C6, C34, C51, C2×C34, C102, C22×C34, C2×C102, C22×C102
►Regular action on 408 points
Generators in S408
(1 241)(2 242)(3 243)(4 244)(5 245)(6 246)(7 247)(8 248)(9 249)(10 250)(11 251)(12 252)(13 253)(14 254)(15 255)(16 256)(17 257)(18 258)(19 259)(20 260)(21 261)(22 262)(23 263)(24 264)(25 265)(26 266)(27 267)(28 268)(29 269)(30 270)(31 271)(32 272)(33 273)(34 274)(35 275)(36 276)(37 277)(38 278)(39 279)(40 280)(41 281)(42 282)(43 283)(44 284)(45 285)(46 286)(47 287)(48 288)(49 289)(50 290)(51 291)(52 292)(53 293)(54 294)(55 295)(56 296)(57 297)(58 298)(59 299)(60 300)(61 301)(62 302)(63 303)(64 304)(65 305)(66 306)(67 205)(68 206)(69 207)(70 208)(71 209)(72 210)(73 211)(74 212)(75 213)(76 214)(77 215)(78 216)(79 217)(80 218)(81 219)(82 220)(83 221)(84 222)(85 223)(86 224)(87 225)(88 226)(89 227)(90 228)(91 229)(92 230)(93 231)(94 232)(95 233)(96 234)(97 235)(98 236)(99 237)(100 238)(101 239)(102 240)(103 357)(104 358)(105 359)(106 360)(107 361)(108 362)(109 363)(110 364)(111 365)(112 366)(113 367)(114 368)(115 369)(116 370)(117 371)(118 372)(119 373)(120 374)(121 375)(122 376)(123 377)(124 378)(125 379)(126 380)(127 381)(128 382)(129 383)(130 384)(131 385)(132 386)(133 387)(134 388)(135 389)(136 390)(137 391)(138 392)(139 393)(140 394)(141 395)(142 396)(143 397)(144 398)(145 399)(146 400)(147 401)(148 402)(149 403)(150 404)(151 405)(152 406)(153 407)(154 408)(155 307)(156 308)(157 309)(158 310)(159 311)(160 312)(161 313)(162 314)(163 315)(164 316)(165 317)(166 318)(167 319)(168 320)(169 321)(170 322)(171 323)(172 324)(173 325)(174 326)(175 327)(176 328)(177 329)(178 330)(179 331)(180 332)(181 333)(182 334)(183 335)(184 336)(185 337)(186 338)(187 339)(188 340)(189 341)(190 342)(191 343)(192 344)(193 345)(194 346)(195 347)(196 348)(197 349)(198 350)(199 351)(200 352)(201 353)(202 354)(203 355)(204 356)
(1 140)(2 141)(3 142)(4 143)(5 144)(6 145)(7 146)(8 147)(9 148)(10 149)(11 150)(12 151)(13 152)(14 153)(15 154)(16 155)(17 156)(18 157)(19 158)(20 159)(21 160)(22 161)(23 162)(24 163)(25 164)(26 165)(27 166)(28 167)(29 168)(30 169)(31 170)(32 171)(33 172)(34 173)(35 174)(36 175)(37 176)(38 177)(39 178)(40 179)(41 180)(42 181)(43 182)(44 183)(45 184)(46 185)(47 186)(48 187)(49 188)(50 189)(51 190)(52 191)(53 192)(54 193)(55 194)(56 195)(57 196)(58 197)(59 198)(60 199)(61 200)(62 201)(63 202)(64 203)(65 204)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 121)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(91 128)(92 129)(93 130)(94 131)(95 132)(96 133)(97 134)(98 135)(99 136)(100 137)(101 138)(102 139)(205 358)(206 359)(207 360)(208 361)(209 362)(210 363)(211 364)(212 365)(213 366)(214 367)(215 368)(216 369)(217 370)(218 371)(219 372)(220 373)(221 374)(222 375)(223 376)(224 377)(225 378)(226 379)(227 380)(228 381)(229 382)(230 383)(231 384)(232 385)(233 386)(234 387)(235 388)(236 389)(237 390)(238 391)(239 392)(240 393)(241 394)(242 395)(243 396)(244 397)(245 398)(246 399)(247 400)(248 401)(249 402)(250 403)(251 404)(252 405)(253 406)(254 407)(255 408)(256 307)(257 308)(258 309)(259 310)(260 311)(261 312)(262 313)(263 314)(264 315)(265 316)(266 317)(267 318)(268 319)(269 320)(270 321)(271 322)(272 323)(273 324)(274 325)(275 326)(276 327)(277 328)(278 329)(279 330)(280 331)(281 332)(282 333)(283 334)(284 335)(285 336)(286 337)(287 338)(288 339)(289 340)(290 341)(291 342)(292 343)(293 344)(294 345)(295 346)(296 347)(297 348)(298 349)(299 350)(300 351)(301 352)(302 353)(303 354)(304 355)(305 356)(306 357)
(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)(307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408)
G:=sub<Sym(408)| (1,241)(2,242)(3,243)(4,244)(5,245)(6,246)(7,247)(8,248)(9,249)(10,250)(11,251)(12,252)(13,253)(14,254)(15,255)(16,256)(17,257)(18,258)(19,259)(20,260)(21,261)(22,262)(23,263)(24,264)(25,265)(26,266)(27,267)(28,268)(29,269)(30,270)(31,271)(32,272)(33,273)(34,274)(35,275)(36,276)(37,277)(38,278)(39,279)(40,280)(41,281)(42,282)(43,283)(44,284)(45,285)(46,286)(47,287)(48,288)(49,289)(50,290)(51,291)(52,292)(53,293)(54,294)(55,295)(56,296)(57,297)(58,298)(59,299)(60,300)(61,301)(62,302)(63,303)(64,304)(65,305)(66,306)(67,205)(68,206)(69,207)(70,208)(71,209)(72,210)(73,211)(74,212)(75,213)(76,214)(77,215)(78,216)(79,217)(80,218)(81,219)(82,220)(83,221)(84,222)(85,223)(86,224)(87,225)(88,226)(89,227)(90,228)(91,229)(92,230)(93,231)(94,232)(95,233)(96,234)(97,235)(98,236)(99,237)(100,238)(101,239)(102,240)(103,357)(104,358)(105,359)(106,360)(107,361)(108,362)(109,363)(110,364)(111,365)(112,366)(113,367)(114,368)(115,369)(116,370)(117,371)(118,372)(119,373)(120,374)(121,375)(122,376)(123,377)(124,378)(125,379)(126,380)(127,381)(128,382)(129,383)(130,384)(131,385)(132,386)(133,387)(134,388)(135,389)(136,390)(137,391)(138,392)(139,393)(140,394)(141,395)(142,396)(143,397)(144,398)(145,399)(146,400)(147,401)(148,402)(149,403)(150,404)(151,405)(152,406)(153,407)(154,408)(155,307)(156,308)(157,309)(158,310)(159,311)(160,312)(161,313)(162,314)(163,315)(164,316)(165,317)(166,318)(167,319)(168,320)(169,321)(170,322)(171,323)(172,324)(173,325)(174,326)(175,327)(176,328)(177,329)(178,330)(179,331)(180,332)(181,333)(182,334)(183,335)(184,336)(185,337)(186,338)(187,339)(188,340)(189,341)(190,342)(191,343)(192,344)(193,345)(194,346)(195,347)(196,348)(197,349)(198,350)(199,351)(200,352)(201,353)(202,354)(203,355)(204,356), (1,140)(2,141)(3,142)(4,143)(5,144)(6,145)(7,146)(8,147)(9,148)(10,149)(11,150)(12,151)(13,152)(14,153)(15,154)(16,155)(17,156)(18,157)(19,158)(20,159)(21,160)(22,161)(23,162)(24,163)(25,164)(26,165)(27,166)(28,167)(29,168)(30,169)(31,170)(32,171)(33,172)(34,173)(35,174)(36,175)(37,176)(38,177)(39,178)(40,179)(41,180)(42,181)(43,182)(44,183)(45,184)(46,185)(47,186)(48,187)(49,188)(50,189)(51,190)(52,191)(53,192)(54,193)(55,194)(56,195)(57,196)(58,197)(59,198)(60,199)(61,200)(62,201)(63,202)(64,203)(65,204)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,121)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,129)(93,130)(94,131)(95,132)(96,133)(97,134)(98,135)(99,136)(100,137)(101,138)(102,139)(205,358)(206,359)(207,360)(208,361)(209,362)(210,363)(211,364)(212,365)(213,366)(214,367)(215,368)(216,369)(217,370)(218,371)(219,372)(220,373)(221,374)(222,375)(223,376)(224,377)(225,378)(226,379)(227,380)(228,381)(229,382)(230,383)(231,384)(232,385)(233,386)(234,387)(235,388)(236,389)(237,390)(238,391)(239,392)(240,393)(241,394)(242,395)(243,396)(244,397)(245,398)(246,399)(247,400)(248,401)(249,402)(250,403)(251,404)(252,405)(253,406)(254,407)(255,408)(256,307)(257,308)(258,309)(259,310)(260,311)(261,312)(262,313)(263,314)(264,315)(265,316)(266,317)(267,318)(268,319)(269,320)(270,321)(271,322)(272,323)(273,324)(274,325)(275,326)(276,327)(277,328)(278,329)(279,330)(280,331)(281,332)(282,333)(283,334)(284,335)(285,336)(286,337)(287,338)(288,339)(289,340)(290,341)(291,342)(292,343)(293,344)(294,345)(295,346)(296,347)(297,348)(298,349)(299,350)(300,351)(301,352)(302,353)(303,354)(304,355)(305,356)(306,357), (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)(307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408)>;
G:=Group( (1,241)(2,242)(3,243)(4,244)(5,245)(6,246)(7,247)(8,248)(9,249)(10,250)(11,251)(12,252)(13,253)(14,254)(15,255)(16,256)(17,257)(18,258)(19,259)(20,260)(21,261)(22,262)(23,263)(24,264)(25,265)(26,266)(27,267)(28,268)(29,269)(30,270)(31,271)(32,272)(33,273)(34,274)(35,275)(36,276)(37,277)(38,278)(39,279)(40,280)(41,281)(42,282)(43,283)(44,284)(45,285)(46,286)(47,287)(48,288)(49,289)(50,290)(51,291)(52,292)(53,293)(54,294)(55,295)(56,296)(57,297)(58,298)(59,299)(60,300)(61,301)(62,302)(63,303)(64,304)(65,305)(66,306)(67,205)(68,206)(69,207)(70,208)(71,209)(72,210)(73,211)(74,212)(75,213)(76,214)(77,215)(78,216)(79,217)(80,218)(81,219)(82,220)(83,221)(84,222)(85,223)(86,224)(87,225)(88,226)(89,227)(90,228)(91,229)(92,230)(93,231)(94,232)(95,233)(96,234)(97,235)(98,236)(99,237)(100,238)(101,239)(102,240)(103,357)(104,358)(105,359)(106,360)(107,361)(108,362)(109,363)(110,364)(111,365)(112,366)(113,367)(114,368)(115,369)(116,370)(117,371)(118,372)(119,373)(120,374)(121,375)(122,376)(123,377)(124,378)(125,379)(126,380)(127,381)(128,382)(129,383)(130,384)(131,385)(132,386)(133,387)(134,388)(135,389)(136,390)(137,391)(138,392)(139,393)(140,394)(141,395)(142,396)(143,397)(144,398)(145,399)(146,400)(147,401)(148,402)(149,403)(150,404)(151,405)(152,406)(153,407)(154,408)(155,307)(156,308)(157,309)(158,310)(159,311)(160,312)(161,313)(162,314)(163,315)(164,316)(165,317)(166,318)(167,319)(168,320)(169,321)(170,322)(171,323)(172,324)(173,325)(174,326)(175,327)(176,328)(177,329)(178,330)(179,331)(180,332)(181,333)(182,334)(183,335)(184,336)(185,337)(186,338)(187,339)(188,340)(189,341)(190,342)(191,343)(192,344)(193,345)(194,346)(195,347)(196,348)(197,349)(198,350)(199,351)(200,352)(201,353)(202,354)(203,355)(204,356), (1,140)(2,141)(3,142)(4,143)(5,144)(6,145)(7,146)(8,147)(9,148)(10,149)(11,150)(12,151)(13,152)(14,153)(15,154)(16,155)(17,156)(18,157)(19,158)(20,159)(21,160)(22,161)(23,162)(24,163)(25,164)(26,165)(27,166)(28,167)(29,168)(30,169)(31,170)(32,171)(33,172)(34,173)(35,174)(36,175)(37,176)(38,177)(39,178)(40,179)(41,180)(42,181)(43,182)(44,183)(45,184)(46,185)(47,186)(48,187)(49,188)(50,189)(51,190)(52,191)(53,192)(54,193)(55,194)(56,195)(57,196)(58,197)(59,198)(60,199)(61,200)(62,201)(63,202)(64,203)(65,204)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,121)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,129)(93,130)(94,131)(95,132)(96,133)(97,134)(98,135)(99,136)(100,137)(101,138)(102,139)(205,358)(206,359)(207,360)(208,361)(209,362)(210,363)(211,364)(212,365)(213,366)(214,367)(215,368)(216,369)(217,370)(218,371)(219,372)(220,373)(221,374)(222,375)(223,376)(224,377)(225,378)(226,379)(227,380)(228,381)(229,382)(230,383)(231,384)(232,385)(233,386)(234,387)(235,388)(236,389)(237,390)(238,391)(239,392)(240,393)(241,394)(242,395)(243,396)(244,397)(245,398)(246,399)(247,400)(248,401)(249,402)(250,403)(251,404)(252,405)(253,406)(254,407)(255,408)(256,307)(257,308)(258,309)(259,310)(260,311)(261,312)(262,313)(263,314)(264,315)(265,316)(266,317)(267,318)(268,319)(269,320)(270,321)(271,322)(272,323)(273,324)(274,325)(275,326)(276,327)(277,328)(278,329)(279,330)(280,331)(281,332)(282,333)(283,334)(284,335)(285,336)(286,337)(287,338)(288,339)(289,340)(290,341)(291,342)(292,343)(293,344)(294,345)(295,346)(296,347)(297,348)(298,349)(299,350)(300,351)(301,352)(302,353)(303,354)(304,355)(305,356)(306,357), (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)(307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408) );
G=PermutationGroup([[(1,241),(2,242),(3,243),(4,244),(5,245),(6,246),(7,247),(8,248),(9,249),(10,250),(11,251),(12,252),(13,253),(14,254),(15,255),(16,256),(17,257),(18,258),(19,259),(20,260),(21,261),(22,262),(23,263),(24,264),(25,265),(26,266),(27,267),(28,268),(29,269),(30,270),(31,271),(32,272),(33,273),(34,274),(35,275),(36,276),(37,277),(38,278),(39,279),(40,280),(41,281),(42,282),(43,283),(44,284),(45,285),(46,286),(47,287),(48,288),(49,289),(50,290),(51,291),(52,292),(53,293),(54,294),(55,295),(56,296),(57,297),(58,298),(59,299),(60,300),(61,301),(62,302),(63,303),(64,304),(65,305),(66,306),(67,205),(68,206),(69,207),(70,208),(71,209),(72,210),(73,211),(74,212),(75,213),(76,214),(77,215),(78,216),(79,217),(80,218),(81,219),(82,220),(83,221),(84,222),(85,223),(86,224),(87,225),(88,226),(89,227),(90,228),(91,229),(92,230),(93,231),(94,232),(95,233),(96,234),(97,235),(98,236),(99,237),(100,238),(101,239),(102,240),(103,357),(104,358),(105,359),(106,360),(107,361),(108,362),(109,363),(110,364),(111,365),(112,366),(113,367),(114,368),(115,369),(116,370),(117,371),(118,372),(119,373),(120,374),(121,375),(122,376),(123,377),(124,378),(125,379),(126,380),(127,381),(128,382),(129,383),(130,384),(131,385),(132,386),(133,387),(134,388),(135,389),(136,390),(137,391),(138,392),(139,393),(140,394),(141,395),(142,396),(143,397),(144,398),(145,399),(146,400),(147,401),(148,402),(149,403),(150,404),(151,405),(152,406),(153,407),(154,408),(155,307),(156,308),(157,309),(158,310),(159,311),(160,312),(161,313),(162,314),(163,315),(164,316),(165,317),(166,318),(167,319),(168,320),(169,321),(170,322),(171,323),(172,324),(173,325),(174,326),(175,327),(176,328),(177,329),(178,330),(179,331),(180,332),(181,333),(182,334),(183,335),(184,336),(185,337),(186,338),(187,339),(188,340),(189,341),(190,342),(191,343),(192,344),(193,345),(194,346),(195,347),(196,348),(197,349),(198,350),(199,351),(200,352),(201,353),(202,354),(203,355),(204,356)], [(1,140),(2,141),(3,142),(4,143),(5,144),(6,145),(7,146),(8,147),(9,148),(10,149),(11,150),(12,151),(13,152),(14,153),(15,154),(16,155),(17,156),(18,157),(19,158),(20,159),(21,160),(22,161),(23,162),(24,163),(25,164),(26,165),(27,166),(28,167),(29,168),(30,169),(31,170),(32,171),(33,172),(34,173),(35,174),(36,175),(37,176),(38,177),(39,178),(40,179),(41,180),(42,181),(43,182),(44,183),(45,184),(46,185),(47,186),(48,187),(49,188),(50,189),(51,190),(52,191),(53,192),(54,193),(55,194),(56,195),(57,196),(58,197),(59,198),(60,199),(61,200),(62,201),(63,202),(64,203),(65,204),(66,103),(67,104),(68,105),(69,106),(70,107),(71,108),(72,109),(73,110),(74,111),(75,112),(76,113),(77,114),(78,115),(79,116),(80,117),(81,118),(82,119),(83,120),(84,121),(85,122),(86,123),(87,124),(88,125),(89,126),(90,127),(91,128),(92,129),(93,130),(94,131),(95,132),(96,133),(97,134),(98,135),(99,136),(100,137),(101,138),(102,139),(205,358),(206,359),(207,360),(208,361),(209,362),(210,363),(211,364),(212,365),(213,366),(214,367),(215,368),(216,369),(217,370),(218,371),(219,372),(220,373),(221,374),(222,375),(223,376),(224,377),(225,378),(226,379),(227,380),(228,381),(229,382),(230,383),(231,384),(232,385),(233,386),(234,387),(235,388),(236,389),(237,390),(238,391),(239,392),(240,393),(241,394),(242,395),(243,396),(244,397),(245,398),(246,399),(247,400),(248,401),(249,402),(250,403),(251,404),(252,405),(253,406),(254,407),(255,408),(256,307),(257,308),(258,309),(259,310),(260,311),(261,312),(262,313),(263,314),(264,315),(265,316),(266,317),(267,318),(268,319),(269,320),(270,321),(271,322),(272,323),(273,324),(274,325),(275,326),(276,327),(277,328),(278,329),(279,330),(280,331),(281,332),(282,333),(283,334),(284,335),(285,336),(286,337),(287,338),(288,339),(289,340),(290,341),(291,342),(292,343),(293,344),(294,345),(295,346),(296,347),(297,348),(298,349),(299,350),(300,351),(301,352),(302,353),(303,354),(304,355),(305,356),(306,357)], [(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),(307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408)]])
408 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 6A | ··· | 6N | 17A | ··· | 17P | 34A | ··· | 34DH | 51A | ··· | 51AF | 102A | ··· | 102HP |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 6 | ··· | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
408 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C17 | C34 | C51 | C102 |
| kernel | C22×C102 | C2×C102 | C22×C34 | C2×C34 | C22×C6 | C2×C6 | C23 | C22 |
| # reps | 1 | 7 | 2 | 14 | 16 | 112 | 32 | 224 |
Matrix representation of C22×C102 ►in GL3(𝔽103) generated by
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 102 |
| 1 | 0 | 0 |
| 0 | 102 | 0 |
| 0 | 0 | 1 |
| 80 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 78 |
G:=sub<GL(3,GF(103))| [1,0,0,0,1,0,0,0,102],[1,0,0,0,102,0,0,0,1],[80,0,0,0,5,0,0,0,78] >;
C22×C102 in GAP, Magma, Sage, TeX
C_2^2\times C_{102} % in TeX
G:=Group("C2^2xC102"); // GroupNames label
G:=SmallGroup(408,46);
// by ID
G=gap.SmallGroup(408,46);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^102=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations