direct product, abelian, monomial, 2-elementary
Aliases: C22×C104, SmallGroup(416,190)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C104 |
| C1 — C22×C104 |
| C1 — C22×C104 |
Generators and relations for C22×C104
G = < a,b,c | a2=b2=c104=1, ab=ba, ac=ca, bc=cb >
Subgroups: 76, all normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, C23, C13, C2×C8, C22×C4, C26, C26, C22×C8, C52, C52, C2×C26, C104, C2×C52, C22×C26, C2×C104, C22×C52, C22×C104
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C13, C2×C8, C22×C4, C26, C22×C8, C52, C2×C26, C104, C2×C52, C22×C26, C2×C104, C22×C52, C22×C104
►Regular action on 416 points
Generators in S416
(1 328)(2 329)(3 330)(4 331)(5 332)(6 333)(7 334)(8 335)(9 336)(10 337)(11 338)(12 339)(13 340)(14 341)(15 342)(16 343)(17 344)(18 345)(19 346)(20 347)(21 348)(22 349)(23 350)(24 351)(25 352)(26 353)(27 354)(28 355)(29 356)(30 357)(31 358)(32 359)(33 360)(34 361)(35 362)(36 363)(37 364)(38 365)(39 366)(40 367)(41 368)(42 369)(43 370)(44 371)(45 372)(46 373)(47 374)(48 375)(49 376)(50 377)(51 378)(52 379)(53 380)(54 381)(55 382)(56 383)(57 384)(58 385)(59 386)(60 387)(61 388)(62 389)(63 390)(64 391)(65 392)(66 393)(67 394)(68 395)(69 396)(70 397)(71 398)(72 399)(73 400)(74 401)(75 402)(76 403)(77 404)(78 405)(79 406)(80 407)(81 408)(82 409)(83 410)(84 411)(85 412)(86 413)(87 414)(88 415)(89 416)(90 313)(91 314)(92 315)(93 316)(94 317)(95 318)(96 319)(97 320)(98 321)(99 322)(100 323)(101 324)(102 325)(103 326)(104 327)(105 215)(106 216)(107 217)(108 218)(109 219)(110 220)(111 221)(112 222)(113 223)(114 224)(115 225)(116 226)(117 227)(118 228)(119 229)(120 230)(121 231)(122 232)(123 233)(124 234)(125 235)(126 236)(127 237)(128 238)(129 239)(130 240)(131 241)(132 242)(133 243)(134 244)(135 245)(136 246)(137 247)(138 248)(139 249)(140 250)(141 251)(142 252)(143 253)(144 254)(145 255)(146 256)(147 257)(148 258)(149 259)(150 260)(151 261)(152 262)(153 263)(154 264)(155 265)(156 266)(157 267)(158 268)(159 269)(160 270)(161 271)(162 272)(163 273)(164 274)(165 275)(166 276)(167 277)(168 278)(169 279)(170 280)(171 281)(172 282)(173 283)(174 284)(175 285)(176 286)(177 287)(178 288)(179 289)(180 290)(181 291)(182 292)(183 293)(184 294)(185 295)(186 296)(187 297)(188 298)(189 299)(190 300)(191 301)(192 302)(193 303)(194 304)(195 305)(196 306)(197 307)(198 308)(199 309)(200 310)(201 311)(202 312)(203 209)(204 210)(205 211)(206 212)(207 213)(208 214)
(1 131)(2 132)(3 133)(4 134)(5 135)(6 136)(7 137)(8 138)(9 139)(10 140)(11 141)(12 142)(13 143)(14 144)(15 145)(16 146)(17 147)(18 148)(19 149)(20 150)(21 151)(22 152)(23 153)(24 154)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 161)(32 162)(33 163)(34 164)(35 165)(36 166)(37 167)(38 168)(39 169)(40 170)(41 171)(42 172)(43 173)(44 174)(45 175)(46 176)(47 177)(48 178)(49 179)(50 180)(51 181)(52 182)(53 183)(54 184)(55 185)(56 186)(57 187)(58 188)(59 189)(60 190)(61 191)(62 192)(63 193)(64 194)(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)(79 105)(80 106)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)(101 127)(102 128)(103 129)(104 130)(209 400)(210 401)(211 402)(212 403)(213 404)(214 405)(215 406)(216 407)(217 408)(218 409)(219 410)(220 411)(221 412)(222 413)(223 414)(224 415)(225 416)(226 313)(227 314)(228 315)(229 316)(230 317)(231 318)(232 319)(233 320)(234 321)(235 322)(236 323)(237 324)(238 325)(239 326)(240 327)(241 328)(242 329)(243 330)(244 331)(245 332)(246 333)(247 334)(248 335)(249 336)(250 337)(251 338)(252 339)(253 340)(254 341)(255 342)(256 343)(257 344)(258 345)(259 346)(260 347)(261 348)(262 349)(263 350)(264 351)(265 352)(266 353)(267 354)(268 355)(269 356)(270 357)(271 358)(272 359)(273 360)(274 361)(275 362)(276 363)(277 364)(278 365)(279 366)(280 367)(281 368)(282 369)(283 370)(284 371)(285 372)(286 373)(287 374)(288 375)(289 376)(290 377)(291 378)(292 379)(293 380)(294 381)(295 382)(296 383)(297 384)(298 385)(299 386)(300 387)(301 388)(302 389)(303 390)(304 391)(305 392)(306 393)(307 394)(308 395)(309 396)(310 397)(311 398)(312 399)
(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)(313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416)
G:=sub<Sym(416)| (1,328)(2,329)(3,330)(4,331)(5,332)(6,333)(7,334)(8,335)(9,336)(10,337)(11,338)(12,339)(13,340)(14,341)(15,342)(16,343)(17,344)(18,345)(19,346)(20,347)(21,348)(22,349)(23,350)(24,351)(25,352)(26,353)(27,354)(28,355)(29,356)(30,357)(31,358)(32,359)(33,360)(34,361)(35,362)(36,363)(37,364)(38,365)(39,366)(40,367)(41,368)(42,369)(43,370)(44,371)(45,372)(46,373)(47,374)(48,375)(49,376)(50,377)(51,378)(52,379)(53,380)(54,381)(55,382)(56,383)(57,384)(58,385)(59,386)(60,387)(61,388)(62,389)(63,390)(64,391)(65,392)(66,393)(67,394)(68,395)(69,396)(70,397)(71,398)(72,399)(73,400)(74,401)(75,402)(76,403)(77,404)(78,405)(79,406)(80,407)(81,408)(82,409)(83,410)(84,411)(85,412)(86,413)(87,414)(88,415)(89,416)(90,313)(91,314)(92,315)(93,316)(94,317)(95,318)(96,319)(97,320)(98,321)(99,322)(100,323)(101,324)(102,325)(103,326)(104,327)(105,215)(106,216)(107,217)(108,218)(109,219)(110,220)(111,221)(112,222)(113,223)(114,224)(115,225)(116,226)(117,227)(118,228)(119,229)(120,230)(121,231)(122,232)(123,233)(124,234)(125,235)(126,236)(127,237)(128,238)(129,239)(130,240)(131,241)(132,242)(133,243)(134,244)(135,245)(136,246)(137,247)(138,248)(139,249)(140,250)(141,251)(142,252)(143,253)(144,254)(145,255)(146,256)(147,257)(148,258)(149,259)(150,260)(151,261)(152,262)(153,263)(154,264)(155,265)(156,266)(157,267)(158,268)(159,269)(160,270)(161,271)(162,272)(163,273)(164,274)(165,275)(166,276)(167,277)(168,278)(169,279)(170,280)(171,281)(172,282)(173,283)(174,284)(175,285)(176,286)(177,287)(178,288)(179,289)(180,290)(181,291)(182,292)(183,293)(184,294)(185,295)(186,296)(187,297)(188,298)(189,299)(190,300)(191,301)(192,302)(193,303)(194,304)(195,305)(196,306)(197,307)(198,308)(199,309)(200,310)(201,311)(202,312)(203,209)(204,210)(205,211)(206,212)(207,213)(208,214), (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,141)(12,142)(13,143)(14,144)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,151)(22,152)(23,153)(24,154)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,175)(46,176)(47,177)(48,178)(49,179)(50,180)(51,181)(52,182)(53,183)(54,184)(55,185)(56,186)(57,187)(58,188)(59,189)(60,190)(61,191)(62,192)(63,193)(64,194)(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)(79,105)(80,106)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,127)(102,128)(103,129)(104,130)(209,400)(210,401)(211,402)(212,403)(213,404)(214,405)(215,406)(216,407)(217,408)(218,409)(219,410)(220,411)(221,412)(222,413)(223,414)(224,415)(225,416)(226,313)(227,314)(228,315)(229,316)(230,317)(231,318)(232,319)(233,320)(234,321)(235,322)(236,323)(237,324)(238,325)(239,326)(240,327)(241,328)(242,329)(243,330)(244,331)(245,332)(246,333)(247,334)(248,335)(249,336)(250,337)(251,338)(252,339)(253,340)(254,341)(255,342)(256,343)(257,344)(258,345)(259,346)(260,347)(261,348)(262,349)(263,350)(264,351)(265,352)(266,353)(267,354)(268,355)(269,356)(270,357)(271,358)(272,359)(273,360)(274,361)(275,362)(276,363)(277,364)(278,365)(279,366)(280,367)(281,368)(282,369)(283,370)(284,371)(285,372)(286,373)(287,374)(288,375)(289,376)(290,377)(291,378)(292,379)(293,380)(294,381)(295,382)(296,383)(297,384)(298,385)(299,386)(300,387)(301,388)(302,389)(303,390)(304,391)(305,392)(306,393)(307,394)(308,395)(309,396)(310,397)(311,398)(312,399), (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)(313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416)>;
G:=Group( (1,328)(2,329)(3,330)(4,331)(5,332)(6,333)(7,334)(8,335)(9,336)(10,337)(11,338)(12,339)(13,340)(14,341)(15,342)(16,343)(17,344)(18,345)(19,346)(20,347)(21,348)(22,349)(23,350)(24,351)(25,352)(26,353)(27,354)(28,355)(29,356)(30,357)(31,358)(32,359)(33,360)(34,361)(35,362)(36,363)(37,364)(38,365)(39,366)(40,367)(41,368)(42,369)(43,370)(44,371)(45,372)(46,373)(47,374)(48,375)(49,376)(50,377)(51,378)(52,379)(53,380)(54,381)(55,382)(56,383)(57,384)(58,385)(59,386)(60,387)(61,388)(62,389)(63,390)(64,391)(65,392)(66,393)(67,394)(68,395)(69,396)(70,397)(71,398)(72,399)(73,400)(74,401)(75,402)(76,403)(77,404)(78,405)(79,406)(80,407)(81,408)(82,409)(83,410)(84,411)(85,412)(86,413)(87,414)(88,415)(89,416)(90,313)(91,314)(92,315)(93,316)(94,317)(95,318)(96,319)(97,320)(98,321)(99,322)(100,323)(101,324)(102,325)(103,326)(104,327)(105,215)(106,216)(107,217)(108,218)(109,219)(110,220)(111,221)(112,222)(113,223)(114,224)(115,225)(116,226)(117,227)(118,228)(119,229)(120,230)(121,231)(122,232)(123,233)(124,234)(125,235)(126,236)(127,237)(128,238)(129,239)(130,240)(131,241)(132,242)(133,243)(134,244)(135,245)(136,246)(137,247)(138,248)(139,249)(140,250)(141,251)(142,252)(143,253)(144,254)(145,255)(146,256)(147,257)(148,258)(149,259)(150,260)(151,261)(152,262)(153,263)(154,264)(155,265)(156,266)(157,267)(158,268)(159,269)(160,270)(161,271)(162,272)(163,273)(164,274)(165,275)(166,276)(167,277)(168,278)(169,279)(170,280)(171,281)(172,282)(173,283)(174,284)(175,285)(176,286)(177,287)(178,288)(179,289)(180,290)(181,291)(182,292)(183,293)(184,294)(185,295)(186,296)(187,297)(188,298)(189,299)(190,300)(191,301)(192,302)(193,303)(194,304)(195,305)(196,306)(197,307)(198,308)(199,309)(200,310)(201,311)(202,312)(203,209)(204,210)(205,211)(206,212)(207,213)(208,214), (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,141)(12,142)(13,143)(14,144)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,151)(22,152)(23,153)(24,154)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,175)(46,176)(47,177)(48,178)(49,179)(50,180)(51,181)(52,182)(53,183)(54,184)(55,185)(56,186)(57,187)(58,188)(59,189)(60,190)(61,191)(62,192)(63,193)(64,194)(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)(79,105)(80,106)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,127)(102,128)(103,129)(104,130)(209,400)(210,401)(211,402)(212,403)(213,404)(214,405)(215,406)(216,407)(217,408)(218,409)(219,410)(220,411)(221,412)(222,413)(223,414)(224,415)(225,416)(226,313)(227,314)(228,315)(229,316)(230,317)(231,318)(232,319)(233,320)(234,321)(235,322)(236,323)(237,324)(238,325)(239,326)(240,327)(241,328)(242,329)(243,330)(244,331)(245,332)(246,333)(247,334)(248,335)(249,336)(250,337)(251,338)(252,339)(253,340)(254,341)(255,342)(256,343)(257,344)(258,345)(259,346)(260,347)(261,348)(262,349)(263,350)(264,351)(265,352)(266,353)(267,354)(268,355)(269,356)(270,357)(271,358)(272,359)(273,360)(274,361)(275,362)(276,363)(277,364)(278,365)(279,366)(280,367)(281,368)(282,369)(283,370)(284,371)(285,372)(286,373)(287,374)(288,375)(289,376)(290,377)(291,378)(292,379)(293,380)(294,381)(295,382)(296,383)(297,384)(298,385)(299,386)(300,387)(301,388)(302,389)(303,390)(304,391)(305,392)(306,393)(307,394)(308,395)(309,396)(310,397)(311,398)(312,399), (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)(313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416) );
G=PermutationGroup([[(1,328),(2,329),(3,330),(4,331),(5,332),(6,333),(7,334),(8,335),(9,336),(10,337),(11,338),(12,339),(13,340),(14,341),(15,342),(16,343),(17,344),(18,345),(19,346),(20,347),(21,348),(22,349),(23,350),(24,351),(25,352),(26,353),(27,354),(28,355),(29,356),(30,357),(31,358),(32,359),(33,360),(34,361),(35,362),(36,363),(37,364),(38,365),(39,366),(40,367),(41,368),(42,369),(43,370),(44,371),(45,372),(46,373),(47,374),(48,375),(49,376),(50,377),(51,378),(52,379),(53,380),(54,381),(55,382),(56,383),(57,384),(58,385),(59,386),(60,387),(61,388),(62,389),(63,390),(64,391),(65,392),(66,393),(67,394),(68,395),(69,396),(70,397),(71,398),(72,399),(73,400),(74,401),(75,402),(76,403),(77,404),(78,405),(79,406),(80,407),(81,408),(82,409),(83,410),(84,411),(85,412),(86,413),(87,414),(88,415),(89,416),(90,313),(91,314),(92,315),(93,316),(94,317),(95,318),(96,319),(97,320),(98,321),(99,322),(100,323),(101,324),(102,325),(103,326),(104,327),(105,215),(106,216),(107,217),(108,218),(109,219),(110,220),(111,221),(112,222),(113,223),(114,224),(115,225),(116,226),(117,227),(118,228),(119,229),(120,230),(121,231),(122,232),(123,233),(124,234),(125,235),(126,236),(127,237),(128,238),(129,239),(130,240),(131,241),(132,242),(133,243),(134,244),(135,245),(136,246),(137,247),(138,248),(139,249),(140,250),(141,251),(142,252),(143,253),(144,254),(145,255),(146,256),(147,257),(148,258),(149,259),(150,260),(151,261),(152,262),(153,263),(154,264),(155,265),(156,266),(157,267),(158,268),(159,269),(160,270),(161,271),(162,272),(163,273),(164,274),(165,275),(166,276),(167,277),(168,278),(169,279),(170,280),(171,281),(172,282),(173,283),(174,284),(175,285),(176,286),(177,287),(178,288),(179,289),(180,290),(181,291),(182,292),(183,293),(184,294),(185,295),(186,296),(187,297),(188,298),(189,299),(190,300),(191,301),(192,302),(193,303),(194,304),(195,305),(196,306),(197,307),(198,308),(199,309),(200,310),(201,311),(202,312),(203,209),(204,210),(205,211),(206,212),(207,213),(208,214)], [(1,131),(2,132),(3,133),(4,134),(5,135),(6,136),(7,137),(8,138),(9,139),(10,140),(11,141),(12,142),(13,143),(14,144),(15,145),(16,146),(17,147),(18,148),(19,149),(20,150),(21,151),(22,152),(23,153),(24,154),(25,155),(26,156),(27,157),(28,158),(29,159),(30,160),(31,161),(32,162),(33,163),(34,164),(35,165),(36,166),(37,167),(38,168),(39,169),(40,170),(41,171),(42,172),(43,173),(44,174),(45,175),(46,176),(47,177),(48,178),(49,179),(50,180),(51,181),(52,182),(53,183),(54,184),(55,185),(56,186),(57,187),(58,188),(59,189),(60,190),(61,191),(62,192),(63,193),(64,194),(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),(79,105),(80,106),(81,107),(82,108),(83,109),(84,110),(85,111),(86,112),(87,113),(88,114),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,121),(96,122),(97,123),(98,124),(99,125),(100,126),(101,127),(102,128),(103,129),(104,130),(209,400),(210,401),(211,402),(212,403),(213,404),(214,405),(215,406),(216,407),(217,408),(218,409),(219,410),(220,411),(221,412),(222,413),(223,414),(224,415),(225,416),(226,313),(227,314),(228,315),(229,316),(230,317),(231,318),(232,319),(233,320),(234,321),(235,322),(236,323),(237,324),(238,325),(239,326),(240,327),(241,328),(242,329),(243,330),(244,331),(245,332),(246,333),(247,334),(248,335),(249,336),(250,337),(251,338),(252,339),(253,340),(254,341),(255,342),(256,343),(257,344),(258,345),(259,346),(260,347),(261,348),(262,349),(263,350),(264,351),(265,352),(266,353),(267,354),(268,355),(269,356),(270,357),(271,358),(272,359),(273,360),(274,361),(275,362),(276,363),(277,364),(278,365),(279,366),(280,367),(281,368),(282,369),(283,370),(284,371),(285,372),(286,373),(287,374),(288,375),(289,376),(290,377),(291,378),(292,379),(293,380),(294,381),(295,382),(296,383),(297,384),(298,385),(299,386),(300,387),(301,388),(302,389),(303,390),(304,391),(305,392),(306,393),(307,394),(308,395),(309,396),(310,397),(311,398),(312,399)], [(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),(313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416)]])
416 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 8A | ··· | 8P | 13A | ··· | 13L | 26A | ··· | 26CF | 52A | ··· | 52CR | 104A | ··· | 104GJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
416 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C13 | C26 | C26 | C52 | C52 | C104 |
| kernel | C22×C104 | C2×C104 | C22×C52 | C2×C52 | C22×C26 | C2×C26 | C22×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 6 | 2 | 16 | 12 | 72 | 12 | 72 | 24 | 192 |
Matrix representation of C22×C104 ►in GL3(𝔽313) generated by
| 1 | 0 | 0 |
| 0 | 312 | 0 |
| 0 | 0 | 312 |
| 1 | 0 | 0 |
| 0 | 312 | 0 |
| 0 | 0 | 1 |
| 172 | 0 | 0 |
| 0 | 39 | 0 |
| 0 | 0 | 39 |
G:=sub<GL(3,GF(313))| [1,0,0,0,312,0,0,0,312],[1,0,0,0,312,0,0,0,1],[172,0,0,0,39,0,0,0,39] >;
C22×C104 in GAP, Magma, Sage, TeX
C_2^2\times C_{104} % in TeX
G:=Group("C2^2xC104"); // GroupNames label
G:=SmallGroup(416,190);
// by ID
G=gap.SmallGroup(416,190);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,624,88]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^104=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations