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G = C22×C104order 416 = 25·13

Abelian group of type [2,2,104]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C104, SmallGroup(416,190)

Series: Derived Chief Lower central Upper central

C1 — C22×C104
C1C2C4C52C104C2×C104 — 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 [×6], C4, C4 [×3], C22 [×7], C8 [×4], C2×C4 [×6], C23, C13, C2×C8 [×6], C22×C4, C26, C26 [×6], C22×C8, C52, C52 [×3], C2×C26 [×7], C104 [×4], C2×C52 [×6], C22×C26, C2×C104 [×6], C22×C52, C22×C104
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C13, C2×C8 [×6], C22×C4, C26 [×7], C22×C8, C52 [×4], C2×C26 [×7], C104 [×4], C2×C52 [×6], C22×C26, C2×C104 [×6], C22×C52, C22×C104

Smallest permutation representation of C22×C104
Regular action on 416 points
Generators in S416
(1 266)(2 267)(3 268)(4 269)(5 270)(6 271)(7 272)(8 273)(9 274)(10 275)(11 276)(12 277)(13 278)(14 279)(15 280)(16 281)(17 282)(18 283)(19 284)(20 285)(21 286)(22 287)(23 288)(24 289)(25 290)(26 291)(27 292)(28 293)(29 294)(30 295)(31 296)(32 297)(33 298)(34 299)(35 300)(36 301)(37 302)(38 303)(39 304)(40 305)(41 306)(42 307)(43 308)(44 309)(45 310)(46 311)(47 312)(48 209)(49 210)(50 211)(51 212)(52 213)(53 214)(54 215)(55 216)(56 217)(57 218)(58 219)(59 220)(60 221)(61 222)(62 223)(63 224)(64 225)(65 226)(66 227)(67 228)(68 229)(69 230)(70 231)(71 232)(72 233)(73 234)(74 235)(75 236)(76 237)(77 238)(78 239)(79 240)(80 241)(81 242)(82 243)(83 244)(84 245)(85 246)(86 247)(87 248)(88 249)(89 250)(90 251)(91 252)(92 253)(93 254)(94 255)(95 256)(96 257)(97 258)(98 259)(99 260)(100 261)(101 262)(102 263)(103 264)(104 265)(105 403)(106 404)(107 405)(108 406)(109 407)(110 408)(111 409)(112 410)(113 411)(114 412)(115 413)(116 414)(117 415)(118 416)(119 313)(120 314)(121 315)(122 316)(123 317)(124 318)(125 319)(126 320)(127 321)(128 322)(129 323)(130 324)(131 325)(132 326)(133 327)(134 328)(135 329)(136 330)(137 331)(138 332)(139 333)(140 334)(141 335)(142 336)(143 337)(144 338)(145 339)(146 340)(147 341)(148 342)(149 343)(150 344)(151 345)(152 346)(153 347)(154 348)(155 349)(156 350)(157 351)(158 352)(159 353)(160 354)(161 355)(162 356)(163 357)(164 358)(165 359)(166 360)(167 361)(168 362)(169 363)(170 364)(171 365)(172 366)(173 367)(174 368)(175 369)(176 370)(177 371)(178 372)(179 373)(180 374)(181 375)(182 376)(183 377)(184 378)(185 379)(186 380)(187 381)(188 382)(189 383)(190 384)(191 385)(192 386)(193 387)(194 388)(195 389)(196 390)(197 391)(198 392)(199 393)(200 394)(201 395)(202 396)(203 397)(204 398)(205 399)(206 400)(207 401)(208 402)
(1 344)(2 345)(3 346)(4 347)(5 348)(6 349)(7 350)(8 351)(9 352)(10 353)(11 354)(12 355)(13 356)(14 357)(15 358)(16 359)(17 360)(18 361)(19 362)(20 363)(21 364)(22 365)(23 366)(24 367)(25 368)(26 369)(27 370)(28 371)(29 372)(30 373)(31 374)(32 375)(33 376)(34 377)(35 378)(36 379)(37 380)(38 381)(39 382)(40 383)(41 384)(42 385)(43 386)(44 387)(45 388)(46 389)(47 390)(48 391)(49 392)(50 393)(51 394)(52 395)(53 396)(54 397)(55 398)(56 399)(57 400)(58 401)(59 402)(60 403)(61 404)(62 405)(63 406)(64 407)(65 408)(66 409)(67 410)(68 411)(69 412)(70 413)(71 414)(72 415)(73 416)(74 313)(75 314)(76 315)(77 316)(78 317)(79 318)(80 319)(81 320)(82 321)(83 322)(84 323)(85 324)(86 325)(87 326)(88 327)(89 328)(90 329)(91 330)(92 331)(93 332)(94 333)(95 334)(96 335)(97 336)(98 337)(99 338)(100 339)(101 340)(102 341)(103 342)(104 343)(105 221)(106 222)(107 223)(108 224)(109 225)(110 226)(111 227)(112 228)(113 229)(114 230)(115 231)(116 232)(117 233)(118 234)(119 235)(120 236)(121 237)(122 238)(123 239)(124 240)(125 241)(126 242)(127 243)(128 244)(129 245)(130 246)(131 247)(132 248)(133 249)(134 250)(135 251)(136 252)(137 253)(138 254)(139 255)(140 256)(141 257)(142 258)(143 259)(144 260)(145 261)(146 262)(147 263)(148 264)(149 265)(150 266)(151 267)(152 268)(153 269)(154 270)(155 271)(156 272)(157 273)(158 274)(159 275)(160 276)(161 277)(162 278)(163 279)(164 280)(165 281)(166 282)(167 283)(168 284)(169 285)(170 286)(171 287)(172 288)(173 289)(174 290)(175 291)(176 292)(177 293)(178 294)(179 295)(180 296)(181 297)(182 298)(183 299)(184 300)(185 301)(186 302)(187 303)(188 304)(189 305)(190 306)(191 307)(192 308)(193 309)(194 310)(195 311)(196 312)(197 209)(198 210)(199 211)(200 212)(201 213)(202 214)(203 215)(204 216)(205 217)(206 218)(207 219)(208 220)
(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)| (1,266)(2,267)(3,268)(4,269)(5,270)(6,271)(7,272)(8,273)(9,274)(10,275)(11,276)(12,277)(13,278)(14,279)(15,280)(16,281)(17,282)(18,283)(19,284)(20,285)(21,286)(22,287)(23,288)(24,289)(25,290)(26,291)(27,292)(28,293)(29,294)(30,295)(31,296)(32,297)(33,298)(34,299)(35,300)(36,301)(37,302)(38,303)(39,304)(40,305)(41,306)(42,307)(43,308)(44,309)(45,310)(46,311)(47,312)(48,209)(49,210)(50,211)(51,212)(52,213)(53,214)(54,215)(55,216)(56,217)(57,218)(58,219)(59,220)(60,221)(61,222)(62,223)(63,224)(64,225)(65,226)(66,227)(67,228)(68,229)(69,230)(70,231)(71,232)(72,233)(73,234)(74,235)(75,236)(76,237)(77,238)(78,239)(79,240)(80,241)(81,242)(82,243)(83,244)(84,245)(85,246)(86,247)(87,248)(88,249)(89,250)(90,251)(91,252)(92,253)(93,254)(94,255)(95,256)(96,257)(97,258)(98,259)(99,260)(100,261)(101,262)(102,263)(103,264)(104,265)(105,403)(106,404)(107,405)(108,406)(109,407)(110,408)(111,409)(112,410)(113,411)(114,412)(115,413)(116,414)(117,415)(118,416)(119,313)(120,314)(121,315)(122,316)(123,317)(124,318)(125,319)(126,320)(127,321)(128,322)(129,323)(130,324)(131,325)(132,326)(133,327)(134,328)(135,329)(136,330)(137,331)(138,332)(139,333)(140,334)(141,335)(142,336)(143,337)(144,338)(145,339)(146,340)(147,341)(148,342)(149,343)(150,344)(151,345)(152,346)(153,347)(154,348)(155,349)(156,350)(157,351)(158,352)(159,353)(160,354)(161,355)(162,356)(163,357)(164,358)(165,359)(166,360)(167,361)(168,362)(169,363)(170,364)(171,365)(172,366)(173,367)(174,368)(175,369)(176,370)(177,371)(178,372)(179,373)(180,374)(181,375)(182,376)(183,377)(184,378)(185,379)(186,380)(187,381)(188,382)(189,383)(190,384)(191,385)(192,386)(193,387)(194,388)(195,389)(196,390)(197,391)(198,392)(199,393)(200,394)(201,395)(202,396)(203,397)(204,398)(205,399)(206,400)(207,401)(208,402), (1,344)(2,345)(3,346)(4,347)(5,348)(6,349)(7,350)(8,351)(9,352)(10,353)(11,354)(12,355)(13,356)(14,357)(15,358)(16,359)(17,360)(18,361)(19,362)(20,363)(21,364)(22,365)(23,366)(24,367)(25,368)(26,369)(27,370)(28,371)(29,372)(30,373)(31,374)(32,375)(33,376)(34,377)(35,378)(36,379)(37,380)(38,381)(39,382)(40,383)(41,384)(42,385)(43,386)(44,387)(45,388)(46,389)(47,390)(48,391)(49,392)(50,393)(51,394)(52,395)(53,396)(54,397)(55,398)(56,399)(57,400)(58,401)(59,402)(60,403)(61,404)(62,405)(63,406)(64,407)(65,408)(66,409)(67,410)(68,411)(69,412)(70,413)(71,414)(72,415)(73,416)(74,313)(75,314)(76,315)(77,316)(78,317)(79,318)(80,319)(81,320)(82,321)(83,322)(84,323)(85,324)(86,325)(87,326)(88,327)(89,328)(90,329)(91,330)(92,331)(93,332)(94,333)(95,334)(96,335)(97,336)(98,337)(99,338)(100,339)(101,340)(102,341)(103,342)(104,343)(105,221)(106,222)(107,223)(108,224)(109,225)(110,226)(111,227)(112,228)(113,229)(114,230)(115,231)(116,232)(117,233)(118,234)(119,235)(120,236)(121,237)(122,238)(123,239)(124,240)(125,241)(126,242)(127,243)(128,244)(129,245)(130,246)(131,247)(132,248)(133,249)(134,250)(135,251)(136,252)(137,253)(138,254)(139,255)(140,256)(141,257)(142,258)(143,259)(144,260)(145,261)(146,262)(147,263)(148,264)(149,265)(150,266)(151,267)(152,268)(153,269)(154,270)(155,271)(156,272)(157,273)(158,274)(159,275)(160,276)(161,277)(162,278)(163,279)(164,280)(165,281)(166,282)(167,283)(168,284)(169,285)(170,286)(171,287)(172,288)(173,289)(174,290)(175,291)(176,292)(177,293)(178,294)(179,295)(180,296)(181,297)(182,298)(183,299)(184,300)(185,301)(186,302)(187,303)(188,304)(189,305)(190,306)(191,307)(192,308)(193,309)(194,310)(195,311)(196,312)(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)(206,218)(207,219)(208,220), (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( (1,266)(2,267)(3,268)(4,269)(5,270)(6,271)(7,272)(8,273)(9,274)(10,275)(11,276)(12,277)(13,278)(14,279)(15,280)(16,281)(17,282)(18,283)(19,284)(20,285)(21,286)(22,287)(23,288)(24,289)(25,290)(26,291)(27,292)(28,293)(29,294)(30,295)(31,296)(32,297)(33,298)(34,299)(35,300)(36,301)(37,302)(38,303)(39,304)(40,305)(41,306)(42,307)(43,308)(44,309)(45,310)(46,311)(47,312)(48,209)(49,210)(50,211)(51,212)(52,213)(53,214)(54,215)(55,216)(56,217)(57,218)(58,219)(59,220)(60,221)(61,222)(62,223)(63,224)(64,225)(65,226)(66,227)(67,228)(68,229)(69,230)(70,231)(71,232)(72,233)(73,234)(74,235)(75,236)(76,237)(77,238)(78,239)(79,240)(80,241)(81,242)(82,243)(83,244)(84,245)(85,246)(86,247)(87,248)(88,249)(89,250)(90,251)(91,252)(92,253)(93,254)(94,255)(95,256)(96,257)(97,258)(98,259)(99,260)(100,261)(101,262)(102,263)(103,264)(104,265)(105,403)(106,404)(107,405)(108,406)(109,407)(110,408)(111,409)(112,410)(113,411)(114,412)(115,413)(116,414)(117,415)(118,416)(119,313)(120,314)(121,315)(122,316)(123,317)(124,318)(125,319)(126,320)(127,321)(128,322)(129,323)(130,324)(131,325)(132,326)(133,327)(134,328)(135,329)(136,330)(137,331)(138,332)(139,333)(140,334)(141,335)(142,336)(143,337)(144,338)(145,339)(146,340)(147,341)(148,342)(149,343)(150,344)(151,345)(152,346)(153,347)(154,348)(155,349)(156,350)(157,351)(158,352)(159,353)(160,354)(161,355)(162,356)(163,357)(164,358)(165,359)(166,360)(167,361)(168,362)(169,363)(170,364)(171,365)(172,366)(173,367)(174,368)(175,369)(176,370)(177,371)(178,372)(179,373)(180,374)(181,375)(182,376)(183,377)(184,378)(185,379)(186,380)(187,381)(188,382)(189,383)(190,384)(191,385)(192,386)(193,387)(194,388)(195,389)(196,390)(197,391)(198,392)(199,393)(200,394)(201,395)(202,396)(203,397)(204,398)(205,399)(206,400)(207,401)(208,402), (1,344)(2,345)(3,346)(4,347)(5,348)(6,349)(7,350)(8,351)(9,352)(10,353)(11,354)(12,355)(13,356)(14,357)(15,358)(16,359)(17,360)(18,361)(19,362)(20,363)(21,364)(22,365)(23,366)(24,367)(25,368)(26,369)(27,370)(28,371)(29,372)(30,373)(31,374)(32,375)(33,376)(34,377)(35,378)(36,379)(37,380)(38,381)(39,382)(40,383)(41,384)(42,385)(43,386)(44,387)(45,388)(46,389)(47,390)(48,391)(49,392)(50,393)(51,394)(52,395)(53,396)(54,397)(55,398)(56,399)(57,400)(58,401)(59,402)(60,403)(61,404)(62,405)(63,406)(64,407)(65,408)(66,409)(67,410)(68,411)(69,412)(70,413)(71,414)(72,415)(73,416)(74,313)(75,314)(76,315)(77,316)(78,317)(79,318)(80,319)(81,320)(82,321)(83,322)(84,323)(85,324)(86,325)(87,326)(88,327)(89,328)(90,329)(91,330)(92,331)(93,332)(94,333)(95,334)(96,335)(97,336)(98,337)(99,338)(100,339)(101,340)(102,341)(103,342)(104,343)(105,221)(106,222)(107,223)(108,224)(109,225)(110,226)(111,227)(112,228)(113,229)(114,230)(115,231)(116,232)(117,233)(118,234)(119,235)(120,236)(121,237)(122,238)(123,239)(124,240)(125,241)(126,242)(127,243)(128,244)(129,245)(130,246)(131,247)(132,248)(133,249)(134,250)(135,251)(136,252)(137,253)(138,254)(139,255)(140,256)(141,257)(142,258)(143,259)(144,260)(145,261)(146,262)(147,263)(148,264)(149,265)(150,266)(151,267)(152,268)(153,269)(154,270)(155,271)(156,272)(157,273)(158,274)(159,275)(160,276)(161,277)(162,278)(163,279)(164,280)(165,281)(166,282)(167,283)(168,284)(169,285)(170,286)(171,287)(172,288)(173,289)(174,290)(175,291)(176,292)(177,293)(178,294)(179,295)(180,296)(181,297)(182,298)(183,299)(184,300)(185,301)(186,302)(187,303)(188,304)(189,305)(190,306)(191,307)(192,308)(193,309)(194,310)(195,311)(196,312)(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)(206,218)(207,219)(208,220), (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([(1,266),(2,267),(3,268),(4,269),(5,270),(6,271),(7,272),(8,273),(9,274),(10,275),(11,276),(12,277),(13,278),(14,279),(15,280),(16,281),(17,282),(18,283),(19,284),(20,285),(21,286),(22,287),(23,288),(24,289),(25,290),(26,291),(27,292),(28,293),(29,294),(30,295),(31,296),(32,297),(33,298),(34,299),(35,300),(36,301),(37,302),(38,303),(39,304),(40,305),(41,306),(42,307),(43,308),(44,309),(45,310),(46,311),(47,312),(48,209),(49,210),(50,211),(51,212),(52,213),(53,214),(54,215),(55,216),(56,217),(57,218),(58,219),(59,220),(60,221),(61,222),(62,223),(63,224),(64,225),(65,226),(66,227),(67,228),(68,229),(69,230),(70,231),(71,232),(72,233),(73,234),(74,235),(75,236),(76,237),(77,238),(78,239),(79,240),(80,241),(81,242),(82,243),(83,244),(84,245),(85,246),(86,247),(87,248),(88,249),(89,250),(90,251),(91,252),(92,253),(93,254),(94,255),(95,256),(96,257),(97,258),(98,259),(99,260),(100,261),(101,262),(102,263),(103,264),(104,265),(105,403),(106,404),(107,405),(108,406),(109,407),(110,408),(111,409),(112,410),(113,411),(114,412),(115,413),(116,414),(117,415),(118,416),(119,313),(120,314),(121,315),(122,316),(123,317),(124,318),(125,319),(126,320),(127,321),(128,322),(129,323),(130,324),(131,325),(132,326),(133,327),(134,328),(135,329),(136,330),(137,331),(138,332),(139,333),(140,334),(141,335),(142,336),(143,337),(144,338),(145,339),(146,340),(147,341),(148,342),(149,343),(150,344),(151,345),(152,346),(153,347),(154,348),(155,349),(156,350),(157,351),(158,352),(159,353),(160,354),(161,355),(162,356),(163,357),(164,358),(165,359),(166,360),(167,361),(168,362),(169,363),(170,364),(171,365),(172,366),(173,367),(174,368),(175,369),(176,370),(177,371),(178,372),(179,373),(180,374),(181,375),(182,376),(183,377),(184,378),(185,379),(186,380),(187,381),(188,382),(189,383),(190,384),(191,385),(192,386),(193,387),(194,388),(195,389),(196,390),(197,391),(198,392),(199,393),(200,394),(201,395),(202,396),(203,397),(204,398),(205,399),(206,400),(207,401),(208,402)], [(1,344),(2,345),(3,346),(4,347),(5,348),(6,349),(7,350),(8,351),(9,352),(10,353),(11,354),(12,355),(13,356),(14,357),(15,358),(16,359),(17,360),(18,361),(19,362),(20,363),(21,364),(22,365),(23,366),(24,367),(25,368),(26,369),(27,370),(28,371),(29,372),(30,373),(31,374),(32,375),(33,376),(34,377),(35,378),(36,379),(37,380),(38,381),(39,382),(40,383),(41,384),(42,385),(43,386),(44,387),(45,388),(46,389),(47,390),(48,391),(49,392),(50,393),(51,394),(52,395),(53,396),(54,397),(55,398),(56,399),(57,400),(58,401),(59,402),(60,403),(61,404),(62,405),(63,406),(64,407),(65,408),(66,409),(67,410),(68,411),(69,412),(70,413),(71,414),(72,415),(73,416),(74,313),(75,314),(76,315),(77,316),(78,317),(79,318),(80,319),(81,320),(82,321),(83,322),(84,323),(85,324),(86,325),(87,326),(88,327),(89,328),(90,329),(91,330),(92,331),(93,332),(94,333),(95,334),(96,335),(97,336),(98,337),(99,338),(100,339),(101,340),(102,341),(103,342),(104,343),(105,221),(106,222),(107,223),(108,224),(109,225),(110,226),(111,227),(112,228),(113,229),(114,230),(115,231),(116,232),(117,233),(118,234),(119,235),(120,236),(121,237),(122,238),(123,239),(124,240),(125,241),(126,242),(127,243),(128,244),(129,245),(130,246),(131,247),(132,248),(133,249),(134,250),(135,251),(136,252),(137,253),(138,254),(139,255),(140,256),(141,257),(142,258),(143,259),(144,260),(145,261),(146,262),(147,263),(148,264),(149,265),(150,266),(151,267),(152,268),(153,269),(154,270),(155,271),(156,272),(157,273),(158,274),(159,275),(160,276),(161,277),(162,278),(163,279),(164,280),(165,281),(166,282),(167,283),(168,284),(169,285),(170,286),(171,287),(172,288),(173,289),(174,290),(175,291),(176,292),(177,293),(178,294),(179,295),(180,296),(181,297),(182,298),(183,299),(184,300),(185,301),(186,302),(187,303),(188,304),(189,305),(190,306),(191,307),(192,308),(193,309),(194,310),(195,311),(196,312),(197,209),(198,210),(199,211),(200,212),(201,213),(202,214),(203,215),(204,216),(205,217),(206,218),(207,219),(208,220)], [(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···2G4A···4H8A···8P13A···13L26A···26CF52A···52CR104A···104GJ
order12···24···48···813···1326···2652···52104···104
size11···11···11···11···11···11···11···1

416 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C8C13C26C26C52C52C104
kernelC22×C104C2×C104C22×C52C2×C52C22×C26C2×C26C22×C8C2×C8C22×C4C2×C4C23C22
# reps16162161272127224192

Matrix representation of C22×C104 in GL3(𝔽313) generated by

100
03120
00312
,
100
03120
001
,
17200
0390
0039
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

׿
×
𝔽