direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Dic22, C55⋊3Q8, C220.3C2, C44.5C10, C20.3D11, C10.13D22, C110.13C22, Dic11.2C10, C4.(C5×D11), C11⋊3(C5×Q8), C22.9(C2×C10), C2.3(C10×D11), (C5×Dic11).2C2, SmallGroup(440,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Dic22
G = < a,b,c | a5=b44=1, c2=b22, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic22
►Regular action on 440 points
Generators in S440
(1 210 147 357 89)(2 211 148 358 90)(3 212 149 359 91)(4 213 150 360 92)(5 214 151 361 93)(6 215 152 362 94)(7 216 153 363 95)(8 217 154 364 96)(9 218 155 365 97)(10 219 156 366 98)(11 220 157 367 99)(12 177 158 368 100)(13 178 159 369 101)(14 179 160 370 102)(15 180 161 371 103)(16 181 162 372 104)(17 182 163 373 105)(18 183 164 374 106)(19 184 165 375 107)(20 185 166 376 108)(21 186 167 377 109)(22 187 168 378 110)(23 188 169 379 111)(24 189 170 380 112)(25 190 171 381 113)(26 191 172 382 114)(27 192 173 383 115)(28 193 174 384 116)(29 194 175 385 117)(30 195 176 386 118)(31 196 133 387 119)(32 197 134 388 120)(33 198 135 389 121)(34 199 136 390 122)(35 200 137 391 123)(36 201 138 392 124)(37 202 139 393 125)(38 203 140 394 126)(39 204 141 395 127)(40 205 142 396 128)(41 206 143 353 129)(42 207 144 354 130)(43 208 145 355 131)(44 209 146 356 132)(45 303 439 311 241)(46 304 440 312 242)(47 305 397 313 243)(48 306 398 314 244)(49 307 399 315 245)(50 308 400 316 246)(51 265 401 317 247)(52 266 402 318 248)(53 267 403 319 249)(54 268 404 320 250)(55 269 405 321 251)(56 270 406 322 252)(57 271 407 323 253)(58 272 408 324 254)(59 273 409 325 255)(60 274 410 326 256)(61 275 411 327 257)(62 276 412 328 258)(63 277 413 329 259)(64 278 414 330 260)(65 279 415 331 261)(66 280 416 332 262)(67 281 417 333 263)(68 282 418 334 264)(69 283 419 335 221)(70 284 420 336 222)(71 285 421 337 223)(72 286 422 338 224)(73 287 423 339 225)(74 288 424 340 226)(75 289 425 341 227)(76 290 426 342 228)(77 291 427 343 229)(78 292 428 344 230)(79 293 429 345 231)(80 294 430 346 232)(81 295 431 347 233)(82 296 432 348 234)(83 297 433 349 235)(84 298 434 350 236)(85 299 435 351 237)(86 300 436 352 238)(87 301 437 309 239)(88 302 438 310 240)
(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)(397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440)
(1 271 23 293)(2 270 24 292)(3 269 25 291)(4 268 26 290)(5 267 27 289)(6 266 28 288)(7 265 29 287)(8 308 30 286)(9 307 31 285)(10 306 32 284)(11 305 33 283)(12 304 34 282)(13 303 35 281)(14 302 36 280)(15 301 37 279)(16 300 38 278)(17 299 39 277)(18 298 40 276)(19 297 41 275)(20 296 42 274)(21 295 43 273)(22 294 44 272)(45 123 67 101)(46 122 68 100)(47 121 69 99)(48 120 70 98)(49 119 71 97)(50 118 72 96)(51 117 73 95)(52 116 74 94)(53 115 75 93)(54 114 76 92)(55 113 77 91)(56 112 78 90)(57 111 79 89)(58 110 80 132)(59 109 81 131)(60 108 82 130)(61 107 83 129)(62 106 84 128)(63 105 85 127)(64 104 86 126)(65 103 87 125)(66 102 88 124)(133 337 155 315)(134 336 156 314)(135 335 157 313)(136 334 158 312)(137 333 159 311)(138 332 160 310)(139 331 161 309)(140 330 162 352)(141 329 163 351)(142 328 164 350)(143 327 165 349)(144 326 166 348)(145 325 167 347)(146 324 168 346)(147 323 169 345)(148 322 170 344)(149 321 171 343)(150 320 172 342)(151 319 173 341)(152 318 174 340)(153 317 175 339)(154 316 176 338)(177 440 199 418)(178 439 200 417)(179 438 201 416)(180 437 202 415)(181 436 203 414)(182 435 204 413)(183 434 205 412)(184 433 206 411)(185 432 207 410)(186 431 208 409)(187 430 209 408)(188 429 210 407)(189 428 211 406)(190 427 212 405)(191 426 213 404)(192 425 214 403)(193 424 215 402)(194 423 216 401)(195 422 217 400)(196 421 218 399)(197 420 219 398)(198 419 220 397)(221 367 243 389)(222 366 244 388)(223 365 245 387)(224 364 246 386)(225 363 247 385)(226 362 248 384)(227 361 249 383)(228 360 250 382)(229 359 251 381)(230 358 252 380)(231 357 253 379)(232 356 254 378)(233 355 255 377)(234 354 256 376)(235 353 257 375)(236 396 258 374)(237 395 259 373)(238 394 260 372)(239 393 261 371)(240 392 262 370)(241 391 263 369)(242 390 264 368)
G:=sub<Sym(440)| (1,210,147,357,89)(2,211,148,358,90)(3,212,149,359,91)(4,213,150,360,92)(5,214,151,361,93)(6,215,152,362,94)(7,216,153,363,95)(8,217,154,364,96)(9,218,155,365,97)(10,219,156,366,98)(11,220,157,367,99)(12,177,158,368,100)(13,178,159,369,101)(14,179,160,370,102)(15,180,161,371,103)(16,181,162,372,104)(17,182,163,373,105)(18,183,164,374,106)(19,184,165,375,107)(20,185,166,376,108)(21,186,167,377,109)(22,187,168,378,110)(23,188,169,379,111)(24,189,170,380,112)(25,190,171,381,113)(26,191,172,382,114)(27,192,173,383,115)(28,193,174,384,116)(29,194,175,385,117)(30,195,176,386,118)(31,196,133,387,119)(32,197,134,388,120)(33,198,135,389,121)(34,199,136,390,122)(35,200,137,391,123)(36,201,138,392,124)(37,202,139,393,125)(38,203,140,394,126)(39,204,141,395,127)(40,205,142,396,128)(41,206,143,353,129)(42,207,144,354,130)(43,208,145,355,131)(44,209,146,356,132)(45,303,439,311,241)(46,304,440,312,242)(47,305,397,313,243)(48,306,398,314,244)(49,307,399,315,245)(50,308,400,316,246)(51,265,401,317,247)(52,266,402,318,248)(53,267,403,319,249)(54,268,404,320,250)(55,269,405,321,251)(56,270,406,322,252)(57,271,407,323,253)(58,272,408,324,254)(59,273,409,325,255)(60,274,410,326,256)(61,275,411,327,257)(62,276,412,328,258)(63,277,413,329,259)(64,278,414,330,260)(65,279,415,331,261)(66,280,416,332,262)(67,281,417,333,263)(68,282,418,334,264)(69,283,419,335,221)(70,284,420,336,222)(71,285,421,337,223)(72,286,422,338,224)(73,287,423,339,225)(74,288,424,340,226)(75,289,425,341,227)(76,290,426,342,228)(77,291,427,343,229)(78,292,428,344,230)(79,293,429,345,231)(80,294,430,346,232)(81,295,431,347,233)(82,296,432,348,234)(83,297,433,349,235)(84,298,434,350,236)(85,299,435,351,237)(86,300,436,352,238)(87,301,437,309,239)(88,302,438,310,240), (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)(397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440), (1,271,23,293)(2,270,24,292)(3,269,25,291)(4,268,26,290)(5,267,27,289)(6,266,28,288)(7,265,29,287)(8,308,30,286)(9,307,31,285)(10,306,32,284)(11,305,33,283)(12,304,34,282)(13,303,35,281)(14,302,36,280)(15,301,37,279)(16,300,38,278)(17,299,39,277)(18,298,40,276)(19,297,41,275)(20,296,42,274)(21,295,43,273)(22,294,44,272)(45,123,67,101)(46,122,68,100)(47,121,69,99)(48,120,70,98)(49,119,71,97)(50,118,72,96)(51,117,73,95)(52,116,74,94)(53,115,75,93)(54,114,76,92)(55,113,77,91)(56,112,78,90)(57,111,79,89)(58,110,80,132)(59,109,81,131)(60,108,82,130)(61,107,83,129)(62,106,84,128)(63,105,85,127)(64,104,86,126)(65,103,87,125)(66,102,88,124)(133,337,155,315)(134,336,156,314)(135,335,157,313)(136,334,158,312)(137,333,159,311)(138,332,160,310)(139,331,161,309)(140,330,162,352)(141,329,163,351)(142,328,164,350)(143,327,165,349)(144,326,166,348)(145,325,167,347)(146,324,168,346)(147,323,169,345)(148,322,170,344)(149,321,171,343)(150,320,172,342)(151,319,173,341)(152,318,174,340)(153,317,175,339)(154,316,176,338)(177,440,199,418)(178,439,200,417)(179,438,201,416)(180,437,202,415)(181,436,203,414)(182,435,204,413)(183,434,205,412)(184,433,206,411)(185,432,207,410)(186,431,208,409)(187,430,209,408)(188,429,210,407)(189,428,211,406)(190,427,212,405)(191,426,213,404)(192,425,214,403)(193,424,215,402)(194,423,216,401)(195,422,217,400)(196,421,218,399)(197,420,219,398)(198,419,220,397)(221,367,243,389)(222,366,244,388)(223,365,245,387)(224,364,246,386)(225,363,247,385)(226,362,248,384)(227,361,249,383)(228,360,250,382)(229,359,251,381)(230,358,252,380)(231,357,253,379)(232,356,254,378)(233,355,255,377)(234,354,256,376)(235,353,257,375)(236,396,258,374)(237,395,259,373)(238,394,260,372)(239,393,261,371)(240,392,262,370)(241,391,263,369)(242,390,264,368)>;
G:=Group( (1,210,147,357,89)(2,211,148,358,90)(3,212,149,359,91)(4,213,150,360,92)(5,214,151,361,93)(6,215,152,362,94)(7,216,153,363,95)(8,217,154,364,96)(9,218,155,365,97)(10,219,156,366,98)(11,220,157,367,99)(12,177,158,368,100)(13,178,159,369,101)(14,179,160,370,102)(15,180,161,371,103)(16,181,162,372,104)(17,182,163,373,105)(18,183,164,374,106)(19,184,165,375,107)(20,185,166,376,108)(21,186,167,377,109)(22,187,168,378,110)(23,188,169,379,111)(24,189,170,380,112)(25,190,171,381,113)(26,191,172,382,114)(27,192,173,383,115)(28,193,174,384,116)(29,194,175,385,117)(30,195,176,386,118)(31,196,133,387,119)(32,197,134,388,120)(33,198,135,389,121)(34,199,136,390,122)(35,200,137,391,123)(36,201,138,392,124)(37,202,139,393,125)(38,203,140,394,126)(39,204,141,395,127)(40,205,142,396,128)(41,206,143,353,129)(42,207,144,354,130)(43,208,145,355,131)(44,209,146,356,132)(45,303,439,311,241)(46,304,440,312,242)(47,305,397,313,243)(48,306,398,314,244)(49,307,399,315,245)(50,308,400,316,246)(51,265,401,317,247)(52,266,402,318,248)(53,267,403,319,249)(54,268,404,320,250)(55,269,405,321,251)(56,270,406,322,252)(57,271,407,323,253)(58,272,408,324,254)(59,273,409,325,255)(60,274,410,326,256)(61,275,411,327,257)(62,276,412,328,258)(63,277,413,329,259)(64,278,414,330,260)(65,279,415,331,261)(66,280,416,332,262)(67,281,417,333,263)(68,282,418,334,264)(69,283,419,335,221)(70,284,420,336,222)(71,285,421,337,223)(72,286,422,338,224)(73,287,423,339,225)(74,288,424,340,226)(75,289,425,341,227)(76,290,426,342,228)(77,291,427,343,229)(78,292,428,344,230)(79,293,429,345,231)(80,294,430,346,232)(81,295,431,347,233)(82,296,432,348,234)(83,297,433,349,235)(84,298,434,350,236)(85,299,435,351,237)(86,300,436,352,238)(87,301,437,309,239)(88,302,438,310,240), (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)(397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440), (1,271,23,293)(2,270,24,292)(3,269,25,291)(4,268,26,290)(5,267,27,289)(6,266,28,288)(7,265,29,287)(8,308,30,286)(9,307,31,285)(10,306,32,284)(11,305,33,283)(12,304,34,282)(13,303,35,281)(14,302,36,280)(15,301,37,279)(16,300,38,278)(17,299,39,277)(18,298,40,276)(19,297,41,275)(20,296,42,274)(21,295,43,273)(22,294,44,272)(45,123,67,101)(46,122,68,100)(47,121,69,99)(48,120,70,98)(49,119,71,97)(50,118,72,96)(51,117,73,95)(52,116,74,94)(53,115,75,93)(54,114,76,92)(55,113,77,91)(56,112,78,90)(57,111,79,89)(58,110,80,132)(59,109,81,131)(60,108,82,130)(61,107,83,129)(62,106,84,128)(63,105,85,127)(64,104,86,126)(65,103,87,125)(66,102,88,124)(133,337,155,315)(134,336,156,314)(135,335,157,313)(136,334,158,312)(137,333,159,311)(138,332,160,310)(139,331,161,309)(140,330,162,352)(141,329,163,351)(142,328,164,350)(143,327,165,349)(144,326,166,348)(145,325,167,347)(146,324,168,346)(147,323,169,345)(148,322,170,344)(149,321,171,343)(150,320,172,342)(151,319,173,341)(152,318,174,340)(153,317,175,339)(154,316,176,338)(177,440,199,418)(178,439,200,417)(179,438,201,416)(180,437,202,415)(181,436,203,414)(182,435,204,413)(183,434,205,412)(184,433,206,411)(185,432,207,410)(186,431,208,409)(187,430,209,408)(188,429,210,407)(189,428,211,406)(190,427,212,405)(191,426,213,404)(192,425,214,403)(193,424,215,402)(194,423,216,401)(195,422,217,400)(196,421,218,399)(197,420,219,398)(198,419,220,397)(221,367,243,389)(222,366,244,388)(223,365,245,387)(224,364,246,386)(225,363,247,385)(226,362,248,384)(227,361,249,383)(228,360,250,382)(229,359,251,381)(230,358,252,380)(231,357,253,379)(232,356,254,378)(233,355,255,377)(234,354,256,376)(235,353,257,375)(236,396,258,374)(237,395,259,373)(238,394,260,372)(239,393,261,371)(240,392,262,370)(241,391,263,369)(242,390,264,368) );
G=PermutationGroup([[(1,210,147,357,89),(2,211,148,358,90),(3,212,149,359,91),(4,213,150,360,92),(5,214,151,361,93),(6,215,152,362,94),(7,216,153,363,95),(8,217,154,364,96),(9,218,155,365,97),(10,219,156,366,98),(11,220,157,367,99),(12,177,158,368,100),(13,178,159,369,101),(14,179,160,370,102),(15,180,161,371,103),(16,181,162,372,104),(17,182,163,373,105),(18,183,164,374,106),(19,184,165,375,107),(20,185,166,376,108),(21,186,167,377,109),(22,187,168,378,110),(23,188,169,379,111),(24,189,170,380,112),(25,190,171,381,113),(26,191,172,382,114),(27,192,173,383,115),(28,193,174,384,116),(29,194,175,385,117),(30,195,176,386,118),(31,196,133,387,119),(32,197,134,388,120),(33,198,135,389,121),(34,199,136,390,122),(35,200,137,391,123),(36,201,138,392,124),(37,202,139,393,125),(38,203,140,394,126),(39,204,141,395,127),(40,205,142,396,128),(41,206,143,353,129),(42,207,144,354,130),(43,208,145,355,131),(44,209,146,356,132),(45,303,439,311,241),(46,304,440,312,242),(47,305,397,313,243),(48,306,398,314,244),(49,307,399,315,245),(50,308,400,316,246),(51,265,401,317,247),(52,266,402,318,248),(53,267,403,319,249),(54,268,404,320,250),(55,269,405,321,251),(56,270,406,322,252),(57,271,407,323,253),(58,272,408,324,254),(59,273,409,325,255),(60,274,410,326,256),(61,275,411,327,257),(62,276,412,328,258),(63,277,413,329,259),(64,278,414,330,260),(65,279,415,331,261),(66,280,416,332,262),(67,281,417,333,263),(68,282,418,334,264),(69,283,419,335,221),(70,284,420,336,222),(71,285,421,337,223),(72,286,422,338,224),(73,287,423,339,225),(74,288,424,340,226),(75,289,425,341,227),(76,290,426,342,228),(77,291,427,343,229),(78,292,428,344,230),(79,293,429,345,231),(80,294,430,346,232),(81,295,431,347,233),(82,296,432,348,234),(83,297,433,349,235),(84,298,434,350,236),(85,299,435,351,237),(86,300,436,352,238),(87,301,437,309,239),(88,302,438,310,240)], [(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),(397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440)], [(1,271,23,293),(2,270,24,292),(3,269,25,291),(4,268,26,290),(5,267,27,289),(6,266,28,288),(7,265,29,287),(8,308,30,286),(9,307,31,285),(10,306,32,284),(11,305,33,283),(12,304,34,282),(13,303,35,281),(14,302,36,280),(15,301,37,279),(16,300,38,278),(17,299,39,277),(18,298,40,276),(19,297,41,275),(20,296,42,274),(21,295,43,273),(22,294,44,272),(45,123,67,101),(46,122,68,100),(47,121,69,99),(48,120,70,98),(49,119,71,97),(50,118,72,96),(51,117,73,95),(52,116,74,94),(53,115,75,93),(54,114,76,92),(55,113,77,91),(56,112,78,90),(57,111,79,89),(58,110,80,132),(59,109,81,131),(60,108,82,130),(61,107,83,129),(62,106,84,128),(63,105,85,127),(64,104,86,126),(65,103,87,125),(66,102,88,124),(133,337,155,315),(134,336,156,314),(135,335,157,313),(136,334,158,312),(137,333,159,311),(138,332,160,310),(139,331,161,309),(140,330,162,352),(141,329,163,351),(142,328,164,350),(143,327,165,349),(144,326,166,348),(145,325,167,347),(146,324,168,346),(147,323,169,345),(148,322,170,344),(149,321,171,343),(150,320,172,342),(151,319,173,341),(152,318,174,340),(153,317,175,339),(154,316,176,338),(177,440,199,418),(178,439,200,417),(179,438,201,416),(180,437,202,415),(181,436,203,414),(182,435,204,413),(183,434,205,412),(184,433,206,411),(185,432,207,410),(186,431,208,409),(187,430,209,408),(188,429,210,407),(189,428,211,406),(190,427,212,405),(191,426,213,404),(192,425,214,403),(193,424,215,402),(194,423,216,401),(195,422,217,400),(196,421,218,399),(197,420,219,398),(198,419,220,397),(221,367,243,389),(222,366,244,388),(223,365,245,387),(224,364,246,386),(225,363,247,385),(226,362,248,384),(227,361,249,383),(228,360,250,382),(229,359,251,381),(230,358,252,380),(231,357,253,379),(232,356,254,378),(233,355,255,377),(234,354,256,376),(235,353,257,375),(236,396,258,374),(237,395,259,373),(238,394,260,372),(239,393,261,371),(240,392,262,370),(241,391,263,369),(242,390,264,368)]])
125 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 11A | ··· | 11E | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 22A | ··· | 22E | 44A | ··· | 44J | 55A | ··· | 55T | 110A | ··· | 110T | 220A | ··· | 220AN |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 11 | ··· | 11 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 | 220 | ··· | 220 |
| size | 1 | 1 | 2 | 22 | 22 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 22 | ··· | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | Q8 | D11 | C5×Q8 | D22 | Dic22 | C5×D11 | C10×D11 | C5×Dic22 |
| kernel | C5×Dic22 | C5×Dic11 | C220 | Dic22 | Dic11 | C44 | C55 | C20 | C11 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 5 | 4 | 5 | 10 | 20 | 20 | 40 |
Matrix representation of C5×Dic22 ►in GL2(𝔽661) generated by
| 247 | 0 |
| 0 | 247 |
| 299 | 53 |
| 608 | 278 |
| 444 | 6 |
| 304 | 217 |
G:=sub<GL(2,GF(661))| [247,0,0,247],[299,608,53,278],[444,304,6,217] >;
C5×Dic22 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_{22} % in TeX
G:=Group("C5xDic22"); // GroupNames label
G:=SmallGroup(440,24);
// by ID
G=gap.SmallGroup(440,24);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-11,100,221,106,10004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^44=1,c^2=b^22,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export