Copied to
clipboard

## G = C5×Dic22order 440 = 23·5·11

### Direct product of C5 and Dic22

Aliases: C5×Dic22, C553Q8, C220.3C2, C44.5C10, C20.3D11, C10.13D22, C110.13C22, Dic11.2C10, C4.(C5×D11), C113(C5×Q8), C22.9(C2×C10), C2.3(C10×D11), (C5×Dic11).2C2, SmallGroup(440,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×Dic22
 Chief series C1 — C11 — C22 — C110 — C5×Dic11 — C5×Dic22
 Lower central C11 — C22 — C5×Dic22
 Upper central C1 — C10 — C20

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
(1 215 84 391 412)(2 216 85 392 413)(3 217 86 393 414)(4 218 87 394 415)(5 219 88 395 416)(6 220 45 396 417)(7 177 46 353 418)(8 178 47 354 419)(9 179 48 355 420)(10 180 49 356 421)(11 181 50 357 422)(12 182 51 358 423)(13 183 52 359 424)(14 184 53 360 425)(15 185 54 361 426)(16 186 55 362 427)(17 187 56 363 428)(18 188 57 364 429)(19 189 58 365 430)(20 190 59 366 431)(21 191 60 367 432)(22 192 61 368 433)(23 193 62 369 434)(24 194 63 370 435)(25 195 64 371 436)(26 196 65 372 437)(27 197 66 373 438)(28 198 67 374 439)(29 199 68 375 440)(30 200 69 376 397)(31 201 70 377 398)(32 202 71 378 399)(33 203 72 379 400)(34 204 73 380 401)(35 205 74 381 402)(36 206 75 382 403)(37 207 76 383 404)(38 208 77 384 405)(39 209 78 385 406)(40 210 79 386 407)(41 211 80 387 408)(42 212 81 388 409)(43 213 82 389 410)(44 214 83 390 411)(89 277 312 252 146)(90 278 313 253 147)(91 279 314 254 148)(92 280 315 255 149)(93 281 316 256 150)(94 282 317 257 151)(95 283 318 258 152)(96 284 319 259 153)(97 285 320 260 154)(98 286 321 261 155)(99 287 322 262 156)(100 288 323 263 157)(101 289 324 264 158)(102 290 325 221 159)(103 291 326 222 160)(104 292 327 223 161)(105 293 328 224 162)(106 294 329 225 163)(107 295 330 226 164)(108 296 331 227 165)(109 297 332 228 166)(110 298 333 229 167)(111 299 334 230 168)(112 300 335 231 169)(113 301 336 232 170)(114 302 337 233 171)(115 303 338 234 172)(116 304 339 235 173)(117 305 340 236 174)(118 306 341 237 175)(119 307 342 238 176)(120 308 343 239 133)(121 265 344 240 134)(122 266 345 241 135)(123 267 346 242 136)(124 268 347 243 137)(125 269 348 244 138)(126 270 349 245 139)(127 271 350 246 140)(128 272 351 247 141)(129 273 352 248 142)(130 274 309 249 143)(131 275 310 250 144)(132 276 311 251 145)
(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)
(1 228 23 250)(2 227 24 249)(3 226 25 248)(4 225 26 247)(5 224 27 246)(6 223 28 245)(7 222 29 244)(8 221 30 243)(9 264 31 242)(10 263 32 241)(11 262 33 240)(12 261 34 239)(13 260 35 238)(14 259 36 237)(15 258 37 236)(16 257 38 235)(17 256 39 234)(18 255 40 233)(19 254 41 232)(20 253 42 231)(21 252 43 230)(22 251 44 229)(45 104 67 126)(46 103 68 125)(47 102 69 124)(48 101 70 123)(49 100 71 122)(50 99 72 121)(51 98 73 120)(52 97 74 119)(53 96 75 118)(54 95 76 117)(55 94 77 116)(56 93 78 115)(57 92 79 114)(58 91 80 113)(59 90 81 112)(60 89 82 111)(61 132 83 110)(62 131 84 109)(63 130 85 108)(64 129 86 107)(65 128 87 106)(66 127 88 105)(133 182 155 204)(134 181 156 203)(135 180 157 202)(136 179 158 201)(137 178 159 200)(138 177 160 199)(139 220 161 198)(140 219 162 197)(141 218 163 196)(142 217 164 195)(143 216 165 194)(144 215 166 193)(145 214 167 192)(146 213 168 191)(147 212 169 190)(148 211 170 189)(149 210 171 188)(150 209 172 187)(151 208 173 186)(152 207 174 185)(153 206 175 184)(154 205 176 183)(265 357 287 379)(266 356 288 378)(267 355 289 377)(268 354 290 376)(269 353 291 375)(270 396 292 374)(271 395 293 373)(272 394 294 372)(273 393 295 371)(274 392 296 370)(275 391 297 369)(276 390 298 368)(277 389 299 367)(278 388 300 366)(279 387 301 365)(280 386 302 364)(281 385 303 363)(282 384 304 362)(283 383 305 361)(284 382 306 360)(285 381 307 359)(286 380 308 358)(309 413 331 435)(310 412 332 434)(311 411 333 433)(312 410 334 432)(313 409 335 431)(314 408 336 430)(315 407 337 429)(316 406 338 428)(317 405 339 427)(318 404 340 426)(319 403 341 425)(320 402 342 424)(321 401 343 423)(322 400 344 422)(323 399 345 421)(324 398 346 420)(325 397 347 419)(326 440 348 418)(327 439 349 417)(328 438 350 416)(329 437 351 415)(330 436 352 414)

G:=sub<Sym(440)| (1,215,84,391,412)(2,216,85,392,413)(3,217,86,393,414)(4,218,87,394,415)(5,219,88,395,416)(6,220,45,396,417)(7,177,46,353,418)(8,178,47,354,419)(9,179,48,355,420)(10,180,49,356,421)(11,181,50,357,422)(12,182,51,358,423)(13,183,52,359,424)(14,184,53,360,425)(15,185,54,361,426)(16,186,55,362,427)(17,187,56,363,428)(18,188,57,364,429)(19,189,58,365,430)(20,190,59,366,431)(21,191,60,367,432)(22,192,61,368,433)(23,193,62,369,434)(24,194,63,370,435)(25,195,64,371,436)(26,196,65,372,437)(27,197,66,373,438)(28,198,67,374,439)(29,199,68,375,440)(30,200,69,376,397)(31,201,70,377,398)(32,202,71,378,399)(33,203,72,379,400)(34,204,73,380,401)(35,205,74,381,402)(36,206,75,382,403)(37,207,76,383,404)(38,208,77,384,405)(39,209,78,385,406)(40,210,79,386,407)(41,211,80,387,408)(42,212,81,388,409)(43,213,82,389,410)(44,214,83,390,411)(89,277,312,252,146)(90,278,313,253,147)(91,279,314,254,148)(92,280,315,255,149)(93,281,316,256,150)(94,282,317,257,151)(95,283,318,258,152)(96,284,319,259,153)(97,285,320,260,154)(98,286,321,261,155)(99,287,322,262,156)(100,288,323,263,157)(101,289,324,264,158)(102,290,325,221,159)(103,291,326,222,160)(104,292,327,223,161)(105,293,328,224,162)(106,294,329,225,163)(107,295,330,226,164)(108,296,331,227,165)(109,297,332,228,166)(110,298,333,229,167)(111,299,334,230,168)(112,300,335,231,169)(113,301,336,232,170)(114,302,337,233,171)(115,303,338,234,172)(116,304,339,235,173)(117,305,340,236,174)(118,306,341,237,175)(119,307,342,238,176)(120,308,343,239,133)(121,265,344,240,134)(122,266,345,241,135)(123,267,346,242,136)(124,268,347,243,137)(125,269,348,244,138)(126,270,349,245,139)(127,271,350,246,140)(128,272,351,247,141)(129,273,352,248,142)(130,274,309,249,143)(131,275,310,250,144)(132,276,311,251,145), (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), (1,228,23,250)(2,227,24,249)(3,226,25,248)(4,225,26,247)(5,224,27,246)(6,223,28,245)(7,222,29,244)(8,221,30,243)(9,264,31,242)(10,263,32,241)(11,262,33,240)(12,261,34,239)(13,260,35,238)(14,259,36,237)(15,258,37,236)(16,257,38,235)(17,256,39,234)(18,255,40,233)(19,254,41,232)(20,253,42,231)(21,252,43,230)(22,251,44,229)(45,104,67,126)(46,103,68,125)(47,102,69,124)(48,101,70,123)(49,100,71,122)(50,99,72,121)(51,98,73,120)(52,97,74,119)(53,96,75,118)(54,95,76,117)(55,94,77,116)(56,93,78,115)(57,92,79,114)(58,91,80,113)(59,90,81,112)(60,89,82,111)(61,132,83,110)(62,131,84,109)(63,130,85,108)(64,129,86,107)(65,128,87,106)(66,127,88,105)(133,182,155,204)(134,181,156,203)(135,180,157,202)(136,179,158,201)(137,178,159,200)(138,177,160,199)(139,220,161,198)(140,219,162,197)(141,218,163,196)(142,217,164,195)(143,216,165,194)(144,215,166,193)(145,214,167,192)(146,213,168,191)(147,212,169,190)(148,211,170,189)(149,210,171,188)(150,209,172,187)(151,208,173,186)(152,207,174,185)(153,206,175,184)(154,205,176,183)(265,357,287,379)(266,356,288,378)(267,355,289,377)(268,354,290,376)(269,353,291,375)(270,396,292,374)(271,395,293,373)(272,394,294,372)(273,393,295,371)(274,392,296,370)(275,391,297,369)(276,390,298,368)(277,389,299,367)(278,388,300,366)(279,387,301,365)(280,386,302,364)(281,385,303,363)(282,384,304,362)(283,383,305,361)(284,382,306,360)(285,381,307,359)(286,380,308,358)(309,413,331,435)(310,412,332,434)(311,411,333,433)(312,410,334,432)(313,409,335,431)(314,408,336,430)(315,407,337,429)(316,406,338,428)(317,405,339,427)(318,404,340,426)(319,403,341,425)(320,402,342,424)(321,401,343,423)(322,400,344,422)(323,399,345,421)(324,398,346,420)(325,397,347,419)(326,440,348,418)(327,439,349,417)(328,438,350,416)(329,437,351,415)(330,436,352,414)>;

G:=Group( (1,215,84,391,412)(2,216,85,392,413)(3,217,86,393,414)(4,218,87,394,415)(5,219,88,395,416)(6,220,45,396,417)(7,177,46,353,418)(8,178,47,354,419)(9,179,48,355,420)(10,180,49,356,421)(11,181,50,357,422)(12,182,51,358,423)(13,183,52,359,424)(14,184,53,360,425)(15,185,54,361,426)(16,186,55,362,427)(17,187,56,363,428)(18,188,57,364,429)(19,189,58,365,430)(20,190,59,366,431)(21,191,60,367,432)(22,192,61,368,433)(23,193,62,369,434)(24,194,63,370,435)(25,195,64,371,436)(26,196,65,372,437)(27,197,66,373,438)(28,198,67,374,439)(29,199,68,375,440)(30,200,69,376,397)(31,201,70,377,398)(32,202,71,378,399)(33,203,72,379,400)(34,204,73,380,401)(35,205,74,381,402)(36,206,75,382,403)(37,207,76,383,404)(38,208,77,384,405)(39,209,78,385,406)(40,210,79,386,407)(41,211,80,387,408)(42,212,81,388,409)(43,213,82,389,410)(44,214,83,390,411)(89,277,312,252,146)(90,278,313,253,147)(91,279,314,254,148)(92,280,315,255,149)(93,281,316,256,150)(94,282,317,257,151)(95,283,318,258,152)(96,284,319,259,153)(97,285,320,260,154)(98,286,321,261,155)(99,287,322,262,156)(100,288,323,263,157)(101,289,324,264,158)(102,290,325,221,159)(103,291,326,222,160)(104,292,327,223,161)(105,293,328,224,162)(106,294,329,225,163)(107,295,330,226,164)(108,296,331,227,165)(109,297,332,228,166)(110,298,333,229,167)(111,299,334,230,168)(112,300,335,231,169)(113,301,336,232,170)(114,302,337,233,171)(115,303,338,234,172)(116,304,339,235,173)(117,305,340,236,174)(118,306,341,237,175)(119,307,342,238,176)(120,308,343,239,133)(121,265,344,240,134)(122,266,345,241,135)(123,267,346,242,136)(124,268,347,243,137)(125,269,348,244,138)(126,270,349,245,139)(127,271,350,246,140)(128,272,351,247,141)(129,273,352,248,142)(130,274,309,249,143)(131,275,310,250,144)(132,276,311,251,145), (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), (1,228,23,250)(2,227,24,249)(3,226,25,248)(4,225,26,247)(5,224,27,246)(6,223,28,245)(7,222,29,244)(8,221,30,243)(9,264,31,242)(10,263,32,241)(11,262,33,240)(12,261,34,239)(13,260,35,238)(14,259,36,237)(15,258,37,236)(16,257,38,235)(17,256,39,234)(18,255,40,233)(19,254,41,232)(20,253,42,231)(21,252,43,230)(22,251,44,229)(45,104,67,126)(46,103,68,125)(47,102,69,124)(48,101,70,123)(49,100,71,122)(50,99,72,121)(51,98,73,120)(52,97,74,119)(53,96,75,118)(54,95,76,117)(55,94,77,116)(56,93,78,115)(57,92,79,114)(58,91,80,113)(59,90,81,112)(60,89,82,111)(61,132,83,110)(62,131,84,109)(63,130,85,108)(64,129,86,107)(65,128,87,106)(66,127,88,105)(133,182,155,204)(134,181,156,203)(135,180,157,202)(136,179,158,201)(137,178,159,200)(138,177,160,199)(139,220,161,198)(140,219,162,197)(141,218,163,196)(142,217,164,195)(143,216,165,194)(144,215,166,193)(145,214,167,192)(146,213,168,191)(147,212,169,190)(148,211,170,189)(149,210,171,188)(150,209,172,187)(151,208,173,186)(152,207,174,185)(153,206,175,184)(154,205,176,183)(265,357,287,379)(266,356,288,378)(267,355,289,377)(268,354,290,376)(269,353,291,375)(270,396,292,374)(271,395,293,373)(272,394,294,372)(273,393,295,371)(274,392,296,370)(275,391,297,369)(276,390,298,368)(277,389,299,367)(278,388,300,366)(279,387,301,365)(280,386,302,364)(281,385,303,363)(282,384,304,362)(283,383,305,361)(284,382,306,360)(285,381,307,359)(286,380,308,358)(309,413,331,435)(310,412,332,434)(311,411,333,433)(312,410,334,432)(313,409,335,431)(314,408,336,430)(315,407,337,429)(316,406,338,428)(317,405,339,427)(318,404,340,426)(319,403,341,425)(320,402,342,424)(321,401,343,423)(322,400,344,422)(323,399,345,421)(324,398,346,420)(325,397,347,419)(326,440,348,418)(327,439,349,417)(328,438,350,416)(329,437,351,415)(330,436,352,414) );

G=PermutationGroup([(1,215,84,391,412),(2,216,85,392,413),(3,217,86,393,414),(4,218,87,394,415),(5,219,88,395,416),(6,220,45,396,417),(7,177,46,353,418),(8,178,47,354,419),(9,179,48,355,420),(10,180,49,356,421),(11,181,50,357,422),(12,182,51,358,423),(13,183,52,359,424),(14,184,53,360,425),(15,185,54,361,426),(16,186,55,362,427),(17,187,56,363,428),(18,188,57,364,429),(19,189,58,365,430),(20,190,59,366,431),(21,191,60,367,432),(22,192,61,368,433),(23,193,62,369,434),(24,194,63,370,435),(25,195,64,371,436),(26,196,65,372,437),(27,197,66,373,438),(28,198,67,374,439),(29,199,68,375,440),(30,200,69,376,397),(31,201,70,377,398),(32,202,71,378,399),(33,203,72,379,400),(34,204,73,380,401),(35,205,74,381,402),(36,206,75,382,403),(37,207,76,383,404),(38,208,77,384,405),(39,209,78,385,406),(40,210,79,386,407),(41,211,80,387,408),(42,212,81,388,409),(43,213,82,389,410),(44,214,83,390,411),(89,277,312,252,146),(90,278,313,253,147),(91,279,314,254,148),(92,280,315,255,149),(93,281,316,256,150),(94,282,317,257,151),(95,283,318,258,152),(96,284,319,259,153),(97,285,320,260,154),(98,286,321,261,155),(99,287,322,262,156),(100,288,323,263,157),(101,289,324,264,158),(102,290,325,221,159),(103,291,326,222,160),(104,292,327,223,161),(105,293,328,224,162),(106,294,329,225,163),(107,295,330,226,164),(108,296,331,227,165),(109,297,332,228,166),(110,298,333,229,167),(111,299,334,230,168),(112,300,335,231,169),(113,301,336,232,170),(114,302,337,233,171),(115,303,338,234,172),(116,304,339,235,173),(117,305,340,236,174),(118,306,341,237,175),(119,307,342,238,176),(120,308,343,239,133),(121,265,344,240,134),(122,266,345,241,135),(123,267,346,242,136),(124,268,347,243,137),(125,269,348,244,138),(126,270,349,245,139),(127,271,350,246,140),(128,272,351,247,141),(129,273,352,248,142),(130,274,309,249,143),(131,275,310,250,144),(132,276,311,251,145)], [(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)], [(1,228,23,250),(2,227,24,249),(3,226,25,248),(4,225,26,247),(5,224,27,246),(6,223,28,245),(7,222,29,244),(8,221,30,243),(9,264,31,242),(10,263,32,241),(11,262,33,240),(12,261,34,239),(13,260,35,238),(14,259,36,237),(15,258,37,236),(16,257,38,235),(17,256,39,234),(18,255,40,233),(19,254,41,232),(20,253,42,231),(21,252,43,230),(22,251,44,229),(45,104,67,126),(46,103,68,125),(47,102,69,124),(48,101,70,123),(49,100,71,122),(50,99,72,121),(51,98,73,120),(52,97,74,119),(53,96,75,118),(54,95,76,117),(55,94,77,116),(56,93,78,115),(57,92,79,114),(58,91,80,113),(59,90,81,112),(60,89,82,111),(61,132,83,110),(62,131,84,109),(63,130,85,108),(64,129,86,107),(65,128,87,106),(66,127,88,105),(133,182,155,204),(134,181,156,203),(135,180,157,202),(136,179,158,201),(137,178,159,200),(138,177,160,199),(139,220,161,198),(140,219,162,197),(141,218,163,196),(142,217,164,195),(143,216,165,194),(144,215,166,193),(145,214,167,192),(146,213,168,191),(147,212,169,190),(148,211,170,189),(149,210,171,188),(150,209,172,187),(151,208,173,186),(152,207,174,185),(153,206,175,184),(154,205,176,183),(265,357,287,379),(266,356,288,378),(267,355,289,377),(268,354,290,376),(269,353,291,375),(270,396,292,374),(271,395,293,373),(272,394,294,372),(273,393,295,371),(274,392,296,370),(275,391,297,369),(276,390,298,368),(277,389,299,367),(278,388,300,366),(279,387,301,365),(280,386,302,364),(281,385,303,363),(282,384,304,362),(283,383,305,361),(284,382,306,360),(285,381,307,359),(286,380,308,358),(309,413,331,435),(310,412,332,434),(311,411,333,433),(312,410,334,432),(313,409,335,431),(314,408,336,430),(315,407,337,429),(316,406,338,428),(317,405,339,427),(318,404,340,426),(319,403,341,425),(320,402,342,424),(321,401,343,423),(322,400,344,422),(323,399,345,421),(324,398,346,420),(325,397,347,419),(326,440,348,418),(327,439,349,417),(328,438,350,416),(329,437,351,415),(330,436,352,414)])

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

׿
×
𝔽