direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic5×C22, C10⋊2C44, C110⋊5C4, C22.16D10, C110.21C22, C5⋊3(C2×C44), C55⋊12(C2×C4), (C2×C10).C22, (C2×C22).2D5, C2.2(D5×C22), C22.(D5×C11), (C2×C110).3C2, C10.4(C2×C22), SmallGroup(440,32)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — Dic5×C22 |
Generators and relations for Dic5×C22
G = < a,b,c | a22=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic5×C22
►Regular action on 440 points
Generators in S440
(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)
(1 195 90 274 59 162 411 297 202 371)(2 196 91 275 60 163 412 298 203 372)(3 197 92 276 61 164 413 299 204 373)(4 198 93 277 62 165 414 300 205 374)(5 177 94 278 63 166 415 301 206 353)(6 178 95 279 64 167 416 302 207 354)(7 179 96 280 65 168 417 303 208 355)(8 180 97 281 66 169 418 304 209 356)(9 181 98 282 45 170 397 305 210 357)(10 182 99 283 46 171 398 306 211 358)(11 183 100 284 47 172 399 307 212 359)(12 184 101 285 48 173 400 308 213 360)(13 185 102 286 49 174 401 287 214 361)(14 186 103 265 50 175 402 288 215 362)(15 187 104 266 51 176 403 289 216 363)(16 188 105 267 52 155 404 290 217 364)(17 189 106 268 53 156 405 291 218 365)(18 190 107 269 54 157 406 292 219 366)(19 191 108 270 55 158 407 293 220 367)(20 192 109 271 56 159 408 294 199 368)(21 193 110 272 57 160 409 295 200 369)(22 194 89 273 58 161 410 296 201 370)(23 68 116 258 319 425 138 221 395 352)(24 69 117 259 320 426 139 222 396 331)(25 70 118 260 321 427 140 223 375 332)(26 71 119 261 322 428 141 224 376 333)(27 72 120 262 323 429 142 225 377 334)(28 73 121 263 324 430 143 226 378 335)(29 74 122 264 325 431 144 227 379 336)(30 75 123 243 326 432 145 228 380 337)(31 76 124 244 327 433 146 229 381 338)(32 77 125 245 328 434 147 230 382 339)(33 78 126 246 329 435 148 231 383 340)(34 79 127 247 330 436 149 232 384 341)(35 80 128 248 309 437 150 233 385 342)(36 81 129 249 310 438 151 234 386 343)(37 82 130 250 311 439 152 235 387 344)(38 83 131 251 312 440 153 236 388 345)(39 84 132 252 313 419 154 237 389 346)(40 85 111 253 314 420 133 238 390 347)(41 86 112 254 315 421 134 239 391 348)(42 87 113 255 316 422 135 240 392 349)(43 88 114 256 317 423 136 241 393 350)(44 67 115 257 318 424 137 242 394 351)
(1 148 162 78)(2 149 163 79)(3 150 164 80)(4 151 165 81)(5 152 166 82)(6 153 167 83)(7 154 168 84)(8 133 169 85)(9 134 170 86)(10 135 171 87)(11 136 172 88)(12 137 173 67)(13 138 174 68)(14 139 175 69)(15 140 176 70)(16 141 155 71)(17 142 156 72)(18 143 157 73)(19 144 158 74)(20 145 159 75)(21 146 160 76)(22 147 161 77)(23 185 425 401)(24 186 426 402)(25 187 427 403)(26 188 428 404)(27 189 429 405)(28 190 430 406)(29 191 431 407)(30 192 432 408)(31 193 433 409)(32 194 434 410)(33 195 435 411)(34 196 436 412)(35 197 437 413)(36 198 438 414)(37 177 439 415)(38 178 440 416)(39 179 419 417)(40 180 420 418)(41 181 421 397)(42 182 422 398)(43 183 423 399)(44 184 424 400)(45 112 357 239)(46 113 358 240)(47 114 359 241)(48 115 360 242)(49 116 361 221)(50 117 362 222)(51 118 363 223)(52 119 364 224)(53 120 365 225)(54 121 366 226)(55 122 367 227)(56 123 368 228)(57 124 369 229)(58 125 370 230)(59 126 371 231)(60 127 372 232)(61 128 373 233)(62 129 374 234)(63 130 353 235)(64 131 354 236)(65 132 355 237)(66 111 356 238)(89 328 296 339)(90 329 297 340)(91 330 298 341)(92 309 299 342)(93 310 300 343)(94 311 301 344)(95 312 302 345)(96 313 303 346)(97 314 304 347)(98 315 305 348)(99 316 306 349)(100 317 307 350)(101 318 308 351)(102 319 287 352)(103 320 288 331)(104 321 289 332)(105 322 290 333)(106 323 291 334)(107 324 292 335)(108 325 293 336)(109 326 294 337)(110 327 295 338)(199 380 271 243)(200 381 272 244)(201 382 273 245)(202 383 274 246)(203 384 275 247)(204 385 276 248)(205 386 277 249)(206 387 278 250)(207 388 279 251)(208 389 280 252)(209 390 281 253)(210 391 282 254)(211 392 283 255)(212 393 284 256)(213 394 285 257)(214 395 286 258)(215 396 265 259)(216 375 266 260)(217 376 267 261)(218 377 268 262)(219 378 269 263)(220 379 270 264)
G:=sub<Sym(440)| (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), (1,195,90,274,59,162,411,297,202,371)(2,196,91,275,60,163,412,298,203,372)(3,197,92,276,61,164,413,299,204,373)(4,198,93,277,62,165,414,300,205,374)(5,177,94,278,63,166,415,301,206,353)(6,178,95,279,64,167,416,302,207,354)(7,179,96,280,65,168,417,303,208,355)(8,180,97,281,66,169,418,304,209,356)(9,181,98,282,45,170,397,305,210,357)(10,182,99,283,46,171,398,306,211,358)(11,183,100,284,47,172,399,307,212,359)(12,184,101,285,48,173,400,308,213,360)(13,185,102,286,49,174,401,287,214,361)(14,186,103,265,50,175,402,288,215,362)(15,187,104,266,51,176,403,289,216,363)(16,188,105,267,52,155,404,290,217,364)(17,189,106,268,53,156,405,291,218,365)(18,190,107,269,54,157,406,292,219,366)(19,191,108,270,55,158,407,293,220,367)(20,192,109,271,56,159,408,294,199,368)(21,193,110,272,57,160,409,295,200,369)(22,194,89,273,58,161,410,296,201,370)(23,68,116,258,319,425,138,221,395,352)(24,69,117,259,320,426,139,222,396,331)(25,70,118,260,321,427,140,223,375,332)(26,71,119,261,322,428,141,224,376,333)(27,72,120,262,323,429,142,225,377,334)(28,73,121,263,324,430,143,226,378,335)(29,74,122,264,325,431,144,227,379,336)(30,75,123,243,326,432,145,228,380,337)(31,76,124,244,327,433,146,229,381,338)(32,77,125,245,328,434,147,230,382,339)(33,78,126,246,329,435,148,231,383,340)(34,79,127,247,330,436,149,232,384,341)(35,80,128,248,309,437,150,233,385,342)(36,81,129,249,310,438,151,234,386,343)(37,82,130,250,311,439,152,235,387,344)(38,83,131,251,312,440,153,236,388,345)(39,84,132,252,313,419,154,237,389,346)(40,85,111,253,314,420,133,238,390,347)(41,86,112,254,315,421,134,239,391,348)(42,87,113,255,316,422,135,240,392,349)(43,88,114,256,317,423,136,241,393,350)(44,67,115,257,318,424,137,242,394,351), (1,148,162,78)(2,149,163,79)(3,150,164,80)(4,151,165,81)(5,152,166,82)(6,153,167,83)(7,154,168,84)(8,133,169,85)(9,134,170,86)(10,135,171,87)(11,136,172,88)(12,137,173,67)(13,138,174,68)(14,139,175,69)(15,140,176,70)(16,141,155,71)(17,142,156,72)(18,143,157,73)(19,144,158,74)(20,145,159,75)(21,146,160,76)(22,147,161,77)(23,185,425,401)(24,186,426,402)(25,187,427,403)(26,188,428,404)(27,189,429,405)(28,190,430,406)(29,191,431,407)(30,192,432,408)(31,193,433,409)(32,194,434,410)(33,195,435,411)(34,196,436,412)(35,197,437,413)(36,198,438,414)(37,177,439,415)(38,178,440,416)(39,179,419,417)(40,180,420,418)(41,181,421,397)(42,182,422,398)(43,183,423,399)(44,184,424,400)(45,112,357,239)(46,113,358,240)(47,114,359,241)(48,115,360,242)(49,116,361,221)(50,117,362,222)(51,118,363,223)(52,119,364,224)(53,120,365,225)(54,121,366,226)(55,122,367,227)(56,123,368,228)(57,124,369,229)(58,125,370,230)(59,126,371,231)(60,127,372,232)(61,128,373,233)(62,129,374,234)(63,130,353,235)(64,131,354,236)(65,132,355,237)(66,111,356,238)(89,328,296,339)(90,329,297,340)(91,330,298,341)(92,309,299,342)(93,310,300,343)(94,311,301,344)(95,312,302,345)(96,313,303,346)(97,314,304,347)(98,315,305,348)(99,316,306,349)(100,317,307,350)(101,318,308,351)(102,319,287,352)(103,320,288,331)(104,321,289,332)(105,322,290,333)(106,323,291,334)(107,324,292,335)(108,325,293,336)(109,326,294,337)(110,327,295,338)(199,380,271,243)(200,381,272,244)(201,382,273,245)(202,383,274,246)(203,384,275,247)(204,385,276,248)(205,386,277,249)(206,387,278,250)(207,388,279,251)(208,389,280,252)(209,390,281,253)(210,391,282,254)(211,392,283,255)(212,393,284,256)(213,394,285,257)(214,395,286,258)(215,396,265,259)(216,375,266,260)(217,376,267,261)(218,377,268,262)(219,378,269,263)(220,379,270,264)>;
G:=Group( (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), (1,195,90,274,59,162,411,297,202,371)(2,196,91,275,60,163,412,298,203,372)(3,197,92,276,61,164,413,299,204,373)(4,198,93,277,62,165,414,300,205,374)(5,177,94,278,63,166,415,301,206,353)(6,178,95,279,64,167,416,302,207,354)(7,179,96,280,65,168,417,303,208,355)(8,180,97,281,66,169,418,304,209,356)(9,181,98,282,45,170,397,305,210,357)(10,182,99,283,46,171,398,306,211,358)(11,183,100,284,47,172,399,307,212,359)(12,184,101,285,48,173,400,308,213,360)(13,185,102,286,49,174,401,287,214,361)(14,186,103,265,50,175,402,288,215,362)(15,187,104,266,51,176,403,289,216,363)(16,188,105,267,52,155,404,290,217,364)(17,189,106,268,53,156,405,291,218,365)(18,190,107,269,54,157,406,292,219,366)(19,191,108,270,55,158,407,293,220,367)(20,192,109,271,56,159,408,294,199,368)(21,193,110,272,57,160,409,295,200,369)(22,194,89,273,58,161,410,296,201,370)(23,68,116,258,319,425,138,221,395,352)(24,69,117,259,320,426,139,222,396,331)(25,70,118,260,321,427,140,223,375,332)(26,71,119,261,322,428,141,224,376,333)(27,72,120,262,323,429,142,225,377,334)(28,73,121,263,324,430,143,226,378,335)(29,74,122,264,325,431,144,227,379,336)(30,75,123,243,326,432,145,228,380,337)(31,76,124,244,327,433,146,229,381,338)(32,77,125,245,328,434,147,230,382,339)(33,78,126,246,329,435,148,231,383,340)(34,79,127,247,330,436,149,232,384,341)(35,80,128,248,309,437,150,233,385,342)(36,81,129,249,310,438,151,234,386,343)(37,82,130,250,311,439,152,235,387,344)(38,83,131,251,312,440,153,236,388,345)(39,84,132,252,313,419,154,237,389,346)(40,85,111,253,314,420,133,238,390,347)(41,86,112,254,315,421,134,239,391,348)(42,87,113,255,316,422,135,240,392,349)(43,88,114,256,317,423,136,241,393,350)(44,67,115,257,318,424,137,242,394,351), (1,148,162,78)(2,149,163,79)(3,150,164,80)(4,151,165,81)(5,152,166,82)(6,153,167,83)(7,154,168,84)(8,133,169,85)(9,134,170,86)(10,135,171,87)(11,136,172,88)(12,137,173,67)(13,138,174,68)(14,139,175,69)(15,140,176,70)(16,141,155,71)(17,142,156,72)(18,143,157,73)(19,144,158,74)(20,145,159,75)(21,146,160,76)(22,147,161,77)(23,185,425,401)(24,186,426,402)(25,187,427,403)(26,188,428,404)(27,189,429,405)(28,190,430,406)(29,191,431,407)(30,192,432,408)(31,193,433,409)(32,194,434,410)(33,195,435,411)(34,196,436,412)(35,197,437,413)(36,198,438,414)(37,177,439,415)(38,178,440,416)(39,179,419,417)(40,180,420,418)(41,181,421,397)(42,182,422,398)(43,183,423,399)(44,184,424,400)(45,112,357,239)(46,113,358,240)(47,114,359,241)(48,115,360,242)(49,116,361,221)(50,117,362,222)(51,118,363,223)(52,119,364,224)(53,120,365,225)(54,121,366,226)(55,122,367,227)(56,123,368,228)(57,124,369,229)(58,125,370,230)(59,126,371,231)(60,127,372,232)(61,128,373,233)(62,129,374,234)(63,130,353,235)(64,131,354,236)(65,132,355,237)(66,111,356,238)(89,328,296,339)(90,329,297,340)(91,330,298,341)(92,309,299,342)(93,310,300,343)(94,311,301,344)(95,312,302,345)(96,313,303,346)(97,314,304,347)(98,315,305,348)(99,316,306,349)(100,317,307,350)(101,318,308,351)(102,319,287,352)(103,320,288,331)(104,321,289,332)(105,322,290,333)(106,323,291,334)(107,324,292,335)(108,325,293,336)(109,326,294,337)(110,327,295,338)(199,380,271,243)(200,381,272,244)(201,382,273,245)(202,383,274,246)(203,384,275,247)(204,385,276,248)(205,386,277,249)(206,387,278,250)(207,388,279,251)(208,389,280,252)(209,390,281,253)(210,391,282,254)(211,392,283,255)(212,393,284,256)(213,394,285,257)(214,395,286,258)(215,396,265,259)(216,375,266,260)(217,376,267,261)(218,377,268,262)(219,378,269,263)(220,379,270,264) );
G=PermutationGroup([[(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)], [(1,195,90,274,59,162,411,297,202,371),(2,196,91,275,60,163,412,298,203,372),(3,197,92,276,61,164,413,299,204,373),(4,198,93,277,62,165,414,300,205,374),(5,177,94,278,63,166,415,301,206,353),(6,178,95,279,64,167,416,302,207,354),(7,179,96,280,65,168,417,303,208,355),(8,180,97,281,66,169,418,304,209,356),(9,181,98,282,45,170,397,305,210,357),(10,182,99,283,46,171,398,306,211,358),(11,183,100,284,47,172,399,307,212,359),(12,184,101,285,48,173,400,308,213,360),(13,185,102,286,49,174,401,287,214,361),(14,186,103,265,50,175,402,288,215,362),(15,187,104,266,51,176,403,289,216,363),(16,188,105,267,52,155,404,290,217,364),(17,189,106,268,53,156,405,291,218,365),(18,190,107,269,54,157,406,292,219,366),(19,191,108,270,55,158,407,293,220,367),(20,192,109,271,56,159,408,294,199,368),(21,193,110,272,57,160,409,295,200,369),(22,194,89,273,58,161,410,296,201,370),(23,68,116,258,319,425,138,221,395,352),(24,69,117,259,320,426,139,222,396,331),(25,70,118,260,321,427,140,223,375,332),(26,71,119,261,322,428,141,224,376,333),(27,72,120,262,323,429,142,225,377,334),(28,73,121,263,324,430,143,226,378,335),(29,74,122,264,325,431,144,227,379,336),(30,75,123,243,326,432,145,228,380,337),(31,76,124,244,327,433,146,229,381,338),(32,77,125,245,328,434,147,230,382,339),(33,78,126,246,329,435,148,231,383,340),(34,79,127,247,330,436,149,232,384,341),(35,80,128,248,309,437,150,233,385,342),(36,81,129,249,310,438,151,234,386,343),(37,82,130,250,311,439,152,235,387,344),(38,83,131,251,312,440,153,236,388,345),(39,84,132,252,313,419,154,237,389,346),(40,85,111,253,314,420,133,238,390,347),(41,86,112,254,315,421,134,239,391,348),(42,87,113,255,316,422,135,240,392,349),(43,88,114,256,317,423,136,241,393,350),(44,67,115,257,318,424,137,242,394,351)], [(1,148,162,78),(2,149,163,79),(3,150,164,80),(4,151,165,81),(5,152,166,82),(6,153,167,83),(7,154,168,84),(8,133,169,85),(9,134,170,86),(10,135,171,87),(11,136,172,88),(12,137,173,67),(13,138,174,68),(14,139,175,69),(15,140,176,70),(16,141,155,71),(17,142,156,72),(18,143,157,73),(19,144,158,74),(20,145,159,75),(21,146,160,76),(22,147,161,77),(23,185,425,401),(24,186,426,402),(25,187,427,403),(26,188,428,404),(27,189,429,405),(28,190,430,406),(29,191,431,407),(30,192,432,408),(31,193,433,409),(32,194,434,410),(33,195,435,411),(34,196,436,412),(35,197,437,413),(36,198,438,414),(37,177,439,415),(38,178,440,416),(39,179,419,417),(40,180,420,418),(41,181,421,397),(42,182,422,398),(43,183,423,399),(44,184,424,400),(45,112,357,239),(46,113,358,240),(47,114,359,241),(48,115,360,242),(49,116,361,221),(50,117,362,222),(51,118,363,223),(52,119,364,224),(53,120,365,225),(54,121,366,226),(55,122,367,227),(56,123,368,228),(57,124,369,229),(58,125,370,230),(59,126,371,231),(60,127,372,232),(61,128,373,233),(62,129,374,234),(63,130,353,235),(64,131,354,236),(65,132,355,237),(66,111,356,238),(89,328,296,339),(90,329,297,340),(91,330,298,341),(92,309,299,342),(93,310,300,343),(94,311,301,344),(95,312,302,345),(96,313,303,346),(97,314,304,347),(98,315,305,348),(99,316,306,349),(100,317,307,350),(101,318,308,351),(102,319,287,352),(103,320,288,331),(104,321,289,332),(105,322,290,333),(106,323,291,334),(107,324,292,335),(108,325,293,336),(109,326,294,337),(110,327,295,338),(199,380,271,243),(200,381,272,244),(201,382,273,245),(202,383,274,246),(203,384,275,247),(204,385,276,248),(205,386,277,249),(206,387,278,250),(207,388,279,251),(208,389,280,252),(209,390,281,253),(210,391,282,254),(211,392,283,255),(212,393,284,256),(213,394,285,257),(214,395,286,258),(215,396,265,259),(216,375,266,260),(217,376,267,261),(218,377,268,262),(219,378,269,263),(220,379,270,264)]])
176 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 10A | ··· | 10F | 11A | ··· | 11J | 22A | ··· | 22AD | 44A | ··· | 44AN | 55A | ··· | 55T | 110A | ··· | 110BH |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
176 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C11 | C22 | C22 | C44 | D5 | Dic5 | D10 | D5×C11 | C11×Dic5 | D5×C22 |
| kernel | Dic5×C22 | C11×Dic5 | C2×C110 | C110 | C2×Dic5 | Dic5 | C2×C10 | C10 | C2×C22 | C22 | C22 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 10 | 20 | 10 | 40 | 2 | 4 | 2 | 20 | 40 | 20 |
Matrix representation of Dic5×C22 ►in GL3(𝔽661) generated by
| 660 | 0 | 0 |
| 0 | 220 | 0 |
| 0 | 0 | 220 |
| 660 | 0 | 0 |
| 0 | 1 | 660 |
| 0 | 59 | 603 |
| 106 | 0 | 0 |
| 0 | 189 | 187 |
| 0 | 371 | 472 |
G:=sub<GL(3,GF(661))| [660,0,0,0,220,0,0,0,220],[660,0,0,0,1,59,0,660,603],[106,0,0,0,189,371,0,187,472] >;
Dic5×C22 in GAP, Magma, Sage, TeX
{\rm Dic}_5\times C_{22} % in TeX
G:=Group("Dic5xC22"); // GroupNames label
G:=SmallGroup(440,32);
// by ID
G=gap.SmallGroup(440,32);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-5,220,8804]);
// Polycyclic
G:=Group<a,b,c|a^22=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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