metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C55⋊Q8, C5⋊1Dic22, C22.7D10, C10.7D22, C11⋊1Dic10, Dic11.D5, C110.7C22, Dic55.2C2, Dic5.1D11, C2.7(D5×D11), (C11×Dic5).1C2, (C5×Dic11).1C2, SmallGroup(440,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C55⋊Q8
G = < a,b,c | a55=b4=1, c2=b2, bab-1=a34, cac-1=a21, cbc-1=b-1 >
Smallest permutation representation of C55⋊Q8
►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)
(1 196 69 147)(2 175 70 126)(3 209 71 160)(4 188 72 139)(5 167 73 118)(6 201 74 152)(7 180 75 131)(8 214 76 165)(9 193 77 144)(10 172 78 123)(11 206 79 157)(12 185 80 136)(13 219 81 115)(14 198 82 149)(15 177 83 128)(16 211 84 162)(17 190 85 141)(18 169 86 120)(19 203 87 154)(20 182 88 133)(21 216 89 112)(22 195 90 146)(23 174 91 125)(24 208 92 159)(25 187 93 138)(26 166 94 117)(27 200 95 151)(28 179 96 130)(29 213 97 164)(30 192 98 143)(31 171 99 122)(32 205 100 156)(33 184 101 135)(34 218 102 114)(35 197 103 148)(36 176 104 127)(37 210 105 161)(38 189 106 140)(39 168 107 119)(40 202 108 153)(41 181 109 132)(42 215 110 111)(43 194 56 145)(44 173 57 124)(45 207 58 158)(46 186 59 137)(47 220 60 116)(48 199 61 150)(49 178 62 129)(50 212 63 163)(51 191 64 142)(52 170 65 121)(53 204 66 155)(54 183 67 134)(55 217 68 113)(221 342 286 428)(222 376 287 407)(223 355 288 386)(224 334 289 420)(225 368 290 399)(226 347 291 433)(227 381 292 412)(228 360 293 391)(229 339 294 425)(230 373 295 404)(231 352 296 438)(232 331 297 417)(233 365 298 396)(234 344 299 430)(235 378 300 409)(236 357 301 388)(237 336 302 422)(238 370 303 401)(239 349 304 435)(240 383 305 414)(241 362 306 393)(242 341 307 427)(243 375 308 406)(244 354 309 440)(245 333 310 419)(246 367 311 398)(247 346 312 432)(248 380 313 411)(249 359 314 390)(250 338 315 424)(251 372 316 403)(252 351 317 437)(253 385 318 416)(254 364 319 395)(255 343 320 429)(256 377 321 408)(257 356 322 387)(258 335 323 421)(259 369 324 400)(260 348 325 434)(261 382 326 413)(262 361 327 392)(263 340 328 426)(264 374 329 405)(265 353 330 439)(266 332 276 418)(267 366 277 397)(268 345 278 431)(269 379 279 410)(270 358 280 389)(271 337 281 423)(272 371 282 402)(273 350 283 436)(274 384 284 415)(275 363 285 394)
(1 303 69 238)(2 324 70 259)(3 290 71 225)(4 311 72 246)(5 277 73 267)(6 298 74 233)(7 319 75 254)(8 285 76 275)(9 306 77 241)(10 327 78 262)(11 293 79 228)(12 314 80 249)(13 280 81 270)(14 301 82 236)(15 322 83 257)(16 288 84 223)(17 309 85 244)(18 330 86 265)(19 296 87 231)(20 317 88 252)(21 283 89 273)(22 304 90 239)(23 325 91 260)(24 291 92 226)(25 312 93 247)(26 278 94 268)(27 299 95 234)(28 320 96 255)(29 286 97 221)(30 307 98 242)(31 328 99 263)(32 294 100 229)(33 315 101 250)(34 281 102 271)(35 302 103 237)(36 323 104 258)(37 289 105 224)(38 310 106 245)(39 276 107 266)(40 297 108 232)(41 318 109 253)(42 284 110 274)(43 305 56 240)(44 326 57 261)(45 292 58 227)(46 313 59 248)(47 279 60 269)(48 300 61 235)(49 321 62 256)(50 287 63 222)(51 308 64 243)(52 329 65 264)(53 295 66 230)(54 316 67 251)(55 282 68 272)(111 415 215 384)(112 436 216 350)(113 402 217 371)(114 423 218 337)(115 389 219 358)(116 410 220 379)(117 431 166 345)(118 397 167 366)(119 418 168 332)(120 439 169 353)(121 405 170 374)(122 426 171 340)(123 392 172 361)(124 413 173 382)(125 434 174 348)(126 400 175 369)(127 421 176 335)(128 387 177 356)(129 408 178 377)(130 429 179 343)(131 395 180 364)(132 416 181 385)(133 437 182 351)(134 403 183 372)(135 424 184 338)(136 390 185 359)(137 411 186 380)(138 432 187 346)(139 398 188 367)(140 419 189 333)(141 440 190 354)(142 406 191 375)(143 427 192 341)(144 393 193 362)(145 414 194 383)(146 435 195 349)(147 401 196 370)(148 422 197 336)(149 388 198 357)(150 409 199 378)(151 430 200 344)(152 396 201 365)(153 417 202 331)(154 438 203 352)(155 404 204 373)(156 425 205 339)(157 391 206 360)(158 412 207 381)(159 433 208 347)(160 399 209 368)(161 420 210 334)(162 386 211 355)(163 407 212 376)(164 428 213 342)(165 394 214 363)
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), (1,196,69,147)(2,175,70,126)(3,209,71,160)(4,188,72,139)(5,167,73,118)(6,201,74,152)(7,180,75,131)(8,214,76,165)(9,193,77,144)(10,172,78,123)(11,206,79,157)(12,185,80,136)(13,219,81,115)(14,198,82,149)(15,177,83,128)(16,211,84,162)(17,190,85,141)(18,169,86,120)(19,203,87,154)(20,182,88,133)(21,216,89,112)(22,195,90,146)(23,174,91,125)(24,208,92,159)(25,187,93,138)(26,166,94,117)(27,200,95,151)(28,179,96,130)(29,213,97,164)(30,192,98,143)(31,171,99,122)(32,205,100,156)(33,184,101,135)(34,218,102,114)(35,197,103,148)(36,176,104,127)(37,210,105,161)(38,189,106,140)(39,168,107,119)(40,202,108,153)(41,181,109,132)(42,215,110,111)(43,194,56,145)(44,173,57,124)(45,207,58,158)(46,186,59,137)(47,220,60,116)(48,199,61,150)(49,178,62,129)(50,212,63,163)(51,191,64,142)(52,170,65,121)(53,204,66,155)(54,183,67,134)(55,217,68,113)(221,342,286,428)(222,376,287,407)(223,355,288,386)(224,334,289,420)(225,368,290,399)(226,347,291,433)(227,381,292,412)(228,360,293,391)(229,339,294,425)(230,373,295,404)(231,352,296,438)(232,331,297,417)(233,365,298,396)(234,344,299,430)(235,378,300,409)(236,357,301,388)(237,336,302,422)(238,370,303,401)(239,349,304,435)(240,383,305,414)(241,362,306,393)(242,341,307,427)(243,375,308,406)(244,354,309,440)(245,333,310,419)(246,367,311,398)(247,346,312,432)(248,380,313,411)(249,359,314,390)(250,338,315,424)(251,372,316,403)(252,351,317,437)(253,385,318,416)(254,364,319,395)(255,343,320,429)(256,377,321,408)(257,356,322,387)(258,335,323,421)(259,369,324,400)(260,348,325,434)(261,382,326,413)(262,361,327,392)(263,340,328,426)(264,374,329,405)(265,353,330,439)(266,332,276,418)(267,366,277,397)(268,345,278,431)(269,379,279,410)(270,358,280,389)(271,337,281,423)(272,371,282,402)(273,350,283,436)(274,384,284,415)(275,363,285,394), (1,303,69,238)(2,324,70,259)(3,290,71,225)(4,311,72,246)(5,277,73,267)(6,298,74,233)(7,319,75,254)(8,285,76,275)(9,306,77,241)(10,327,78,262)(11,293,79,228)(12,314,80,249)(13,280,81,270)(14,301,82,236)(15,322,83,257)(16,288,84,223)(17,309,85,244)(18,330,86,265)(19,296,87,231)(20,317,88,252)(21,283,89,273)(22,304,90,239)(23,325,91,260)(24,291,92,226)(25,312,93,247)(26,278,94,268)(27,299,95,234)(28,320,96,255)(29,286,97,221)(30,307,98,242)(31,328,99,263)(32,294,100,229)(33,315,101,250)(34,281,102,271)(35,302,103,237)(36,323,104,258)(37,289,105,224)(38,310,106,245)(39,276,107,266)(40,297,108,232)(41,318,109,253)(42,284,110,274)(43,305,56,240)(44,326,57,261)(45,292,58,227)(46,313,59,248)(47,279,60,269)(48,300,61,235)(49,321,62,256)(50,287,63,222)(51,308,64,243)(52,329,65,264)(53,295,66,230)(54,316,67,251)(55,282,68,272)(111,415,215,384)(112,436,216,350)(113,402,217,371)(114,423,218,337)(115,389,219,358)(116,410,220,379)(117,431,166,345)(118,397,167,366)(119,418,168,332)(120,439,169,353)(121,405,170,374)(122,426,171,340)(123,392,172,361)(124,413,173,382)(125,434,174,348)(126,400,175,369)(127,421,176,335)(128,387,177,356)(129,408,178,377)(130,429,179,343)(131,395,180,364)(132,416,181,385)(133,437,182,351)(134,403,183,372)(135,424,184,338)(136,390,185,359)(137,411,186,380)(138,432,187,346)(139,398,188,367)(140,419,189,333)(141,440,190,354)(142,406,191,375)(143,427,192,341)(144,393,193,362)(145,414,194,383)(146,435,195,349)(147,401,196,370)(148,422,197,336)(149,388,198,357)(150,409,199,378)(151,430,200,344)(152,396,201,365)(153,417,202,331)(154,438,203,352)(155,404,204,373)(156,425,205,339)(157,391,206,360)(158,412,207,381)(159,433,208,347)(160,399,209,368)(161,420,210,334)(162,386,211,355)(163,407,212,376)(164,428,213,342)(165,394,214,363)>;
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), (1,196,69,147)(2,175,70,126)(3,209,71,160)(4,188,72,139)(5,167,73,118)(6,201,74,152)(7,180,75,131)(8,214,76,165)(9,193,77,144)(10,172,78,123)(11,206,79,157)(12,185,80,136)(13,219,81,115)(14,198,82,149)(15,177,83,128)(16,211,84,162)(17,190,85,141)(18,169,86,120)(19,203,87,154)(20,182,88,133)(21,216,89,112)(22,195,90,146)(23,174,91,125)(24,208,92,159)(25,187,93,138)(26,166,94,117)(27,200,95,151)(28,179,96,130)(29,213,97,164)(30,192,98,143)(31,171,99,122)(32,205,100,156)(33,184,101,135)(34,218,102,114)(35,197,103,148)(36,176,104,127)(37,210,105,161)(38,189,106,140)(39,168,107,119)(40,202,108,153)(41,181,109,132)(42,215,110,111)(43,194,56,145)(44,173,57,124)(45,207,58,158)(46,186,59,137)(47,220,60,116)(48,199,61,150)(49,178,62,129)(50,212,63,163)(51,191,64,142)(52,170,65,121)(53,204,66,155)(54,183,67,134)(55,217,68,113)(221,342,286,428)(222,376,287,407)(223,355,288,386)(224,334,289,420)(225,368,290,399)(226,347,291,433)(227,381,292,412)(228,360,293,391)(229,339,294,425)(230,373,295,404)(231,352,296,438)(232,331,297,417)(233,365,298,396)(234,344,299,430)(235,378,300,409)(236,357,301,388)(237,336,302,422)(238,370,303,401)(239,349,304,435)(240,383,305,414)(241,362,306,393)(242,341,307,427)(243,375,308,406)(244,354,309,440)(245,333,310,419)(246,367,311,398)(247,346,312,432)(248,380,313,411)(249,359,314,390)(250,338,315,424)(251,372,316,403)(252,351,317,437)(253,385,318,416)(254,364,319,395)(255,343,320,429)(256,377,321,408)(257,356,322,387)(258,335,323,421)(259,369,324,400)(260,348,325,434)(261,382,326,413)(262,361,327,392)(263,340,328,426)(264,374,329,405)(265,353,330,439)(266,332,276,418)(267,366,277,397)(268,345,278,431)(269,379,279,410)(270,358,280,389)(271,337,281,423)(272,371,282,402)(273,350,283,436)(274,384,284,415)(275,363,285,394), (1,303,69,238)(2,324,70,259)(3,290,71,225)(4,311,72,246)(5,277,73,267)(6,298,74,233)(7,319,75,254)(8,285,76,275)(9,306,77,241)(10,327,78,262)(11,293,79,228)(12,314,80,249)(13,280,81,270)(14,301,82,236)(15,322,83,257)(16,288,84,223)(17,309,85,244)(18,330,86,265)(19,296,87,231)(20,317,88,252)(21,283,89,273)(22,304,90,239)(23,325,91,260)(24,291,92,226)(25,312,93,247)(26,278,94,268)(27,299,95,234)(28,320,96,255)(29,286,97,221)(30,307,98,242)(31,328,99,263)(32,294,100,229)(33,315,101,250)(34,281,102,271)(35,302,103,237)(36,323,104,258)(37,289,105,224)(38,310,106,245)(39,276,107,266)(40,297,108,232)(41,318,109,253)(42,284,110,274)(43,305,56,240)(44,326,57,261)(45,292,58,227)(46,313,59,248)(47,279,60,269)(48,300,61,235)(49,321,62,256)(50,287,63,222)(51,308,64,243)(52,329,65,264)(53,295,66,230)(54,316,67,251)(55,282,68,272)(111,415,215,384)(112,436,216,350)(113,402,217,371)(114,423,218,337)(115,389,219,358)(116,410,220,379)(117,431,166,345)(118,397,167,366)(119,418,168,332)(120,439,169,353)(121,405,170,374)(122,426,171,340)(123,392,172,361)(124,413,173,382)(125,434,174,348)(126,400,175,369)(127,421,176,335)(128,387,177,356)(129,408,178,377)(130,429,179,343)(131,395,180,364)(132,416,181,385)(133,437,182,351)(134,403,183,372)(135,424,184,338)(136,390,185,359)(137,411,186,380)(138,432,187,346)(139,398,188,367)(140,419,189,333)(141,440,190,354)(142,406,191,375)(143,427,192,341)(144,393,193,362)(145,414,194,383)(146,435,195,349)(147,401,196,370)(148,422,197,336)(149,388,198,357)(150,409,199,378)(151,430,200,344)(152,396,201,365)(153,417,202,331)(154,438,203,352)(155,404,204,373)(156,425,205,339)(157,391,206,360)(158,412,207,381)(159,433,208,347)(160,399,209,368)(161,420,210,334)(162,386,211,355)(163,407,212,376)(164,428,213,342)(165,394,214,363) );
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)], [(1,196,69,147),(2,175,70,126),(3,209,71,160),(4,188,72,139),(5,167,73,118),(6,201,74,152),(7,180,75,131),(8,214,76,165),(9,193,77,144),(10,172,78,123),(11,206,79,157),(12,185,80,136),(13,219,81,115),(14,198,82,149),(15,177,83,128),(16,211,84,162),(17,190,85,141),(18,169,86,120),(19,203,87,154),(20,182,88,133),(21,216,89,112),(22,195,90,146),(23,174,91,125),(24,208,92,159),(25,187,93,138),(26,166,94,117),(27,200,95,151),(28,179,96,130),(29,213,97,164),(30,192,98,143),(31,171,99,122),(32,205,100,156),(33,184,101,135),(34,218,102,114),(35,197,103,148),(36,176,104,127),(37,210,105,161),(38,189,106,140),(39,168,107,119),(40,202,108,153),(41,181,109,132),(42,215,110,111),(43,194,56,145),(44,173,57,124),(45,207,58,158),(46,186,59,137),(47,220,60,116),(48,199,61,150),(49,178,62,129),(50,212,63,163),(51,191,64,142),(52,170,65,121),(53,204,66,155),(54,183,67,134),(55,217,68,113),(221,342,286,428),(222,376,287,407),(223,355,288,386),(224,334,289,420),(225,368,290,399),(226,347,291,433),(227,381,292,412),(228,360,293,391),(229,339,294,425),(230,373,295,404),(231,352,296,438),(232,331,297,417),(233,365,298,396),(234,344,299,430),(235,378,300,409),(236,357,301,388),(237,336,302,422),(238,370,303,401),(239,349,304,435),(240,383,305,414),(241,362,306,393),(242,341,307,427),(243,375,308,406),(244,354,309,440),(245,333,310,419),(246,367,311,398),(247,346,312,432),(248,380,313,411),(249,359,314,390),(250,338,315,424),(251,372,316,403),(252,351,317,437),(253,385,318,416),(254,364,319,395),(255,343,320,429),(256,377,321,408),(257,356,322,387),(258,335,323,421),(259,369,324,400),(260,348,325,434),(261,382,326,413),(262,361,327,392),(263,340,328,426),(264,374,329,405),(265,353,330,439),(266,332,276,418),(267,366,277,397),(268,345,278,431),(269,379,279,410),(270,358,280,389),(271,337,281,423),(272,371,282,402),(273,350,283,436),(274,384,284,415),(275,363,285,394)], [(1,303,69,238),(2,324,70,259),(3,290,71,225),(4,311,72,246),(5,277,73,267),(6,298,74,233),(7,319,75,254),(8,285,76,275),(9,306,77,241),(10,327,78,262),(11,293,79,228),(12,314,80,249),(13,280,81,270),(14,301,82,236),(15,322,83,257),(16,288,84,223),(17,309,85,244),(18,330,86,265),(19,296,87,231),(20,317,88,252),(21,283,89,273),(22,304,90,239),(23,325,91,260),(24,291,92,226),(25,312,93,247),(26,278,94,268),(27,299,95,234),(28,320,96,255),(29,286,97,221),(30,307,98,242),(31,328,99,263),(32,294,100,229),(33,315,101,250),(34,281,102,271),(35,302,103,237),(36,323,104,258),(37,289,105,224),(38,310,106,245),(39,276,107,266),(40,297,108,232),(41,318,109,253),(42,284,110,274),(43,305,56,240),(44,326,57,261),(45,292,58,227),(46,313,59,248),(47,279,60,269),(48,300,61,235),(49,321,62,256),(50,287,63,222),(51,308,64,243),(52,329,65,264),(53,295,66,230),(54,316,67,251),(55,282,68,272),(111,415,215,384),(112,436,216,350),(113,402,217,371),(114,423,218,337),(115,389,219,358),(116,410,220,379),(117,431,166,345),(118,397,167,366),(119,418,168,332),(120,439,169,353),(121,405,170,374),(122,426,171,340),(123,392,172,361),(124,413,173,382),(125,434,174,348),(126,400,175,369),(127,421,176,335),(128,387,177,356),(129,408,178,377),(130,429,179,343),(131,395,180,364),(132,416,181,385),(133,437,182,351),(134,403,183,372),(135,424,184,338),(136,390,185,359),(137,411,186,380),(138,432,187,346),(139,398,188,367),(140,419,189,333),(141,440,190,354),(142,406,191,375),(143,427,192,341),(144,393,193,362),(145,414,194,383),(146,435,195,349),(147,401,196,370),(148,422,197,336),(149,388,198,357),(150,409,199,378),(151,430,200,344),(152,396,201,365),(153,417,202,331),(154,438,203,352),(155,404,204,373),(156,425,205,339),(157,391,206,360),(158,412,207,381),(159,433,208,347),(160,399,209,368),(161,420,210,334),(162,386,211,355),(163,407,212,376),(164,428,213,342),(165,394,214,363)]])
53 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 10A | 10B | 11A | ··· | 11E | 20A | 20B | 20C | 20D | 22A | ··· | 22E | 44A | ··· | 44J | 55A | ··· | 55J | 110A | ··· | 110J |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 10 | 10 | 11 | ··· | 11 | 20 | 20 | 20 | 20 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 10 | 22 | 110 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D5 | D10 | D11 | Dic10 | D22 | Dic22 | D5×D11 | C55⋊Q8 |
| kernel | C55⋊Q8 | C11×Dic5 | C5×Dic11 | Dic55 | C55 | Dic11 | C22 | Dic5 | C11 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 4 | 5 | 10 | 10 | 10 |
Matrix representation of C55⋊Q8 ►in GL4(𝔽661) generated by
| 0 | 1 | 0 | 0 |
| 660 | 603 | 0 | 0 |
| 0 | 0 | 18 | 123 |
| 0 | 0 | 572 | 567 |
| 511 | 427 | 0 | 0 |
| 534 | 150 | 0 | 0 |
| 0 | 0 | 542 | 317 |
| 0 | 0 | 437 | 119 |
| 624 | 409 | 0 | 0 |
| 252 | 37 | 0 | 0 |
| 0 | 0 | 460 | 626 |
| 0 | 0 | 550 | 201 |
G:=sub<GL(4,GF(661))| [0,660,0,0,1,603,0,0,0,0,18,572,0,0,123,567],[511,534,0,0,427,150,0,0,0,0,542,437,0,0,317,119],[624,252,0,0,409,37,0,0,0,0,460,550,0,0,626,201] >;
C55⋊Q8 in GAP, Magma, Sage, TeX
C_{55}\rtimes Q_8 % in TeX
G:=Group("C55:Q8"); // GroupNames label
G:=SmallGroup(440,23);
// by ID
G=gap.SmallGroup(440,23);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,20,61,26,328,10004]);
// Polycyclic
G:=Group<a,b,c|a^55=b^4=1,c^2=b^2,b*a*b^-1=a^34,c*a*c^-1=a^21,c*b*c^-1=b^-1>;
// generators/relations
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