direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C55, C4.C110, C20.3C22, C220.7C2, C44.7C10, C110.24C22, C10.7(C2×C22), C2.2(C2×C110), C22.15(C2×C10), SmallGroup(440,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C55
G = < a,b,c | a55=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C55
►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 64 411 282)(2 65 412 283)(3 66 413 284)(4 67 414 285)(5 68 415 286)(6 69 416 287)(7 70 417 288)(8 71 418 289)(9 72 419 290)(10 73 420 291)(11 74 421 292)(12 75 422 293)(13 76 423 294)(14 77 424 295)(15 78 425 296)(16 79 426 297)(17 80 427 298)(18 81 428 299)(19 82 429 300)(20 83 430 301)(21 84 431 302)(22 85 432 303)(23 86 433 304)(24 87 434 305)(25 88 435 306)(26 89 436 307)(27 90 437 308)(28 91 438 309)(29 92 439 310)(30 93 440 311)(31 94 386 312)(32 95 387 313)(33 96 388 314)(34 97 389 315)(35 98 390 316)(36 99 391 317)(37 100 392 318)(38 101 393 319)(39 102 394 320)(40 103 395 321)(41 104 396 322)(42 105 397 323)(43 106 398 324)(44 107 399 325)(45 108 400 326)(46 109 401 327)(47 110 402 328)(48 56 403 329)(49 57 404 330)(50 58 405 276)(51 59 406 277)(52 60 407 278)(53 61 408 279)(54 62 409 280)(55 63 410 281)(111 373 229 189)(112 374 230 190)(113 375 231 191)(114 376 232 192)(115 377 233 193)(116 378 234 194)(117 379 235 195)(118 380 236 196)(119 381 237 197)(120 382 238 198)(121 383 239 199)(122 384 240 200)(123 385 241 201)(124 331 242 202)(125 332 243 203)(126 333 244 204)(127 334 245 205)(128 335 246 206)(129 336 247 207)(130 337 248 208)(131 338 249 209)(132 339 250 210)(133 340 251 211)(134 341 252 212)(135 342 253 213)(136 343 254 214)(137 344 255 215)(138 345 256 216)(139 346 257 217)(140 347 258 218)(141 348 259 219)(142 349 260 220)(143 350 261 166)(144 351 262 167)(145 352 263 168)(146 353 264 169)(147 354 265 170)(148 355 266 171)(149 356 267 172)(150 357 268 173)(151 358 269 174)(152 359 270 175)(153 360 271 176)(154 361 272 177)(155 362 273 178)(156 363 274 179)(157 364 275 180)(158 365 221 181)(159 366 222 182)(160 367 223 183)(161 368 224 184)(162 369 225 185)(163 370 226 186)(164 371 227 187)(165 372 228 188)
(1 168 411 352)(2 169 412 353)(3 170 413 354)(4 171 414 355)(5 172 415 356)(6 173 416 357)(7 174 417 358)(8 175 418 359)(9 176 419 360)(10 177 420 361)(11 178 421 362)(12 179 422 363)(13 180 423 364)(14 181 424 365)(15 182 425 366)(16 183 426 367)(17 184 427 368)(18 185 428 369)(19 186 429 370)(20 187 430 371)(21 188 431 372)(22 189 432 373)(23 190 433 374)(24 191 434 375)(25 192 435 376)(26 193 436 377)(27 194 437 378)(28 195 438 379)(29 196 439 380)(30 197 440 381)(31 198 386 382)(32 199 387 383)(33 200 388 384)(34 201 389 385)(35 202 390 331)(36 203 391 332)(37 204 392 333)(38 205 393 334)(39 206 394 335)(40 207 395 336)(41 208 396 337)(42 209 397 338)(43 210 398 339)(44 211 399 340)(45 212 400 341)(46 213 401 342)(47 214 402 343)(48 215 403 344)(49 216 404 345)(50 217 405 346)(51 218 406 347)(52 219 407 348)(53 220 408 349)(54 166 409 350)(55 167 410 351)(56 255 329 137)(57 256 330 138)(58 257 276 139)(59 258 277 140)(60 259 278 141)(61 260 279 142)(62 261 280 143)(63 262 281 144)(64 263 282 145)(65 264 283 146)(66 265 284 147)(67 266 285 148)(68 267 286 149)(69 268 287 150)(70 269 288 151)(71 270 289 152)(72 271 290 153)(73 272 291 154)(74 273 292 155)(75 274 293 156)(76 275 294 157)(77 221 295 158)(78 222 296 159)(79 223 297 160)(80 224 298 161)(81 225 299 162)(82 226 300 163)(83 227 301 164)(84 228 302 165)(85 229 303 111)(86 230 304 112)(87 231 305 113)(88 232 306 114)(89 233 307 115)(90 234 308 116)(91 235 309 117)(92 236 310 118)(93 237 311 119)(94 238 312 120)(95 239 313 121)(96 240 314 122)(97 241 315 123)(98 242 316 124)(99 243 317 125)(100 244 318 126)(101 245 319 127)(102 246 320 128)(103 247 321 129)(104 248 322 130)(105 249 323 131)(106 250 324 132)(107 251 325 133)(108 252 326 134)(109 253 327 135)(110 254 328 136)
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), (1,64,411,282)(2,65,412,283)(3,66,413,284)(4,67,414,285)(5,68,415,286)(6,69,416,287)(7,70,417,288)(8,71,418,289)(9,72,419,290)(10,73,420,291)(11,74,421,292)(12,75,422,293)(13,76,423,294)(14,77,424,295)(15,78,425,296)(16,79,426,297)(17,80,427,298)(18,81,428,299)(19,82,429,300)(20,83,430,301)(21,84,431,302)(22,85,432,303)(23,86,433,304)(24,87,434,305)(25,88,435,306)(26,89,436,307)(27,90,437,308)(28,91,438,309)(29,92,439,310)(30,93,440,311)(31,94,386,312)(32,95,387,313)(33,96,388,314)(34,97,389,315)(35,98,390,316)(36,99,391,317)(37,100,392,318)(38,101,393,319)(39,102,394,320)(40,103,395,321)(41,104,396,322)(42,105,397,323)(43,106,398,324)(44,107,399,325)(45,108,400,326)(46,109,401,327)(47,110,402,328)(48,56,403,329)(49,57,404,330)(50,58,405,276)(51,59,406,277)(52,60,407,278)(53,61,408,279)(54,62,409,280)(55,63,410,281)(111,373,229,189)(112,374,230,190)(113,375,231,191)(114,376,232,192)(115,377,233,193)(116,378,234,194)(117,379,235,195)(118,380,236,196)(119,381,237,197)(120,382,238,198)(121,383,239,199)(122,384,240,200)(123,385,241,201)(124,331,242,202)(125,332,243,203)(126,333,244,204)(127,334,245,205)(128,335,246,206)(129,336,247,207)(130,337,248,208)(131,338,249,209)(132,339,250,210)(133,340,251,211)(134,341,252,212)(135,342,253,213)(136,343,254,214)(137,344,255,215)(138,345,256,216)(139,346,257,217)(140,347,258,218)(141,348,259,219)(142,349,260,220)(143,350,261,166)(144,351,262,167)(145,352,263,168)(146,353,264,169)(147,354,265,170)(148,355,266,171)(149,356,267,172)(150,357,268,173)(151,358,269,174)(152,359,270,175)(153,360,271,176)(154,361,272,177)(155,362,273,178)(156,363,274,179)(157,364,275,180)(158,365,221,181)(159,366,222,182)(160,367,223,183)(161,368,224,184)(162,369,225,185)(163,370,226,186)(164,371,227,187)(165,372,228,188), (1,168,411,352)(2,169,412,353)(3,170,413,354)(4,171,414,355)(5,172,415,356)(6,173,416,357)(7,174,417,358)(8,175,418,359)(9,176,419,360)(10,177,420,361)(11,178,421,362)(12,179,422,363)(13,180,423,364)(14,181,424,365)(15,182,425,366)(16,183,426,367)(17,184,427,368)(18,185,428,369)(19,186,429,370)(20,187,430,371)(21,188,431,372)(22,189,432,373)(23,190,433,374)(24,191,434,375)(25,192,435,376)(26,193,436,377)(27,194,437,378)(28,195,438,379)(29,196,439,380)(30,197,440,381)(31,198,386,382)(32,199,387,383)(33,200,388,384)(34,201,389,385)(35,202,390,331)(36,203,391,332)(37,204,392,333)(38,205,393,334)(39,206,394,335)(40,207,395,336)(41,208,396,337)(42,209,397,338)(43,210,398,339)(44,211,399,340)(45,212,400,341)(46,213,401,342)(47,214,402,343)(48,215,403,344)(49,216,404,345)(50,217,405,346)(51,218,406,347)(52,219,407,348)(53,220,408,349)(54,166,409,350)(55,167,410,351)(56,255,329,137)(57,256,330,138)(58,257,276,139)(59,258,277,140)(60,259,278,141)(61,260,279,142)(62,261,280,143)(63,262,281,144)(64,263,282,145)(65,264,283,146)(66,265,284,147)(67,266,285,148)(68,267,286,149)(69,268,287,150)(70,269,288,151)(71,270,289,152)(72,271,290,153)(73,272,291,154)(74,273,292,155)(75,274,293,156)(76,275,294,157)(77,221,295,158)(78,222,296,159)(79,223,297,160)(80,224,298,161)(81,225,299,162)(82,226,300,163)(83,227,301,164)(84,228,302,165)(85,229,303,111)(86,230,304,112)(87,231,305,113)(88,232,306,114)(89,233,307,115)(90,234,308,116)(91,235,309,117)(92,236,310,118)(93,237,311,119)(94,238,312,120)(95,239,313,121)(96,240,314,122)(97,241,315,123)(98,242,316,124)(99,243,317,125)(100,244,318,126)(101,245,319,127)(102,246,320,128)(103,247,321,129)(104,248,322,130)(105,249,323,131)(106,250,324,132)(107,251,325,133)(108,252,326,134)(109,253,327,135)(110,254,328,136)>;
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), (1,64,411,282)(2,65,412,283)(3,66,413,284)(4,67,414,285)(5,68,415,286)(6,69,416,287)(7,70,417,288)(8,71,418,289)(9,72,419,290)(10,73,420,291)(11,74,421,292)(12,75,422,293)(13,76,423,294)(14,77,424,295)(15,78,425,296)(16,79,426,297)(17,80,427,298)(18,81,428,299)(19,82,429,300)(20,83,430,301)(21,84,431,302)(22,85,432,303)(23,86,433,304)(24,87,434,305)(25,88,435,306)(26,89,436,307)(27,90,437,308)(28,91,438,309)(29,92,439,310)(30,93,440,311)(31,94,386,312)(32,95,387,313)(33,96,388,314)(34,97,389,315)(35,98,390,316)(36,99,391,317)(37,100,392,318)(38,101,393,319)(39,102,394,320)(40,103,395,321)(41,104,396,322)(42,105,397,323)(43,106,398,324)(44,107,399,325)(45,108,400,326)(46,109,401,327)(47,110,402,328)(48,56,403,329)(49,57,404,330)(50,58,405,276)(51,59,406,277)(52,60,407,278)(53,61,408,279)(54,62,409,280)(55,63,410,281)(111,373,229,189)(112,374,230,190)(113,375,231,191)(114,376,232,192)(115,377,233,193)(116,378,234,194)(117,379,235,195)(118,380,236,196)(119,381,237,197)(120,382,238,198)(121,383,239,199)(122,384,240,200)(123,385,241,201)(124,331,242,202)(125,332,243,203)(126,333,244,204)(127,334,245,205)(128,335,246,206)(129,336,247,207)(130,337,248,208)(131,338,249,209)(132,339,250,210)(133,340,251,211)(134,341,252,212)(135,342,253,213)(136,343,254,214)(137,344,255,215)(138,345,256,216)(139,346,257,217)(140,347,258,218)(141,348,259,219)(142,349,260,220)(143,350,261,166)(144,351,262,167)(145,352,263,168)(146,353,264,169)(147,354,265,170)(148,355,266,171)(149,356,267,172)(150,357,268,173)(151,358,269,174)(152,359,270,175)(153,360,271,176)(154,361,272,177)(155,362,273,178)(156,363,274,179)(157,364,275,180)(158,365,221,181)(159,366,222,182)(160,367,223,183)(161,368,224,184)(162,369,225,185)(163,370,226,186)(164,371,227,187)(165,372,228,188), (1,168,411,352)(2,169,412,353)(3,170,413,354)(4,171,414,355)(5,172,415,356)(6,173,416,357)(7,174,417,358)(8,175,418,359)(9,176,419,360)(10,177,420,361)(11,178,421,362)(12,179,422,363)(13,180,423,364)(14,181,424,365)(15,182,425,366)(16,183,426,367)(17,184,427,368)(18,185,428,369)(19,186,429,370)(20,187,430,371)(21,188,431,372)(22,189,432,373)(23,190,433,374)(24,191,434,375)(25,192,435,376)(26,193,436,377)(27,194,437,378)(28,195,438,379)(29,196,439,380)(30,197,440,381)(31,198,386,382)(32,199,387,383)(33,200,388,384)(34,201,389,385)(35,202,390,331)(36,203,391,332)(37,204,392,333)(38,205,393,334)(39,206,394,335)(40,207,395,336)(41,208,396,337)(42,209,397,338)(43,210,398,339)(44,211,399,340)(45,212,400,341)(46,213,401,342)(47,214,402,343)(48,215,403,344)(49,216,404,345)(50,217,405,346)(51,218,406,347)(52,219,407,348)(53,220,408,349)(54,166,409,350)(55,167,410,351)(56,255,329,137)(57,256,330,138)(58,257,276,139)(59,258,277,140)(60,259,278,141)(61,260,279,142)(62,261,280,143)(63,262,281,144)(64,263,282,145)(65,264,283,146)(66,265,284,147)(67,266,285,148)(68,267,286,149)(69,268,287,150)(70,269,288,151)(71,270,289,152)(72,271,290,153)(73,272,291,154)(74,273,292,155)(75,274,293,156)(76,275,294,157)(77,221,295,158)(78,222,296,159)(79,223,297,160)(80,224,298,161)(81,225,299,162)(82,226,300,163)(83,227,301,164)(84,228,302,165)(85,229,303,111)(86,230,304,112)(87,231,305,113)(88,232,306,114)(89,233,307,115)(90,234,308,116)(91,235,309,117)(92,236,310,118)(93,237,311,119)(94,238,312,120)(95,239,313,121)(96,240,314,122)(97,241,315,123)(98,242,316,124)(99,243,317,125)(100,244,318,126)(101,245,319,127)(102,246,320,128)(103,247,321,129)(104,248,322,130)(105,249,323,131)(106,250,324,132)(107,251,325,133)(108,252,326,134)(109,253,327,135)(110,254,328,136) );
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)], [(1,64,411,282),(2,65,412,283),(3,66,413,284),(4,67,414,285),(5,68,415,286),(6,69,416,287),(7,70,417,288),(8,71,418,289),(9,72,419,290),(10,73,420,291),(11,74,421,292),(12,75,422,293),(13,76,423,294),(14,77,424,295),(15,78,425,296),(16,79,426,297),(17,80,427,298),(18,81,428,299),(19,82,429,300),(20,83,430,301),(21,84,431,302),(22,85,432,303),(23,86,433,304),(24,87,434,305),(25,88,435,306),(26,89,436,307),(27,90,437,308),(28,91,438,309),(29,92,439,310),(30,93,440,311),(31,94,386,312),(32,95,387,313),(33,96,388,314),(34,97,389,315),(35,98,390,316),(36,99,391,317),(37,100,392,318),(38,101,393,319),(39,102,394,320),(40,103,395,321),(41,104,396,322),(42,105,397,323),(43,106,398,324),(44,107,399,325),(45,108,400,326),(46,109,401,327),(47,110,402,328),(48,56,403,329),(49,57,404,330),(50,58,405,276),(51,59,406,277),(52,60,407,278),(53,61,408,279),(54,62,409,280),(55,63,410,281),(111,373,229,189),(112,374,230,190),(113,375,231,191),(114,376,232,192),(115,377,233,193),(116,378,234,194),(117,379,235,195),(118,380,236,196),(119,381,237,197),(120,382,238,198),(121,383,239,199),(122,384,240,200),(123,385,241,201),(124,331,242,202),(125,332,243,203),(126,333,244,204),(127,334,245,205),(128,335,246,206),(129,336,247,207),(130,337,248,208),(131,338,249,209),(132,339,250,210),(133,340,251,211),(134,341,252,212),(135,342,253,213),(136,343,254,214),(137,344,255,215),(138,345,256,216),(139,346,257,217),(140,347,258,218),(141,348,259,219),(142,349,260,220),(143,350,261,166),(144,351,262,167),(145,352,263,168),(146,353,264,169),(147,354,265,170),(148,355,266,171),(149,356,267,172),(150,357,268,173),(151,358,269,174),(152,359,270,175),(153,360,271,176),(154,361,272,177),(155,362,273,178),(156,363,274,179),(157,364,275,180),(158,365,221,181),(159,366,222,182),(160,367,223,183),(161,368,224,184),(162,369,225,185),(163,370,226,186),(164,371,227,187),(165,372,228,188)], [(1,168,411,352),(2,169,412,353),(3,170,413,354),(4,171,414,355),(5,172,415,356),(6,173,416,357),(7,174,417,358),(8,175,418,359),(9,176,419,360),(10,177,420,361),(11,178,421,362),(12,179,422,363),(13,180,423,364),(14,181,424,365),(15,182,425,366),(16,183,426,367),(17,184,427,368),(18,185,428,369),(19,186,429,370),(20,187,430,371),(21,188,431,372),(22,189,432,373),(23,190,433,374),(24,191,434,375),(25,192,435,376),(26,193,436,377),(27,194,437,378),(28,195,438,379),(29,196,439,380),(30,197,440,381),(31,198,386,382),(32,199,387,383),(33,200,388,384),(34,201,389,385),(35,202,390,331),(36,203,391,332),(37,204,392,333),(38,205,393,334),(39,206,394,335),(40,207,395,336),(41,208,396,337),(42,209,397,338),(43,210,398,339),(44,211,399,340),(45,212,400,341),(46,213,401,342),(47,214,402,343),(48,215,403,344),(49,216,404,345),(50,217,405,346),(51,218,406,347),(52,219,407,348),(53,220,408,349),(54,166,409,350),(55,167,410,351),(56,255,329,137),(57,256,330,138),(58,257,276,139),(59,258,277,140),(60,259,278,141),(61,260,279,142),(62,261,280,143),(63,262,281,144),(64,263,282,145),(65,264,283,146),(66,265,284,147),(67,266,285,148),(68,267,286,149),(69,268,287,150),(70,269,288,151),(71,270,289,152),(72,271,290,153),(73,272,291,154),(74,273,292,155),(75,274,293,156),(76,275,294,157),(77,221,295,158),(78,222,296,159),(79,223,297,160),(80,224,298,161),(81,225,299,162),(82,226,300,163),(83,227,301,164),(84,228,302,165),(85,229,303,111),(86,230,304,112),(87,231,305,113),(88,232,306,114),(89,233,307,115),(90,234,308,116),(91,235,309,117),(92,236,310,118),(93,237,311,119),(94,238,312,120),(95,239,313,121),(96,240,314,122),(97,241,315,123),(98,242,316,124),(99,243,317,125),(100,244,318,126),(101,245,319,127),(102,246,320,128),(103,247,321,129),(104,248,322,130),(105,249,323,131),(106,250,324,132),(107,251,325,133),(108,252,326,134),(109,253,327,135),(110,254,328,136)]])
275 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 11A | ··· | 11J | 20A | ··· | 20L | 22A | ··· | 22J | 44A | ··· | 44AD | 55A | ··· | 55AN | 110A | ··· | 110AN | 220A | ··· | 220DP |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 11 | ··· | 11 | 20 | ··· | 20 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 | 220 | ··· | 220 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
275 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | - | |||||||||
| image | C1 | C2 | C5 | C10 | C11 | C22 | C55 | C110 | Q8 | C5×Q8 | Q8×C11 | Q8×C55 |
| kernel | Q8×C55 | C220 | Q8×C11 | C44 | C5×Q8 | C20 | Q8 | C4 | C55 | C11 | C5 | C1 |
| # reps | 1 | 3 | 4 | 12 | 10 | 30 | 40 | 120 | 1 | 4 | 10 | 40 |
Matrix representation of Q8×C55 ►in GL2(𝔽661) generated by
| 630 | 0 |
| 0 | 630 |
| 0 | 1 |
| 660 | 0 |
| 357 | 633 |
| 633 | 304 |
G:=sub<GL(2,GF(661))| [630,0,0,630],[0,660,1,0],[357,633,633,304] >;
Q8×C55 in GAP, Magma, Sage, TeX
Q_8\times C_{55} % in TeX
G:=Group("Q8xC55"); // GroupNames label
G:=SmallGroup(440,41);
// by ID
G=gap.SmallGroup(440,41);
# by ID
G:=PCGroup([5,-2,-2,-5,-11,-2,1100,2221,1106]);
// Polycyclic
G:=Group<a,b,c|a^55=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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