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G = Q8×C55order 440 = 23·5·11

Direct product of C55 and Q8

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

C1C2 — Q8×C55
C1C2C22C110C220 — Q8×C55
C1C2 — Q8×C55
C1C110 — Q8×C55

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

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

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

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

275 conjugacy classes

class 1  2 4A4B4C5A5B5C5D10A10B10C10D11A···11J20A···20L22A···22J44A···44AD55A···55AN110A···110AN220A···220DP
order1244455551010101011···1120···2022···2244···4455···55110···110220···220
size11222111111111···12···21···12···21···11···12···2

275 irreducible representations

dim111111112222
type++-
imageC1C2C5C10C11C22C55C110Q8C5×Q8Q8×C11Q8×C55
kernelQ8×C55C220Q8×C11C44C5×Q8C20Q8C4C55C11C5C1
# reps13412103040120141040

Matrix representation of Q8×C55 in GL2(𝔽661) generated by

6300
0630
,
01
6600
,
357633
633304
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

Export

Subgroup lattice of Q8×C55 in TeX

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