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## G = C2×Dic55order 440 = 23·5·11

### Direct product of C2 and Dic55

Aliases: C2×Dic55, C22⋊Dic5, C1103C4, C22.D55, C2.2D110, C102Dic11, C22.11D10, C10.11D22, C110.11C22, (C2×C22).D5, C5510(C2×C4), (C2×C10).D11, C112(C2×Dic5), C53(C2×Dic11), (C2×C110).1C2, SmallGroup(440,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C55 — C2×Dic55
 Chief series C1 — C11 — C55 — C110 — Dic55 — C2×Dic55
 Lower central C55 — C2×Dic55
 Upper central C1 — C22

Generators and relations for C2×Dic55
G = < a,b,c | a2=b110=1, c2=b55, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C2×Dic55
Regular action on 440 points
Generators in S440
(1 188)(2 189)(3 190)(4 191)(5 192)(6 193)(7 194)(8 195)(9 196)(10 197)(11 198)(12 199)(13 200)(14 201)(15 202)(16 203)(17 204)(18 205)(19 206)(20 207)(21 208)(22 209)(23 210)(24 211)(25 212)(26 213)(27 214)(28 215)(29 216)(30 217)(31 218)(32 219)(33 220)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 121)(45 122)(46 123)(47 124)(48 125)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 132)(56 133)(57 134)(58 135)(59 136)(60 137)(61 138)(62 139)(63 140)(64 141)(65 142)(66 143)(67 144)(68 145)(69 146)(70 147)(71 148)(72 149)(73 150)(74 151)(75 152)(76 153)(77 154)(78 155)(79 156)(80 157)(81 158)(82 159)(83 160)(84 161)(85 162)(86 163)(87 164)(88 165)(89 166)(90 167)(91 168)(92 169)(93 170)(94 171)(95 172)(96 173)(97 174)(98 175)(99 176)(100 177)(101 178)(102 179)(103 180)(104 181)(105 182)(106 183)(107 184)(108 185)(109 186)(110 187)(221 353)(222 354)(223 355)(224 356)(225 357)(226 358)(227 359)(228 360)(229 361)(230 362)(231 363)(232 364)(233 365)(234 366)(235 367)(236 368)(237 369)(238 370)(239 371)(240 372)(241 373)(242 374)(243 375)(244 376)(245 377)(246 378)(247 379)(248 380)(249 381)(250 382)(251 383)(252 384)(253 385)(254 386)(255 387)(256 388)(257 389)(258 390)(259 391)(260 392)(261 393)(262 394)(263 395)(264 396)(265 397)(266 398)(267 399)(268 400)(269 401)(270 402)(271 403)(272 404)(273 405)(274 406)(275 407)(276 408)(277 409)(278 410)(279 411)(280 412)(281 413)(282 414)(283 415)(284 416)(285 417)(286 418)(287 419)(288 420)(289 421)(290 422)(291 423)(292 424)(293 425)(294 426)(295 427)(296 428)(297 429)(298 430)(299 431)(300 432)(301 433)(302 434)(303 435)(304 436)(305 437)(306 438)(307 439)(308 440)(309 331)(310 332)(311 333)(312 334)(313 335)(314 336)(315 337)(316 338)(317 339)(318 340)(319 341)(320 342)(321 343)(322 344)(323 345)(324 346)(325 347)(326 348)(327 349)(328 350)(329 351)(330 352)
(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)
(1 375 56 430)(2 374 57 429)(3 373 58 428)(4 372 59 427)(5 371 60 426)(6 370 61 425)(7 369 62 424)(8 368 63 423)(9 367 64 422)(10 366 65 421)(11 365 66 420)(12 364 67 419)(13 363 68 418)(14 362 69 417)(15 361 70 416)(16 360 71 415)(17 359 72 414)(18 358 73 413)(19 357 74 412)(20 356 75 411)(21 355 76 410)(22 354 77 409)(23 353 78 408)(24 352 79 407)(25 351 80 406)(26 350 81 405)(27 349 82 404)(28 348 83 403)(29 347 84 402)(30 346 85 401)(31 345 86 400)(32 344 87 399)(33 343 88 398)(34 342 89 397)(35 341 90 396)(36 340 91 395)(37 339 92 394)(38 338 93 393)(39 337 94 392)(40 336 95 391)(41 335 96 390)(42 334 97 389)(43 333 98 388)(44 332 99 387)(45 331 100 386)(46 440 101 385)(47 439 102 384)(48 438 103 383)(49 437 104 382)(50 436 105 381)(51 435 106 380)(52 434 107 379)(53 433 108 378)(54 432 109 377)(55 431 110 376)(111 320 166 265)(112 319 167 264)(113 318 168 263)(114 317 169 262)(115 316 170 261)(116 315 171 260)(117 314 172 259)(118 313 173 258)(119 312 174 257)(120 311 175 256)(121 310 176 255)(122 309 177 254)(123 308 178 253)(124 307 179 252)(125 306 180 251)(126 305 181 250)(127 304 182 249)(128 303 183 248)(129 302 184 247)(130 301 185 246)(131 300 186 245)(132 299 187 244)(133 298 188 243)(134 297 189 242)(135 296 190 241)(136 295 191 240)(137 294 192 239)(138 293 193 238)(139 292 194 237)(140 291 195 236)(141 290 196 235)(142 289 197 234)(143 288 198 233)(144 287 199 232)(145 286 200 231)(146 285 201 230)(147 284 202 229)(148 283 203 228)(149 282 204 227)(150 281 205 226)(151 280 206 225)(152 279 207 224)(153 278 208 223)(154 277 209 222)(155 276 210 221)(156 275 211 330)(157 274 212 329)(158 273 213 328)(159 272 214 327)(160 271 215 326)(161 270 216 325)(162 269 217 324)(163 268 218 323)(164 267 219 322)(165 266 220 321)

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

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

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

116 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 10A ··· 10F 11A ··· 11E 22A ··· 22O 55A ··· 55T 110A ··· 110BH order 1 2 2 2 4 4 4 4 5 5 10 ··· 10 11 ··· 11 22 ··· 22 55 ··· 55 110 ··· 110 size 1 1 1 1 55 55 55 55 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

116 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + - + + - + + - + image C1 C2 C2 C4 D5 Dic5 D10 D11 Dic11 D22 D55 Dic55 D110 kernel C2×Dic55 Dic55 C2×C110 C110 C2×C22 C22 C22 C2×C10 C10 C10 C22 C2 C2 # reps 1 2 1 4 2 4 2 5 10 5 20 40 20

Matrix representation of C2×Dic55 in GL3(𝔽661) generated by

 1 0 0 0 660 0 0 0 660
,
 660 0 0 0 484 653 0 8 146
,
 106 0 0 0 518 525 0 267 143
G:=sub<GL(3,GF(661))| [1,0,0,0,660,0,0,0,660],[660,0,0,0,484,8,0,653,146],[106,0,0,0,518,267,0,525,143] >;

C2×Dic55 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{55}
% in TeX

G:=Group("C2xDic55");
// GroupNames label

G:=SmallGroup(440,37);
// by ID

G=gap.SmallGroup(440,37);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-11,20,643,10004]);
// Polycyclic

G:=Group<a,b,c|a^2=b^110=1,c^2=b^55,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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