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G = C522Q8order 416 = 25·13

1st semidirect product of C52 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C522Q8, C4.4D52, C42Dic26, C52.27D4, C42.4D13, C131(C4⋊Q8), (C4×C52).2C2, C2.4(C2×D52), C26.1(C2×D4), C26.2(C2×Q8), (C2×C4).73D26, C523C4.4C2, C2.4(C2×Dic26), (C2×C52).85C22, (C2×C26).10C23, (C2×Dic26).3C2, (C2×Dic13).1C22, C22.34(C22×D13), SmallGroup(416,90)

Series: Derived Chief Lower central Upper central

C1C2×C26 — C522Q8
C1C13C26C2×C26C2×Dic13C2×Dic26 — C522Q8
C13C2×C26 — C522Q8
C1C22C42

Generators and relations for C522Q8
 G = < a,b,c | a52=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 408 in 68 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2 [×2], C4 [×6], C4 [×4], C22, C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C13, C42, C4⋊C4 [×4], C2×Q8 [×2], C26, C26 [×2], C4⋊Q8, Dic13 [×4], C52 [×6], C2×C26, Dic26 [×4], C2×Dic13 [×4], C2×C52, C2×C52 [×2], C523C4 [×4], C4×C52, C2×Dic26 [×2], C522Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, C2×D4, C2×Q8 [×2], D13, C4⋊Q8, D26 [×3], Dic26 [×4], D52 [×2], C22×D13, C2×Dic26 [×2], C2×D52, C522Q8

Smallest permutation representation of C522Q8
Regular action on 416 points
Generators in S416
(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)
(1 182 316 144)(2 183 317 145)(3 184 318 146)(4 185 319 147)(5 186 320 148)(6 187 321 149)(7 188 322 150)(8 189 323 151)(9 190 324 152)(10 191 325 153)(11 192 326 154)(12 193 327 155)(13 194 328 156)(14 195 329 105)(15 196 330 106)(16 197 331 107)(17 198 332 108)(18 199 333 109)(19 200 334 110)(20 201 335 111)(21 202 336 112)(22 203 337 113)(23 204 338 114)(24 205 339 115)(25 206 340 116)(26 207 341 117)(27 208 342 118)(28 157 343 119)(29 158 344 120)(30 159 345 121)(31 160 346 122)(32 161 347 123)(33 162 348 124)(34 163 349 125)(35 164 350 126)(36 165 351 127)(37 166 352 128)(38 167 353 129)(39 168 354 130)(40 169 355 131)(41 170 356 132)(42 171 357 133)(43 172 358 134)(44 173 359 135)(45 174 360 136)(46 175 361 137)(47 176 362 138)(48 177 363 139)(49 178 364 140)(50 179 313 141)(51 180 314 142)(52 181 315 143)(53 222 394 266)(54 223 395 267)(55 224 396 268)(56 225 397 269)(57 226 398 270)(58 227 399 271)(59 228 400 272)(60 229 401 273)(61 230 402 274)(62 231 403 275)(63 232 404 276)(64 233 405 277)(65 234 406 278)(66 235 407 279)(67 236 408 280)(68 237 409 281)(69 238 410 282)(70 239 411 283)(71 240 412 284)(72 241 413 285)(73 242 414 286)(74 243 415 287)(75 244 416 288)(76 245 365 289)(77 246 366 290)(78 247 367 291)(79 248 368 292)(80 249 369 293)(81 250 370 294)(82 251 371 295)(83 252 372 296)(84 253 373 297)(85 254 374 298)(86 255 375 299)(87 256 376 300)(88 257 377 301)(89 258 378 302)(90 259 379 303)(91 260 380 304)(92 209 381 305)(93 210 382 306)(94 211 383 307)(95 212 384 308)(96 213 385 309)(97 214 386 310)(98 215 387 311)(99 216 388 312)(100 217 389 261)(101 218 390 262)(102 219 391 263)(103 220 392 264)(104 221 393 265)
(1 60 316 401)(2 59 317 400)(3 58 318 399)(4 57 319 398)(5 56 320 397)(6 55 321 396)(7 54 322 395)(8 53 323 394)(9 104 324 393)(10 103 325 392)(11 102 326 391)(12 101 327 390)(13 100 328 389)(14 99 329 388)(15 98 330 387)(16 97 331 386)(17 96 332 385)(18 95 333 384)(19 94 334 383)(20 93 335 382)(21 92 336 381)(22 91 337 380)(23 90 338 379)(24 89 339 378)(25 88 340 377)(26 87 341 376)(27 86 342 375)(28 85 343 374)(29 84 344 373)(30 83 345 372)(31 82 346 371)(32 81 347 370)(33 80 348 369)(34 79 349 368)(35 78 350 367)(36 77 351 366)(37 76 352 365)(38 75 353 416)(39 74 354 415)(40 73 355 414)(41 72 356 413)(42 71 357 412)(43 70 358 411)(44 69 359 410)(45 68 360 409)(46 67 361 408)(47 66 362 407)(48 65 363 406)(49 64 364 405)(50 63 313 404)(51 62 314 403)(52 61 315 402)(105 216 195 312)(106 215 196 311)(107 214 197 310)(108 213 198 309)(109 212 199 308)(110 211 200 307)(111 210 201 306)(112 209 202 305)(113 260 203 304)(114 259 204 303)(115 258 205 302)(116 257 206 301)(117 256 207 300)(118 255 208 299)(119 254 157 298)(120 253 158 297)(121 252 159 296)(122 251 160 295)(123 250 161 294)(124 249 162 293)(125 248 163 292)(126 247 164 291)(127 246 165 290)(128 245 166 289)(129 244 167 288)(130 243 168 287)(131 242 169 286)(132 241 170 285)(133 240 171 284)(134 239 172 283)(135 238 173 282)(136 237 174 281)(137 236 175 280)(138 235 176 279)(139 234 177 278)(140 233 178 277)(141 232 179 276)(142 231 180 275)(143 230 181 274)(144 229 182 273)(145 228 183 272)(146 227 184 271)(147 226 185 270)(148 225 186 269)(149 224 187 268)(150 223 188 267)(151 222 189 266)(152 221 190 265)(153 220 191 264)(154 219 192 263)(155 218 193 262)(156 217 194 261)

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

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

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

110 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J13A···13F26A···26R52A···52BT
order12224···4444413···1326···2652···52
size11112···2525252522···22···22···2

110 irreducible representations

dim1111222222
type+++++-++-+
imageC1C2C2C2D4Q8D13D26Dic26D52
kernelC522Q8C523C4C4×C52C2×Dic26C52C52C42C2×C4C4C4
# reps1412246184824

Matrix representation of C522Q8 in GL4(𝔽53) generated by

231800
35900
00716
005213
,
253400
192800
0010
0001
,
413000
41200
004212
004311
G:=sub<GL(4,GF(53))| [23,35,0,0,18,9,0,0,0,0,7,52,0,0,16,13],[25,19,0,0,34,28,0,0,0,0,1,0,0,0,0,1],[41,4,0,0,30,12,0,0,0,0,42,43,0,0,12,11] >;

C522Q8 in GAP, Magma, Sage, TeX

C_{52}\rtimes_2Q_8
% in TeX

G:=Group("C52:2Q8");
// GroupNames label

G:=SmallGroup(416,90);
// by ID

G=gap.SmallGroup(416,90);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,217,103,218,50,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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