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## G = C52⋊Q8order 416 = 25·13

### The semidirect product of C52 and Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C52⋊Q8
 Chief series C1 — C13 — C26 — C2×C26 — C2×Dic13 — C4×Dic13 — C52⋊Q8
 Lower central C13 — C2×C26 — C52⋊Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C52⋊Q8
G = < a,b,c | a52=b4=1, c2=b2, bab-1=a27, cac-1=a25, cbc-1=b-1 >

Subgroups: 408 in 68 conjugacy classes, 37 normal (19 characteristic)
C1, C2 [×3], C4 [×2], C4 [×8], C22, C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C13, C42, C4⋊C4, C4⋊C4 [×3], C2×Q8 [×2], C26 [×3], C4⋊Q8, Dic13 [×4], Dic13 [×2], C52 [×2], C52 [×2], C2×C26, Dic26 [×4], C2×Dic13 [×2], C2×Dic13 [×2], C2×C52, C2×C52 [×2], C4×Dic13, C26.D4 [×2], C523C4, C13×C4⋊C4, C2×Dic26 [×2], C52⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, C2×D4, C2×Q8 [×2], D13, C4⋊Q8, D26 [×3], Dic26 [×2], C22×D13, C2×Dic26, D4×D13, Q8×D13, C52⋊Q8

Smallest permutation representation of C52⋊Q8
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)
(1 170 325 375)(2 197 326 402)(3 172 327 377)(4 199 328 404)(5 174 329 379)(6 201 330 406)(7 176 331 381)(8 203 332 408)(9 178 333 383)(10 205 334 410)(11 180 335 385)(12 207 336 412)(13 182 337 387)(14 157 338 414)(15 184 339 389)(16 159 340 416)(17 186 341 391)(18 161 342 366)(19 188 343 393)(20 163 344 368)(21 190 345 395)(22 165 346 370)(23 192 347 397)(24 167 348 372)(25 194 349 399)(26 169 350 374)(27 196 351 401)(28 171 352 376)(29 198 353 403)(30 173 354 378)(31 200 355 405)(32 175 356 380)(33 202 357 407)(34 177 358 382)(35 204 359 409)(36 179 360 384)(37 206 361 411)(38 181 362 386)(39 208 363 413)(40 183 364 388)(41 158 313 415)(42 185 314 390)(43 160 315 365)(44 187 316 392)(45 162 317 367)(46 189 318 394)(47 164 319 369)(48 191 320 396)(49 166 321 371)(50 193 322 398)(51 168 323 373)(52 195 324 400)(53 243 134 279)(54 218 135 306)(55 245 136 281)(56 220 137 308)(57 247 138 283)(58 222 139 310)(59 249 140 285)(60 224 141 312)(61 251 142 287)(62 226 143 262)(63 253 144 289)(64 228 145 264)(65 255 146 291)(66 230 147 266)(67 257 148 293)(68 232 149 268)(69 259 150 295)(70 234 151 270)(71 209 152 297)(72 236 153 272)(73 211 154 299)(74 238 155 274)(75 213 156 301)(76 240 105 276)(77 215 106 303)(78 242 107 278)(79 217 108 305)(80 244 109 280)(81 219 110 307)(82 246 111 282)(83 221 112 309)(84 248 113 284)(85 223 114 311)(86 250 115 286)(87 225 116 261)(88 252 117 288)(89 227 118 263)(90 254 119 290)(91 229 120 265)(92 256 121 292)(93 231 122 267)(94 258 123 294)(95 233 124 269)(96 260 125 296)(97 235 126 271)(98 210 127 298)(99 237 128 273)(100 212 129 300)(101 239 130 275)(102 214 131 302)(103 241 132 277)(104 216 133 304)
(1 70 325 151)(2 95 326 124)(3 68 327 149)(4 93 328 122)(5 66 329 147)(6 91 330 120)(7 64 331 145)(8 89 332 118)(9 62 333 143)(10 87 334 116)(11 60 335 141)(12 85 336 114)(13 58 337 139)(14 83 338 112)(15 56 339 137)(16 81 340 110)(17 54 341 135)(18 79 342 108)(19 104 343 133)(20 77 344 106)(21 102 345 131)(22 75 346 156)(23 100 347 129)(24 73 348 154)(25 98 349 127)(26 71 350 152)(27 96 351 125)(28 69 352 150)(29 94 353 123)(30 67 354 148)(31 92 355 121)(32 65 356 146)(33 90 357 119)(34 63 358 144)(35 88 359 117)(36 61 360 142)(37 86 361 115)(38 59 362 140)(39 84 363 113)(40 57 364 138)(41 82 313 111)(42 55 314 136)(43 80 315 109)(44 53 316 134)(45 78 317 107)(46 103 318 132)(47 76 319 105)(48 101 320 130)(49 74 321 155)(50 99 322 128)(51 72 323 153)(52 97 324 126)(157 309 414 221)(158 282 415 246)(159 307 416 219)(160 280 365 244)(161 305 366 217)(162 278 367 242)(163 303 368 215)(164 276 369 240)(165 301 370 213)(166 274 371 238)(167 299 372 211)(168 272 373 236)(169 297 374 209)(170 270 375 234)(171 295 376 259)(172 268 377 232)(173 293 378 257)(174 266 379 230)(175 291 380 255)(176 264 381 228)(177 289 382 253)(178 262 383 226)(179 287 384 251)(180 312 385 224)(181 285 386 249)(182 310 387 222)(183 283 388 247)(184 308 389 220)(185 281 390 245)(186 306 391 218)(187 279 392 243)(188 304 393 216)(189 277 394 241)(190 302 395 214)(191 275 396 239)(192 300 397 212)(193 273 398 237)(194 298 399 210)(195 271 400 235)(196 296 401 260)(197 269 402 233)(198 294 403 258)(199 267 404 231)(200 292 405 256)(201 265 406 229)(202 290 407 254)(203 263 408 227)(204 288 409 252)(205 261 410 225)(206 286 411 250)(207 311 412 223)(208 284 413 248)```

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 2 4 4 4 4 4 4 4 4 4 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 2 2 4 4 26 26 26 26 52 52 2 ··· 2 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + - - + + - + - image C1 C2 C2 C2 C2 C2 D4 Q8 Q8 D13 D26 Dic26 D4×D13 Q8×D13 kernel C52⋊Q8 C4×Dic13 C26.D4 C52⋊3C4 C13×C4⋊C4 C2×Dic26 Dic13 Dic13 C52 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 1 2 1 1 2 2 2 2 6 18 24 6 6

Matrix representation of C52⋊Q8 in GL6(𝔽53)

 6 39 0 0 0 0 14 47 0 0 0 0 0 0 4 27 0 0 0 0 21 36 0 0 0 0 0 0 49 6 0 0 0 0 6 4
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 52 0 0 0 0 0 0 52 0 0 0 0 0 0 0 1 0 0 0 0 52 0
,
 52 0 0 0 0 0 0 52 0 0 0 0 0 0 1 52 0 0 0 0 0 52 0 0 0 0 0 0 49 6 0 0 0 0 6 4

`G:=sub<GL(6,GF(53))| [6,14,0,0,0,0,39,47,0,0,0,0,0,0,4,21,0,0,0,0,27,36,0,0,0,0,0,0,49,6,0,0,0,0,6,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,0,52,0,0,0,0,1,0],[52,0,0,0,0,0,0,52,0,0,0,0,0,0,1,0,0,0,0,0,52,52,0,0,0,0,0,0,49,6,0,0,0,0,6,4] >;`

C52⋊Q8 in GAP, Magma, Sage, TeX

`C_{52}\rtimes Q_8`
`% in TeX`

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

`G:=SmallGroup(416,109);`
`// by ID`

`G=gap.SmallGroup(416,109);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,103,218,188,50,13829]);`
`// Polycyclic`

`G:=Group<a,b,c|a^52=b^4=1,c^2=b^2,b*a*b^-1=a^27,c*a*c^-1=a^25,c*b*c^-1=b^-1>;`
`// generators/relations`

׿
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