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## G = Q8×C52order 416 = 25·13

### Direct product of C52 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8×C52
G = < a,b,c | a52=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 76 in 70 conjugacy classes, 64 normal (16 characteristic)
C1, C2 [×3], C4 [×8], C4 [×3], C22, C2×C4, C2×C4 [×6], Q8 [×4], C13, C42 [×3], C4⋊C4 [×3], C2×Q8, C26 [×3], C4×Q8, C52 [×8], C52 [×3], C2×C26, C2×C52, C2×C52 [×6], Q8×C13 [×4], C4×C52 [×3], C13×C4⋊C4 [×3], Q8×C26, Q8×C52
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, C13, C22×C4, C2×Q8, C4○D4, C26 [×7], C4×Q8, C52 [×4], C2×C26 [×7], C2×C52 [×6], Q8×C13 [×2], C22×C26, C22×C52, Q8×C26, C13×C4○D4, Q8×C52

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

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

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

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

260 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4P 13A ··· 13L 26A ··· 26AJ 52A ··· 52AV 52AW ··· 52GJ order 1 2 2 2 4 4 4 4 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 1 1 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 1 ··· 1 2 ··· 2

260 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C2 C4 C13 C26 C26 C26 C52 Q8 C4○D4 Q8×C13 C13×C4○D4 kernel Q8×C52 C4×C52 C13×C4⋊C4 Q8×C26 Q8×C13 C4×Q8 C42 C4⋊C4 C2×Q8 Q8 C52 C26 C4 C2 # reps 1 3 3 1 8 12 36 36 12 96 2 2 24 24

Matrix representation of Q8×C52 in GL3(𝔽53) generated by

 23 0 0 0 10 0 0 0 10
,
 52 0 0 0 23 0 0 0 30
,
 1 0 0 0 0 52 0 1 0
G:=sub<GL(3,GF(53))| [23,0,0,0,10,0,0,0,10],[52,0,0,0,23,0,0,0,30],[1,0,0,0,0,1,0,52,0] >;

Q8×C52 in GAP, Magma, Sage, TeX

Q_8\times C_{52}
% in TeX

G:=Group("Q8xC52");
// GroupNames label

G:=SmallGroup(416,180);
// by ID

G=gap.SmallGroup(416,180);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1248,1273,631,1418]);
// Polycyclic

G:=Group<a,b,c|a^52=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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