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G = C52.Q8order 416 = 25·13

2nd non-split extension by C52 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52.2Q8, C26.4SD16, C4.2Dic26, C132C82C4, C4⋊C4.2D13, C132(C4.Q8), C52.23(C2×C4), (C2×C26).29D4, C4.12(C4×D13), (C2×C4).34D26, C26.10(C4⋊C4), C523C4.9C2, C2.1(Q8⋊D13), (C2×C52).9C22, C2.1(D4.D13), C2.4(C26.D4), C22.13(C13⋊D4), (C13×C4⋊C4).2C2, (C2×C132C8).2C2, SmallGroup(416,15)

Series: Derived Chief Lower central Upper central

C1C52 — C52.Q8
C1C13C26C2×C26C2×C52C2×C132C8 — C52.Q8
C13C26C52 — C52.Q8
C1C22C2×C4C4⋊C4

Generators and relations for C52.Q8
 G = < a,b,c | a52=b4=1, c2=a13b2, bab-1=a27, cac-1=a25, cbc-1=a39b-1 >

4C4
52C4
2C2×C4
13C8
13C8
26C2×C4
4C52
4Dic13
13C2×C8
13C4⋊C4
2C2×Dic13
2C2×C52
13C4.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 68 173 393)(2 95 174 368)(3 70 175 395)(4 97 176 370)(5 72 177 397)(6 99 178 372)(7 74 179 399)(8 101 180 374)(9 76 181 401)(10 103 182 376)(11 78 183 403)(12 53 184 378)(13 80 185 405)(14 55 186 380)(15 82 187 407)(16 57 188 382)(17 84 189 409)(18 59 190 384)(19 86 191 411)(20 61 192 386)(21 88 193 413)(22 63 194 388)(23 90 195 415)(24 65 196 390)(25 92 197 365)(26 67 198 392)(27 94 199 367)(28 69 200 394)(29 96 201 369)(30 71 202 396)(31 98 203 371)(32 73 204 398)(33 100 205 373)(34 75 206 400)(35 102 207 375)(36 77 208 402)(37 104 157 377)(38 79 158 404)(39 54 159 379)(40 81 160 406)(41 56 161 381)(42 83 162 408)(43 58 163 383)(44 85 164 410)(45 60 165 385)(46 87 166 412)(47 62 167 387)(48 89 168 414)(49 64 169 389)(50 91 170 416)(51 66 171 391)(52 93 172 366)(105 218 359 282)(106 245 360 309)(107 220 361 284)(108 247 362 311)(109 222 363 286)(110 249 364 261)(111 224 313 288)(112 251 314 263)(113 226 315 290)(114 253 316 265)(115 228 317 292)(116 255 318 267)(117 230 319 294)(118 257 320 269)(119 232 321 296)(120 259 322 271)(121 234 323 298)(122 209 324 273)(123 236 325 300)(124 211 326 275)(125 238 327 302)(126 213 328 277)(127 240 329 304)(128 215 330 279)(129 242 331 306)(130 217 332 281)(131 244 333 308)(132 219 334 283)(133 246 335 310)(134 221 336 285)(135 248 337 312)(136 223 338 287)(137 250 339 262)(138 225 340 289)(139 252 341 264)(140 227 342 291)(141 254 343 266)(142 229 344 293)(143 256 345 268)(144 231 346 295)(145 258 347 270)(146 233 348 297)(147 260 349 272)(148 235 350 299)(149 210 351 274)(150 237 352 301)(151 212 353 276)(152 239 354 303)(153 214 355 278)(154 241 356 305)(155 216 357 280)(156 243 358 307)
(1 243 186 268 27 217 160 294)(2 216 187 293 28 242 161 267)(3 241 188 266 29 215 162 292)(4 214 189 291 30 240 163 265)(5 239 190 264 31 213 164 290)(6 212 191 289 32 238 165 263)(7 237 192 262 33 211 166 288)(8 210 193 287 34 236 167 261)(9 235 194 312 35 209 168 286)(10 260 195 285 36 234 169 311)(11 233 196 310 37 259 170 284)(12 258 197 283 38 232 171 309)(13 231 198 308 39 257 172 282)(14 256 199 281 40 230 173 307)(15 229 200 306 41 255 174 280)(16 254 201 279 42 228 175 305)(17 227 202 304 43 253 176 278)(18 252 203 277 44 226 177 303)(19 225 204 302 45 251 178 276)(20 250 205 275 46 224 179 301)(21 223 206 300 47 249 180 274)(22 248 207 273 48 222 181 299)(23 221 208 298 49 247 182 272)(24 246 157 271 50 220 183 297)(25 219 158 296 51 245 184 270)(26 244 159 269 52 218 185 295)(53 132 391 347 79 106 365 321)(54 105 392 320 80 131 366 346)(55 130 393 345 81 156 367 319)(56 155 394 318 82 129 368 344)(57 128 395 343 83 154 369 317)(58 153 396 316 84 127 370 342)(59 126 397 341 85 152 371 315)(60 151 398 314 86 125 372 340)(61 124 399 339 87 150 373 313)(62 149 400 364 88 123 374 338)(63 122 401 337 89 148 375 363)(64 147 402 362 90 121 376 336)(65 120 403 335 91 146 377 361)(66 145 404 360 92 119 378 334)(67 118 405 333 93 144 379 359)(68 143 406 358 94 117 380 332)(69 116 407 331 95 142 381 357)(70 141 408 356 96 115 382 330)(71 114 409 329 97 140 383 355)(72 139 410 354 98 113 384 328)(73 112 411 327 99 138 385 353)(74 137 412 352 100 111 386 326)(75 110 413 325 101 136 387 351)(76 135 414 350 102 109 388 324)(77 108 415 323 103 134 389 349)(78 133 416 348 104 107 390 322)

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

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

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

74 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F8A8B8C8D13A···13F26A···26R52A···52AJ
order1222444444888813···1326···2652···52
size111122445252262626262···22···24···4

74 irreducible representations

dim111112222222244
type++++-+++--+
imageC1C2C2C2C4Q8D4SD16D13D26Dic26C4×D13C13⋊D4D4.D13Q8⋊D13
kernelC52.Q8C2×C132C8C523C4C13×C4⋊C4C132C8C52C2×C26C26C4⋊C4C2×C4C4C4C22C2C2
# reps111141146612121266

Matrix representation of C52.Q8 in GL4(𝔽313) generated by

1465100
20614700
001311
001312
,
1074100
4920600
00240221
0019473
,
11518700
10019800
000183
0065183
G:=sub<GL(4,GF(313))| [146,206,0,0,51,147,0,0,0,0,1,1,0,0,311,312],[107,49,0,0,41,206,0,0,0,0,240,194,0,0,221,73],[115,100,0,0,187,198,0,0,0,0,0,65,0,0,183,183] >;

C52.Q8 in GAP, Magma, Sage, TeX

C_{52}.Q_8
% in TeX

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

G:=SmallGroup(416,15);
// by ID

G=gap.SmallGroup(416,15);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,313,31,297,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=b^4=1,c^2=a^13*b^2,b*a*b^-1=a^27,c*a*c^-1=a^25,c*b*c^-1=a^39*b^-1>;
// generators/relations

Export

Subgroup lattice of C52.Q8 in TeX

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