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G = C52.8Q8order 416 = 25·13

5th non-split extension by C52 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52.8Q8, C52.51D4, Dic131C8, C4.8Dic26, C26.6M4(2), C134(C4⋊C8), (C2×C8).1D13, C2.4(C8×D13), (C2×C104).1C2, C26.13(C2×C8), (C2×C4).91D26, C26.11(C4⋊C4), C22.9(C4×D13), C2.1(C8⋊D13), C4.26(C13⋊D4), (C2×Dic13).3C4, (C4×Dic13).5C2, (C2×C52).105C22, C2.1(C26.D4), (C2×C132C8).9C2, (C2×C26).30(C2×C4), SmallGroup(416,21)

Series: Derived Chief Lower central Upper central

C1C26 — C52.8Q8
C1C13C26C52C2×C52C4×Dic13 — C52.8Q8
C13C26 — C52.8Q8
C1C2×C4C2×C8

Generators and relations for C52.8Q8
 G = < a,b,c | a52=1, b4=a26, c2=a39b2, ab=ba, cac-1=a25, cbc-1=a39b3 >

13C4
13C4
26C4
2C8
13C2×C4
13C2×C4
26C8
2Dic13
13C2×C8
13C42
2C104
2C132C8
13C4⋊C8

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

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

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

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

116 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H13A···13F26A···26R52A···52X104A···104AV
order1222444444448888888813···1326···2652···52104···104
size11111111262626262222262626262···22···22···22···2

116 irreducible representations

dim1111112222222222
type+++++-++-
imageC1C2C2C2C4C8D4Q8M4(2)D13D26Dic26C13⋊D4C4×D13C8×D13C8⋊D13
kernelC52.8Q8C2×C132C8C4×Dic13C2×C104C2×Dic13Dic13C52C52C26C2×C8C2×C4C4C4C22C2C2
# reps111148112661212122424

Matrix representation of C52.8Q8 in GL3(𝔽313) generated by

2500
02519
0294236
,
30800
0265111
020248
,
100
02427
017471
G:=sub<GL(3,GF(313))| [25,0,0,0,25,294,0,19,236],[308,0,0,0,265,202,0,111,48],[1,0,0,0,242,174,0,7,71] >;

C52.8Q8 in GAP, Magma, Sage, TeX

C_{52}._8Q_8
% in TeX

G:=Group("C52.8Q8");
// GroupNames label

G:=SmallGroup(416,21);
// by ID

G=gap.SmallGroup(416,21);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,121,31,86,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=1,b^4=a^26,c^2=a^39*b^2,a*b=b*a,c*a*c^-1=a^25,c*b*c^-1=a^39*b^3>;
// generators/relations

Export

Subgroup lattice of C52.8Q8 in TeX

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