direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C52, C42.3C26, C4⋊C4.6C26, (C4×C52).9C2, C4.4(C2×C52), C2.2(Q8×C26), C52.52(C2×C4), (C2×Q8).5C26, C26.19(C2×Q8), C2.5(C22×C52), (Q8×C26).10C2, C26.40(C4○D4), (C2×C26).74C23, C26.46(C22×C4), (C2×C52).122C22, C22.8(C22×C26), C2.3(C13×C4○D4), (C13×C4⋊C4).13C2, (C2×C4).16(C2×C26), SmallGroup(416,180)
Series: Derived ►Chief ►Lower central ►Upper central
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, C4, C4, C22, C2×C4, C2×C4, Q8, C13, C42, C4⋊C4, C2×Q8, C26, C4×Q8, C52, C52, C2×C26, C2×C52, C2×C52, Q8×C13, C4×C52, C13×C4⋊C4, Q8×C26, Q8×C52
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C13, C22×C4, C2×Q8, C4○D4, C26, C4×Q8, C52, C2×C26, C2×C52, Q8×C13, C22×C26, C22×C52, Q8×C26, C13×C4○D4, 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)
(1 244 368 202)(2 245 369 203)(3 246 370 204)(4 247 371 205)(5 248 372 206)(6 249 373 207)(7 250 374 208)(8 251 375 157)(9 252 376 158)(10 253 377 159)(11 254 378 160)(12 255 379 161)(13 256 380 162)(14 257 381 163)(15 258 382 164)(16 259 383 165)(17 260 384 166)(18 209 385 167)(19 210 386 168)(20 211 387 169)(21 212 388 170)(22 213 389 171)(23 214 390 172)(24 215 391 173)(25 216 392 174)(26 217 393 175)(27 218 394 176)(28 219 395 177)(29 220 396 178)(30 221 397 179)(31 222 398 180)(32 223 399 181)(33 224 400 182)(34 225 401 183)(35 226 402 184)(36 227 403 185)(37 228 404 186)(38 229 405 187)(39 230 406 188)(40 231 407 189)(41 232 408 190)(42 233 409 191)(43 234 410 192)(44 235 411 193)(45 236 412 194)(46 237 413 195)(47 238 414 196)(48 239 415 197)(49 240 416 198)(50 241 365 199)(51 242 366 200)(52 243 367 201)(53 153 303 321)(54 154 304 322)(55 155 305 323)(56 156 306 324)(57 105 307 325)(58 106 308 326)(59 107 309 327)(60 108 310 328)(61 109 311 329)(62 110 312 330)(63 111 261 331)(64 112 262 332)(65 113 263 333)(66 114 264 334)(67 115 265 335)(68 116 266 336)(69 117 267 337)(70 118 268 338)(71 119 269 339)(72 120 270 340)(73 121 271 341)(74 122 272 342)(75 123 273 343)(76 124 274 344)(77 125 275 345)(78 126 276 346)(79 127 277 347)(80 128 278 348)(81 129 279 349)(82 130 280 350)(83 131 281 351)(84 132 282 352)(85 133 283 353)(86 134 284 354)(87 135 285 355)(88 136 286 356)(89 137 287 357)(90 138 288 358)(91 139 289 359)(92 140 290 360)(93 141 291 361)(94 142 292 362)(95 143 293 363)(96 144 294 364)(97 145 295 313)(98 146 296 314)(99 147 297 315)(100 148 298 316)(101 149 299 317)(102 150 300 318)(103 151 301 319)(104 152 302 320)
(1 357 368 137)(2 358 369 138)(3 359 370 139)(4 360 371 140)(5 361 372 141)(6 362 373 142)(7 363 374 143)(8 364 375 144)(9 313 376 145)(10 314 377 146)(11 315 378 147)(12 316 379 148)(13 317 380 149)(14 318 381 150)(15 319 382 151)(16 320 383 152)(17 321 384 153)(18 322 385 154)(19 323 386 155)(20 324 387 156)(21 325 388 105)(22 326 389 106)(23 327 390 107)(24 328 391 108)(25 329 392 109)(26 330 393 110)(27 331 394 111)(28 332 395 112)(29 333 396 113)(30 334 397 114)(31 335 398 115)(32 336 399 116)(33 337 400 117)(34 338 401 118)(35 339 402 119)(36 340 403 120)(37 341 404 121)(38 342 405 122)(39 343 406 123)(40 344 407 124)(41 345 408 125)(42 346 409 126)(43 347 410 127)(44 348 411 128)(45 349 412 129)(46 350 413 130)(47 351 414 131)(48 352 415 132)(49 353 416 133)(50 354 365 134)(51 355 366 135)(52 356 367 136)(53 260 303 166)(54 209 304 167)(55 210 305 168)(56 211 306 169)(57 212 307 170)(58 213 308 171)(59 214 309 172)(60 215 310 173)(61 216 311 174)(62 217 312 175)(63 218 261 176)(64 219 262 177)(65 220 263 178)(66 221 264 179)(67 222 265 180)(68 223 266 181)(69 224 267 182)(70 225 268 183)(71 226 269 184)(72 227 270 185)(73 228 271 186)(74 229 272 187)(75 230 273 188)(76 231 274 189)(77 232 275 190)(78 233 276 191)(79 234 277 192)(80 235 278 193)(81 236 279 194)(82 237 280 195)(83 238 281 196)(84 239 282 197)(85 240 283 198)(86 241 284 199)(87 242 285 200)(88 243 286 201)(89 244 287 202)(90 245 288 203)(91 246 289 204)(92 247 290 205)(93 248 291 206)(94 249 292 207)(95 250 293 208)(96 251 294 157)(97 252 295 158)(98 253 296 159)(99 254 297 160)(100 255 298 161)(101 256 299 162)(102 257 300 163)(103 258 301 164)(104 259 302 165)
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), (1,244,368,202)(2,245,369,203)(3,246,370,204)(4,247,371,205)(5,248,372,206)(6,249,373,207)(7,250,374,208)(8,251,375,157)(9,252,376,158)(10,253,377,159)(11,254,378,160)(12,255,379,161)(13,256,380,162)(14,257,381,163)(15,258,382,164)(16,259,383,165)(17,260,384,166)(18,209,385,167)(19,210,386,168)(20,211,387,169)(21,212,388,170)(22,213,389,171)(23,214,390,172)(24,215,391,173)(25,216,392,174)(26,217,393,175)(27,218,394,176)(28,219,395,177)(29,220,396,178)(30,221,397,179)(31,222,398,180)(32,223,399,181)(33,224,400,182)(34,225,401,183)(35,226,402,184)(36,227,403,185)(37,228,404,186)(38,229,405,187)(39,230,406,188)(40,231,407,189)(41,232,408,190)(42,233,409,191)(43,234,410,192)(44,235,411,193)(45,236,412,194)(46,237,413,195)(47,238,414,196)(48,239,415,197)(49,240,416,198)(50,241,365,199)(51,242,366,200)(52,243,367,201)(53,153,303,321)(54,154,304,322)(55,155,305,323)(56,156,306,324)(57,105,307,325)(58,106,308,326)(59,107,309,327)(60,108,310,328)(61,109,311,329)(62,110,312,330)(63,111,261,331)(64,112,262,332)(65,113,263,333)(66,114,264,334)(67,115,265,335)(68,116,266,336)(69,117,267,337)(70,118,268,338)(71,119,269,339)(72,120,270,340)(73,121,271,341)(74,122,272,342)(75,123,273,343)(76,124,274,344)(77,125,275,345)(78,126,276,346)(79,127,277,347)(80,128,278,348)(81,129,279,349)(82,130,280,350)(83,131,281,351)(84,132,282,352)(85,133,283,353)(86,134,284,354)(87,135,285,355)(88,136,286,356)(89,137,287,357)(90,138,288,358)(91,139,289,359)(92,140,290,360)(93,141,291,361)(94,142,292,362)(95,143,293,363)(96,144,294,364)(97,145,295,313)(98,146,296,314)(99,147,297,315)(100,148,298,316)(101,149,299,317)(102,150,300,318)(103,151,301,319)(104,152,302,320), (1,357,368,137)(2,358,369,138)(3,359,370,139)(4,360,371,140)(5,361,372,141)(6,362,373,142)(7,363,374,143)(8,364,375,144)(9,313,376,145)(10,314,377,146)(11,315,378,147)(12,316,379,148)(13,317,380,149)(14,318,381,150)(15,319,382,151)(16,320,383,152)(17,321,384,153)(18,322,385,154)(19,323,386,155)(20,324,387,156)(21,325,388,105)(22,326,389,106)(23,327,390,107)(24,328,391,108)(25,329,392,109)(26,330,393,110)(27,331,394,111)(28,332,395,112)(29,333,396,113)(30,334,397,114)(31,335,398,115)(32,336,399,116)(33,337,400,117)(34,338,401,118)(35,339,402,119)(36,340,403,120)(37,341,404,121)(38,342,405,122)(39,343,406,123)(40,344,407,124)(41,345,408,125)(42,346,409,126)(43,347,410,127)(44,348,411,128)(45,349,412,129)(46,350,413,130)(47,351,414,131)(48,352,415,132)(49,353,416,133)(50,354,365,134)(51,355,366,135)(52,356,367,136)(53,260,303,166)(54,209,304,167)(55,210,305,168)(56,211,306,169)(57,212,307,170)(58,213,308,171)(59,214,309,172)(60,215,310,173)(61,216,311,174)(62,217,312,175)(63,218,261,176)(64,219,262,177)(65,220,263,178)(66,221,264,179)(67,222,265,180)(68,223,266,181)(69,224,267,182)(70,225,268,183)(71,226,269,184)(72,227,270,185)(73,228,271,186)(74,229,272,187)(75,230,273,188)(76,231,274,189)(77,232,275,190)(78,233,276,191)(79,234,277,192)(80,235,278,193)(81,236,279,194)(82,237,280,195)(83,238,281,196)(84,239,282,197)(85,240,283,198)(86,241,284,199)(87,242,285,200)(88,243,286,201)(89,244,287,202)(90,245,288,203)(91,246,289,204)(92,247,290,205)(93,248,291,206)(94,249,292,207)(95,250,293,208)(96,251,294,157)(97,252,295,158)(98,253,296,159)(99,254,297,160)(100,255,298,161)(101,256,299,162)(102,257,300,163)(103,258,301,164)(104,259,302,165)>;
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), (1,244,368,202)(2,245,369,203)(3,246,370,204)(4,247,371,205)(5,248,372,206)(6,249,373,207)(7,250,374,208)(8,251,375,157)(9,252,376,158)(10,253,377,159)(11,254,378,160)(12,255,379,161)(13,256,380,162)(14,257,381,163)(15,258,382,164)(16,259,383,165)(17,260,384,166)(18,209,385,167)(19,210,386,168)(20,211,387,169)(21,212,388,170)(22,213,389,171)(23,214,390,172)(24,215,391,173)(25,216,392,174)(26,217,393,175)(27,218,394,176)(28,219,395,177)(29,220,396,178)(30,221,397,179)(31,222,398,180)(32,223,399,181)(33,224,400,182)(34,225,401,183)(35,226,402,184)(36,227,403,185)(37,228,404,186)(38,229,405,187)(39,230,406,188)(40,231,407,189)(41,232,408,190)(42,233,409,191)(43,234,410,192)(44,235,411,193)(45,236,412,194)(46,237,413,195)(47,238,414,196)(48,239,415,197)(49,240,416,198)(50,241,365,199)(51,242,366,200)(52,243,367,201)(53,153,303,321)(54,154,304,322)(55,155,305,323)(56,156,306,324)(57,105,307,325)(58,106,308,326)(59,107,309,327)(60,108,310,328)(61,109,311,329)(62,110,312,330)(63,111,261,331)(64,112,262,332)(65,113,263,333)(66,114,264,334)(67,115,265,335)(68,116,266,336)(69,117,267,337)(70,118,268,338)(71,119,269,339)(72,120,270,340)(73,121,271,341)(74,122,272,342)(75,123,273,343)(76,124,274,344)(77,125,275,345)(78,126,276,346)(79,127,277,347)(80,128,278,348)(81,129,279,349)(82,130,280,350)(83,131,281,351)(84,132,282,352)(85,133,283,353)(86,134,284,354)(87,135,285,355)(88,136,286,356)(89,137,287,357)(90,138,288,358)(91,139,289,359)(92,140,290,360)(93,141,291,361)(94,142,292,362)(95,143,293,363)(96,144,294,364)(97,145,295,313)(98,146,296,314)(99,147,297,315)(100,148,298,316)(101,149,299,317)(102,150,300,318)(103,151,301,319)(104,152,302,320), (1,357,368,137)(2,358,369,138)(3,359,370,139)(4,360,371,140)(5,361,372,141)(6,362,373,142)(7,363,374,143)(8,364,375,144)(9,313,376,145)(10,314,377,146)(11,315,378,147)(12,316,379,148)(13,317,380,149)(14,318,381,150)(15,319,382,151)(16,320,383,152)(17,321,384,153)(18,322,385,154)(19,323,386,155)(20,324,387,156)(21,325,388,105)(22,326,389,106)(23,327,390,107)(24,328,391,108)(25,329,392,109)(26,330,393,110)(27,331,394,111)(28,332,395,112)(29,333,396,113)(30,334,397,114)(31,335,398,115)(32,336,399,116)(33,337,400,117)(34,338,401,118)(35,339,402,119)(36,340,403,120)(37,341,404,121)(38,342,405,122)(39,343,406,123)(40,344,407,124)(41,345,408,125)(42,346,409,126)(43,347,410,127)(44,348,411,128)(45,349,412,129)(46,350,413,130)(47,351,414,131)(48,352,415,132)(49,353,416,133)(50,354,365,134)(51,355,366,135)(52,356,367,136)(53,260,303,166)(54,209,304,167)(55,210,305,168)(56,211,306,169)(57,212,307,170)(58,213,308,171)(59,214,309,172)(60,215,310,173)(61,216,311,174)(62,217,312,175)(63,218,261,176)(64,219,262,177)(65,220,263,178)(66,221,264,179)(67,222,265,180)(68,223,266,181)(69,224,267,182)(70,225,268,183)(71,226,269,184)(72,227,270,185)(73,228,271,186)(74,229,272,187)(75,230,273,188)(76,231,274,189)(77,232,275,190)(78,233,276,191)(79,234,277,192)(80,235,278,193)(81,236,279,194)(82,237,280,195)(83,238,281,196)(84,239,282,197)(85,240,283,198)(86,241,284,199)(87,242,285,200)(88,243,286,201)(89,244,287,202)(90,245,288,203)(91,246,289,204)(92,247,290,205)(93,248,291,206)(94,249,292,207)(95,250,293,208)(96,251,294,157)(97,252,295,158)(98,253,296,159)(99,254,297,160)(100,255,298,161)(101,256,299,162)(102,257,300,163)(103,258,301,164)(104,259,302,165) );
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)], [(1,244,368,202),(2,245,369,203),(3,246,370,204),(4,247,371,205),(5,248,372,206),(6,249,373,207),(7,250,374,208),(8,251,375,157),(9,252,376,158),(10,253,377,159),(11,254,378,160),(12,255,379,161),(13,256,380,162),(14,257,381,163),(15,258,382,164),(16,259,383,165),(17,260,384,166),(18,209,385,167),(19,210,386,168),(20,211,387,169),(21,212,388,170),(22,213,389,171),(23,214,390,172),(24,215,391,173),(25,216,392,174),(26,217,393,175),(27,218,394,176),(28,219,395,177),(29,220,396,178),(30,221,397,179),(31,222,398,180),(32,223,399,181),(33,224,400,182),(34,225,401,183),(35,226,402,184),(36,227,403,185),(37,228,404,186),(38,229,405,187),(39,230,406,188),(40,231,407,189),(41,232,408,190),(42,233,409,191),(43,234,410,192),(44,235,411,193),(45,236,412,194),(46,237,413,195),(47,238,414,196),(48,239,415,197),(49,240,416,198),(50,241,365,199),(51,242,366,200),(52,243,367,201),(53,153,303,321),(54,154,304,322),(55,155,305,323),(56,156,306,324),(57,105,307,325),(58,106,308,326),(59,107,309,327),(60,108,310,328),(61,109,311,329),(62,110,312,330),(63,111,261,331),(64,112,262,332),(65,113,263,333),(66,114,264,334),(67,115,265,335),(68,116,266,336),(69,117,267,337),(70,118,268,338),(71,119,269,339),(72,120,270,340),(73,121,271,341),(74,122,272,342),(75,123,273,343),(76,124,274,344),(77,125,275,345),(78,126,276,346),(79,127,277,347),(80,128,278,348),(81,129,279,349),(82,130,280,350),(83,131,281,351),(84,132,282,352),(85,133,283,353),(86,134,284,354),(87,135,285,355),(88,136,286,356),(89,137,287,357),(90,138,288,358),(91,139,289,359),(92,140,290,360),(93,141,291,361),(94,142,292,362),(95,143,293,363),(96,144,294,364),(97,145,295,313),(98,146,296,314),(99,147,297,315),(100,148,298,316),(101,149,299,317),(102,150,300,318),(103,151,301,319),(104,152,302,320)], [(1,357,368,137),(2,358,369,138),(3,359,370,139),(4,360,371,140),(5,361,372,141),(6,362,373,142),(7,363,374,143),(8,364,375,144),(9,313,376,145),(10,314,377,146),(11,315,378,147),(12,316,379,148),(13,317,380,149),(14,318,381,150),(15,319,382,151),(16,320,383,152),(17,321,384,153),(18,322,385,154),(19,323,386,155),(20,324,387,156),(21,325,388,105),(22,326,389,106),(23,327,390,107),(24,328,391,108),(25,329,392,109),(26,330,393,110),(27,331,394,111),(28,332,395,112),(29,333,396,113),(30,334,397,114),(31,335,398,115),(32,336,399,116),(33,337,400,117),(34,338,401,118),(35,339,402,119),(36,340,403,120),(37,341,404,121),(38,342,405,122),(39,343,406,123),(40,344,407,124),(41,345,408,125),(42,346,409,126),(43,347,410,127),(44,348,411,128),(45,349,412,129),(46,350,413,130),(47,351,414,131),(48,352,415,132),(49,353,416,133),(50,354,365,134),(51,355,366,135),(52,356,367,136),(53,260,303,166),(54,209,304,167),(55,210,305,168),(56,211,306,169),(57,212,307,170),(58,213,308,171),(59,214,309,172),(60,215,310,173),(61,216,311,174),(62,217,312,175),(63,218,261,176),(64,219,262,177),(65,220,263,178),(66,221,264,179),(67,222,265,180),(68,223,266,181),(69,224,267,182),(70,225,268,183),(71,226,269,184),(72,227,270,185),(73,228,271,186),(74,229,272,187),(75,230,273,188),(76,231,274,189),(77,232,275,190),(78,233,276,191),(79,234,277,192),(80,235,278,193),(81,236,279,194),(82,237,280,195),(83,238,281,196),(84,239,282,197),(85,240,283,198),(86,241,284,199),(87,242,285,200),(88,243,286,201),(89,244,287,202),(90,245,288,203),(91,246,289,204),(92,247,290,205),(93,248,291,206),(94,249,292,207),(95,250,293,208),(96,251,294,157),(97,252,295,158),(98,253,296,159),(99,254,297,160),(100,255,298,161),(101,256,299,162),(102,257,300,163),(103,258,301,164),(104,259,302,165)]])
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