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## G = Q8×C50order 400 = 24·52

### Direct product of C50 and Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C50
 Chief series C1 — C5 — C10 — C50 — C100 — Q8×C25 — Q8×C50
 Lower central C1 — C2 — Q8×C50
 Upper central C1 — C2×C50 — Q8×C50

Generators and relations for Q8×C50
G = < a,b,c | a50=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 57, all normal (12 characteristic)
C1, C2, C2, C4, C22, C5, C2×C4, Q8, C10, C10, C2×Q8, C20, C2×C10, C25, C2×C20, C5×Q8, C50, C50, Q8×C10, C100, C2×C50, C2×C100, Q8×C25, Q8×C50
Quotients: C1, C2, C22, C5, Q8, C23, C10, C2×Q8, C2×C10, C25, C5×Q8, C22×C10, C50, Q8×C10, C2×C50, Q8×C25, C22×C50, Q8×C50

Smallest permutation representation of Q8×C50
Regular action on 400 points
Generators in S400
(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)
(1 84 130 222)(2 85 131 223)(3 86 132 224)(4 87 133 225)(5 88 134 226)(6 89 135 227)(7 90 136 228)(8 91 137 229)(9 92 138 230)(10 93 139 231)(11 94 140 232)(12 95 141 233)(13 96 142 234)(14 97 143 235)(15 98 144 236)(16 99 145 237)(17 100 146 238)(18 51 147 239)(19 52 148 240)(20 53 149 241)(21 54 150 242)(22 55 101 243)(23 56 102 244)(24 57 103 245)(25 58 104 246)(26 59 105 247)(27 60 106 248)(28 61 107 249)(29 62 108 250)(30 63 109 201)(31 64 110 202)(32 65 111 203)(33 66 112 204)(34 67 113 205)(35 68 114 206)(36 69 115 207)(37 70 116 208)(38 71 117 209)(39 72 118 210)(40 73 119 211)(41 74 120 212)(42 75 121 213)(43 76 122 214)(44 77 123 215)(45 78 124 216)(46 79 125 217)(47 80 126 218)(48 81 127 219)(49 82 128 220)(50 83 129 221)(151 368 305 287)(152 369 306 288)(153 370 307 289)(154 371 308 290)(155 372 309 291)(156 373 310 292)(157 374 311 293)(158 375 312 294)(159 376 313 295)(160 377 314 296)(161 378 315 297)(162 379 316 298)(163 380 317 299)(164 381 318 300)(165 382 319 251)(166 383 320 252)(167 384 321 253)(168 385 322 254)(169 386 323 255)(170 387 324 256)(171 388 325 257)(172 389 326 258)(173 390 327 259)(174 391 328 260)(175 392 329 261)(176 393 330 262)(177 394 331 263)(178 395 332 264)(179 396 333 265)(180 397 334 266)(181 398 335 267)(182 399 336 268)(183 400 337 269)(184 351 338 270)(185 352 339 271)(186 353 340 272)(187 354 341 273)(188 355 342 274)(189 356 343 275)(190 357 344 276)(191 358 345 277)(192 359 346 278)(193 360 347 279)(194 361 348 280)(195 362 349 281)(196 363 350 282)(197 364 301 283)(198 365 302 284)(199 366 303 285)(200 367 304 286)
(1 326 130 172)(2 327 131 173)(3 328 132 174)(4 329 133 175)(5 330 134 176)(6 331 135 177)(7 332 136 178)(8 333 137 179)(9 334 138 180)(10 335 139 181)(11 336 140 182)(12 337 141 183)(13 338 142 184)(14 339 143 185)(15 340 144 186)(16 341 145 187)(17 342 146 188)(18 343 147 189)(19 344 148 190)(20 345 149 191)(21 346 150 192)(22 347 101 193)(23 348 102 194)(24 349 103 195)(25 350 104 196)(26 301 105 197)(27 302 106 198)(28 303 107 199)(29 304 108 200)(30 305 109 151)(31 306 110 152)(32 307 111 153)(33 308 112 154)(34 309 113 155)(35 310 114 156)(36 311 115 157)(37 312 116 158)(38 313 117 159)(39 314 118 160)(40 315 119 161)(41 316 120 162)(42 317 121 163)(43 318 122 164)(44 319 123 165)(45 320 124 166)(46 321 125 167)(47 322 126 168)(48 323 127 169)(49 324 128 170)(50 325 129 171)(51 356 239 275)(52 357 240 276)(53 358 241 277)(54 359 242 278)(55 360 243 279)(56 361 244 280)(57 362 245 281)(58 363 246 282)(59 364 247 283)(60 365 248 284)(61 366 249 285)(62 367 250 286)(63 368 201 287)(64 369 202 288)(65 370 203 289)(66 371 204 290)(67 372 205 291)(68 373 206 292)(69 374 207 293)(70 375 208 294)(71 376 209 295)(72 377 210 296)(73 378 211 297)(74 379 212 298)(75 380 213 299)(76 381 214 300)(77 382 215 251)(78 383 216 252)(79 384 217 253)(80 385 218 254)(81 386 219 255)(82 387 220 256)(83 388 221 257)(84 389 222 258)(85 390 223 259)(86 391 224 260)(87 392 225 261)(88 393 226 262)(89 394 227 263)(90 395 228 264)(91 396 229 265)(92 397 230 266)(93 398 231 267)(94 399 232 268)(95 400 233 269)(96 351 234 270)(97 352 235 271)(98 353 236 272)(99 354 237 273)(100 355 238 274)

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

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

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

250 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 5A 5B 5C 5D 10A ··· 10L 20A ··· 20X 25A ··· 25T 50A ··· 50BH 100A ··· 100DP order 1 2 2 2 4 ··· 4 5 5 5 5 10 ··· 10 20 ··· 20 25 ··· 25 50 ··· 50 100 ··· 100 size 1 1 1 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

250 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 type + + + - image C1 C2 C2 C5 C10 C10 C25 C50 C50 Q8 C5×Q8 Q8×C25 kernel Q8×C50 C2×C100 Q8×C25 Q8×C10 C2×C20 C5×Q8 C2×Q8 C2×C4 Q8 C50 C10 C2 # reps 1 3 4 4 12 16 20 60 80 2 8 40

Matrix representation of Q8×C50 in GL3(𝔽101) generated by

 100 0 0 0 96 0 0 0 96
,
 1 0 0 0 0 1 0 100 0
,
 100 0 0 0 79 11 0 11 22
G:=sub<GL(3,GF(101))| [100,0,0,0,96,0,0,0,96],[1,0,0,0,0,100,0,1,0],[100,0,0,0,79,11,0,11,22] >;

Q8×C50 in GAP, Magma, Sage, TeX

Q_8\times C_{50}
% in TeX

G:=Group("Q8xC50");
// GroupNames label

G:=SmallGroup(400,47);
// by ID

G=gap.SmallGroup(400,47);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,240,505,247,261]);
// Polycyclic

G:=Group<a,b,c|a^50=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽