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## G = Q8×C5×C10order 400 = 24·52

### Direct product of C5×C10 and Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C5×C10
 Chief series C1 — C2 — C10 — C5×C10 — C5×C20 — Q8×C52 — Q8×C5×C10
 Lower central C1 — C2 — Q8×C5×C10
 Upper central C1 — C102 — Q8×C5×C10

Generators and relations for Q8×C5×C10
G = < a,b,c,d | a5=b10=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 152, all normal (8 characteristic)
C1, C2, C2, C4, C22, C5, C2×C4, Q8, C10, C2×Q8, C20, C2×C10, C52, C2×C20, C5×Q8, C5×C10, C5×C10, Q8×C10, C5×C20, C102, C10×C20, Q8×C52, Q8×C5×C10
Quotients: C1, C2, C22, C5, Q8, C23, C10, C2×Q8, C2×C10, C52, C5×Q8, C22×C10, C5×C10, Q8×C10, C102, Q8×C52, C2×C102, Q8×C5×C10

Smallest permutation representation of Q8×C5×C10
Regular action on 400 points
Generators in S400
(1 108 97 42 81)(2 109 98 43 82)(3 110 99 44 83)(4 101 100 45 84)(5 102 91 46 85)(6 103 92 47 86)(7 104 93 48 87)(8 105 94 49 88)(9 106 95 50 89)(10 107 96 41 90)(11 32 72 27 66)(12 33 73 28 67)(13 34 74 29 68)(14 35 75 30 69)(15 36 76 21 70)(16 37 77 22 61)(17 38 78 23 62)(18 39 79 24 63)(19 40 80 25 64)(20 31 71 26 65)(51 397 342 381 336)(52 398 343 382 337)(53 399 344 383 338)(54 400 345 384 339)(55 391 346 385 340)(56 392 347 386 331)(57 393 348 387 332)(58 394 349 388 333)(59 395 350 389 334)(60 396 341 390 335)(111 160 172 127 166)(112 151 173 128 167)(113 152 174 129 168)(114 153 175 130 169)(115 154 176 121 170)(116 155 177 122 161)(117 156 178 123 162)(118 157 179 124 163)(119 158 180 125 164)(120 159 171 126 165)(131 203 192 147 186)(132 204 193 148 187)(133 205 194 149 188)(134 206 195 150 189)(135 207 196 141 190)(136 208 197 142 181)(137 209 198 143 182)(138 210 199 144 183)(139 201 200 145 184)(140 202 191 146 185)(211 260 272 227 266)(212 251 273 228 267)(213 252 274 229 268)(214 253 275 230 269)(215 254 276 221 270)(216 255 277 222 261)(217 256 278 223 262)(218 257 279 224 263)(219 258 280 225 264)(220 259 271 226 265)(231 303 292 247 286)(232 304 293 248 287)(233 305 294 249 288)(234 306 295 250 289)(235 307 296 241 290)(236 308 297 242 281)(237 309 298 243 282)(238 310 299 244 283)(239 301 300 245 284)(240 302 291 246 285)(311 360 372 327 366)(312 351 373 328 367)(313 352 374 329 368)(314 353 375 330 369)(315 354 376 321 370)(316 355 377 322 361)(317 356 378 323 362)(318 357 379 324 363)(319 358 380 325 364)(320 359 371 326 365)
(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)
(1 131 40 153)(2 132 31 154)(3 133 32 155)(4 134 33 156)(5 135 34 157)(6 136 35 158)(7 137 36 159)(8 138 37 160)(9 139 38 151)(10 140 39 152)(11 116 83 188)(12 117 84 189)(13 118 85 190)(14 119 86 181)(15 120 87 182)(16 111 88 183)(17 112 89 184)(18 113 90 185)(19 114 81 186)(20 115 82 187)(21 126 93 198)(22 127 94 199)(23 128 95 200)(24 129 96 191)(25 130 97 192)(26 121 98 193)(27 122 99 194)(28 123 100 195)(29 124 91 196)(30 125 92 197)(41 146 63 168)(42 147 64 169)(43 148 65 170)(44 149 66 161)(45 150 67 162)(46 141 68 163)(47 142 69 164)(48 143 70 165)(49 144 61 166)(50 145 62 167)(51 303 380 275)(52 304 371 276)(53 305 372 277)(54 306 373 278)(55 307 374 279)(56 308 375 280)(57 309 376 271)(58 310 377 272)(59 301 378 273)(60 302 379 274)(71 176 109 204)(72 177 110 205)(73 178 101 206)(74 179 102 207)(75 180 103 208)(76 171 104 209)(77 172 105 210)(78 173 106 201)(79 174 107 202)(80 175 108 203)(211 388 283 316)(212 389 284 317)(213 390 285 318)(214 381 286 319)(215 382 287 320)(216 383 288 311)(217 384 289 312)(218 385 290 313)(219 386 281 314)(220 387 282 315)(221 398 293 326)(222 399 294 327)(223 400 295 328)(224 391 296 329)(225 392 297 330)(226 393 298 321)(227 394 299 322)(228 395 300 323)(229 396 291 324)(230 397 292 325)(231 358 253 336)(232 359 254 337)(233 360 255 338)(234 351 256 339)(235 352 257 340)(236 353 258 331)(237 354 259 332)(238 355 260 333)(239 356 251 334)(240 357 252 335)(241 368 263 346)(242 369 264 347)(243 370 265 348)(244 361 266 349)(245 362 267 350)(246 363 268 341)(247 364 269 342)(248 365 270 343)(249 366 261 344)(250 367 262 345)
(1 231 40 253)(2 232 31 254)(3 233 32 255)(4 234 33 256)(5 235 34 257)(6 236 35 258)(7 237 36 259)(8 238 37 260)(9 239 38 251)(10 240 39 252)(11 216 83 288)(12 217 84 289)(13 218 85 290)(14 219 86 281)(15 220 87 282)(16 211 88 283)(17 212 89 284)(18 213 90 285)(19 214 81 286)(20 215 82 287)(21 226 93 298)(22 227 94 299)(23 228 95 300)(24 229 96 291)(25 230 97 292)(26 221 98 293)(27 222 99 294)(28 223 100 295)(29 224 91 296)(30 225 92 297)(41 246 63 268)(42 247 64 269)(43 248 65 270)(44 249 66 261)(45 250 67 262)(46 241 68 263)(47 242 69 264)(48 243 70 265)(49 244 61 266)(50 245 62 267)(51 175 380 203)(52 176 371 204)(53 177 372 205)(54 178 373 206)(55 179 374 207)(56 180 375 208)(57 171 376 209)(58 172 377 210)(59 173 378 201)(60 174 379 202)(71 276 109 304)(72 277 110 305)(73 278 101 306)(74 279 102 307)(75 280 103 308)(76 271 104 309)(77 272 105 310)(78 273 106 301)(79 274 107 302)(80 275 108 303)(111 316 183 388)(112 317 184 389)(113 318 185 390)(114 319 186 381)(115 320 187 382)(116 311 188 383)(117 312 189 384)(118 313 190 385)(119 314 181 386)(120 315 182 387)(121 326 193 398)(122 327 194 399)(123 328 195 400)(124 329 196 391)(125 330 197 392)(126 321 198 393)(127 322 199 394)(128 323 200 395)(129 324 191 396)(130 325 192 397)(131 336 153 358)(132 337 154 359)(133 338 155 360)(134 339 156 351)(135 340 157 352)(136 331 158 353)(137 332 159 354)(138 333 160 355)(139 334 151 356)(140 335 152 357)(141 346 163 368)(142 347 164 369)(143 348 165 370)(144 349 166 361)(145 350 167 362)(146 341 168 363)(147 342 169 364)(148 343 170 365)(149 344 161 366)(150 345 162 367)

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

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

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

250 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 5A ··· 5X 10A ··· 10BT 20A ··· 20EN order 1 2 2 2 4 ··· 4 5 ··· 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

250 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + - image C1 C2 C2 C5 C10 C10 Q8 C5×Q8 kernel Q8×C5×C10 C10×C20 Q8×C52 Q8×C10 C2×C20 C5×Q8 C5×C10 C10 # reps 1 3 4 24 72 96 2 48

Matrix representation of Q8×C5×C10 in GL3(𝔽41) generated by

 18 0 0 0 16 0 0 0 16
,
 4 0 0 0 40 0 0 0 40
,
 40 0 0 0 0 1 0 40 0
,
 1 0 0 0 35 2 0 2 6
G:=sub<GL(3,GF(41))| [18,0,0,0,16,0,0,0,16],[4,0,0,0,40,0,0,0,40],[40,0,0,0,0,40,0,1,0],[1,0,0,0,35,2,0,2,6] >;

Q8×C5×C10 in GAP, Magma, Sage, TeX

Q_8\times C_5\times C_{10}
% in TeX

G:=Group("Q8xC5xC10");
// GroupNames label

G:=SmallGroup(400,203);
// by ID

G=gap.SmallGroup(400,203);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-5,-2,1200,2425,1207]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^10=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
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𝔽