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## G = C2×C52⋊7C8order 400 = 24·52

### Direct product of C2 and C52⋊7C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C52⋊7C8
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C52⋊7C8 — C2×C52⋊7C8
 Lower central C52 — C2×C52⋊7C8
 Upper central C1 — C2×C4

Generators and relations for C2×C527C8
G = < a,b,c,d | a2=b5=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 232 in 88 conjugacy classes, 67 normal (13 characteristic)
C1, C2, C2, C4, C22, C5, C8, C2×C4, C10, C2×C8, C20, C2×C10, C52, C52C8, C2×C20, C5×C10, C5×C10, C2×C52C8, C5×C20, C102, C527C8, C10×C20, C2×C527C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D5, C2×C8, Dic5, D10, C52C8, C2×Dic5, C5⋊D5, C2×C52C8, C526C4, C2×C5⋊D5, C527C8, C2×C526C4, C2×C527C8

Smallest permutation representation of C2×C527C8
Regular action on 400 points
Generators in S400
(1 248)(2 241)(3 242)(4 243)(5 244)(6 245)(7 246)(8 247)(9 250)(10 251)(11 252)(12 253)(13 254)(14 255)(15 256)(16 249)(17 258)(18 259)(19 260)(20 261)(21 262)(22 263)(23 264)(24 257)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 121)(32 122)(33 157)(34 158)(35 159)(36 160)(37 153)(38 154)(39 155)(40 156)(41 165)(42 166)(43 167)(44 168)(45 161)(46 162)(47 163)(48 164)(49 197)(50 198)(51 199)(52 200)(53 193)(54 194)(55 195)(56 196)(57 134)(58 135)(59 136)(60 129)(61 130)(62 131)(63 132)(64 133)(65 273)(66 274)(67 275)(68 276)(69 277)(70 278)(71 279)(72 280)(73 221)(74 222)(75 223)(76 224)(77 217)(78 218)(79 219)(80 220)(81 205)(82 206)(83 207)(84 208)(85 201)(86 202)(87 203)(88 204)(89 213)(90 214)(91 215)(92 216)(93 209)(94 210)(95 211)(96 212)(97 288)(98 281)(99 282)(100 283)(101 284)(102 285)(103 286)(104 287)(105 331)(106 332)(107 333)(108 334)(109 335)(110 336)(111 329)(112 330)(113 339)(114 340)(115 341)(116 342)(117 343)(118 344)(119 337)(120 338)(137 363)(138 364)(139 365)(140 366)(141 367)(142 368)(143 361)(144 362)(145 347)(146 348)(147 349)(148 350)(149 351)(150 352)(151 345)(152 346)(169 395)(170 396)(171 397)(172 398)(173 399)(174 400)(175 393)(176 394)(177 378)(178 379)(179 380)(180 381)(181 382)(182 383)(183 384)(184 377)(185 386)(186 387)(187 388)(188 389)(189 390)(190 391)(191 392)(192 385)(225 309)(226 310)(227 311)(228 312)(229 305)(230 306)(231 307)(232 308)(233 317)(234 318)(235 319)(236 320)(237 313)(238 314)(239 315)(240 316)(265 325)(266 326)(267 327)(268 328)(269 321)(270 322)(271 323)(272 324)(289 374)(290 375)(291 376)(292 369)(293 370)(294 371)(295 372)(296 373)(297 353)(298 354)(299 355)(300 356)(301 357)(302 358)(303 359)(304 360)
(1 261 155 97 207)(2 208 98 156 262)(3 263 157 99 201)(4 202 100 158 264)(5 257 159 101 203)(6 204 102 160 258)(7 259 153 103 205)(8 206 104 154 260)(9 233 197 65 151)(10 152 66 198 234)(11 235 199 67 145)(12 146 68 200 236)(13 237 193 69 147)(14 148 70 194 238)(15 239 195 71 149)(16 150 72 196 240)(17 245 88 285 36)(18 37 286 81 246)(19 247 82 287 38)(20 39 288 83 248)(21 241 84 281 40)(22 33 282 85 242)(23 243 86 283 34)(24 35 284 87 244)(25 140 129 385 230)(26 231 386 130 141)(27 142 131 387 232)(28 225 388 132 143)(29 144 133 389 226)(30 227 390 134 137)(31 138 135 391 228)(32 229 392 136 139)(41 294 213 331 265)(42 266 332 214 295)(43 296 215 333 267)(44 268 334 216 289)(45 290 209 335 269)(46 270 336 210 291)(47 292 211 329 271)(48 272 330 212 293)(49 273 345 250 317)(50 318 251 346 274)(51 275 347 252 319)(52 320 253 348 276)(53 277 349 254 313)(54 314 255 350 278)(55 279 351 256 315)(56 316 249 352 280)(57 363 128 311 189)(58 190 312 121 364)(59 365 122 305 191)(60 192 306 123 366)(61 367 124 307 185)(62 186 308 125 368)(63 361 126 309 187)(64 188 310 127 362)(73 341 298 395 180)(74 181 396 299 342)(75 343 300 397 182)(76 183 398 301 344)(77 337 302 399 184)(78 177 400 303 338)(79 339 304 393 178)(80 179 394 297 340)(89 105 325 165 371)(90 372 166 326 106)(91 107 327 167 373)(92 374 168 328 108)(93 109 321 161 375)(94 376 162 322 110)(95 111 323 163 369)(96 370 164 324 112)(113 360 175 379 219)(114 220 380 176 353)(115 354 169 381 221)(116 222 382 170 355)(117 356 171 383 223)(118 224 384 172 357)(119 358 173 377 217)(120 218 378 174 359)
(1 161 343 389 346)(2 347 390 344 162)(3 163 337 391 348)(4 349 392 338 164)(5 165 339 385 350)(6 351 386 340 166)(7 167 341 387 352)(8 345 388 342 168)(9 63 222 268 82)(10 83 269 223 64)(11 57 224 270 84)(12 85 271 217 58)(13 59 218 272 86)(14 87 265 219 60)(15 61 220 266 88)(16 81 267 221 62)(17 71 307 353 295)(18 296 354 308 72)(19 65 309 355 289)(20 290 356 310 66)(21 67 311 357 291)(22 292 358 312 68)(23 69 305 359 293)(24 294 360 306 70)(25 54 159 89 393)(26 394 90 160 55)(27 56 153 91 395)(28 396 92 154 49)(29 50 155 93 397)(30 398 94 156 51)(31 52 157 95 399)(32 400 96 158 53)(33 211 173 121 200)(34 193 122 174 212)(35 213 175 123 194)(36 195 124 176 214)(37 215 169 125 196)(38 197 126 170 216)(39 209 171 127 198)(40 199 128 172 210)(41 113 192 148 244)(42 245 149 185 114)(43 115 186 150 246)(44 247 151 187 116)(45 117 188 152 248)(46 241 145 189 118)(47 119 190 146 242)(48 243 147 191 120)(73 131 249 205 327)(74 328 206 250 132)(75 133 251 207 321)(76 322 208 252 134)(77 135 253 201 323)(78 324 202 254 136)(79 129 255 203 325)(80 326 204 256 130)(97 109 182 144 318)(98 319 137 183 110)(99 111 184 138 320)(100 313 139 177 112)(101 105 178 140 314)(102 315 141 179 106)(103 107 180 142 316)(104 317 143 181 108)(225 299 374 260 273)(226 274 261 375 300)(227 301 376 262 275)(228 276 263 369 302)(229 303 370 264 277)(230 278 257 371 304)(231 297 372 258 279)(232 280 259 373 298)(233 361 382 334 287)(234 288 335 383 362)(235 363 384 336 281)(236 282 329 377 364)(237 365 378 330 283)(238 284 331 379 366)(239 367 380 332 285)(240 286 333 381 368)
(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)

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A ··· 5L 8A ··· 8H 10A ··· 10AJ 20A ··· 20AV order 1 2 2 2 4 4 4 4 5 ··· 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 1 1 1 1 2 ··· 2 25 ··· 25 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 D5 Dic5 D10 Dic5 C5⋊2C8 kernel C2×C52⋊7C8 C52⋊7C8 C10×C20 C5×C20 C102 C5×C10 C2×C20 C20 C20 C2×C10 C10 # reps 1 2 1 2 2 8 12 12 12 12 48

Matrix representation of C2×C527C8 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 1 0 0 5 35 0 0 0 0 16 0 0 0 24 18
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 24 18
,
 1 0 0 0 36 40 0 0 0 0 16 15 0 0 37 25
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,5,0,0,1,35,0,0,0,0,16,24,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,16,24,0,0,0,18],[1,36,0,0,0,40,0,0,0,0,16,37,0,0,15,25] >;

C2×C527C8 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes_7C_8
% in TeX

G:=Group("C2xC5^2:7C8");
// GroupNames label

G:=SmallGroup(400,97);
// by ID

G=gap.SmallGroup(400,97);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,50,1924,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^5=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽