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## G = C2×C52⋊4C8order 400 = 24·52

### Direct product of C2 and C52⋊4C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C52⋊4C8
 Chief series C1 — C5 — C52 — C5×C10 — C52⋊6C4 — C52⋊4C8 — C2×C52⋊4C8
 Lower central C52 — C2×C52⋊4C8
 Upper central C1 — C22

Generators and relations for C2×C524C8
G = < a,b,c,d | a2=b5=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c3 >

Subgroups: 376 in 88 conjugacy classes, 46 normal (10 characteristic)
C1, C2, C2, C4, C22, C5, C8, C2×C4, C10, C2×C8, Dic5, C2×C10, C52, C5⋊C8, C2×Dic5, C5×C10, C5×C10, C2×C5⋊C8, C526C4, C102, C524C8, C2×C526C4, C2×C524C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C2×C8, F5, C5⋊C8, C2×F5, C2×C5⋊C8, C5⋊F5, C524C8, C2×C5⋊F5, C2×C524C8

Smallest permutation representation of C2×C524C8
Regular action on 400 points
Generators in S400
(1 244)(2 245)(3 246)(4 247)(5 248)(6 241)(7 242)(8 243)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 105)(17 279)(18 280)(19 273)(20 274)(21 275)(22 276)(23 277)(24 278)(25 310)(26 311)(27 312)(28 305)(29 306)(30 307)(31 308)(32 309)(33 228)(34 229)(35 230)(36 231)(37 232)(38 225)(39 226)(40 227)(41 236)(42 237)(43 238)(44 239)(45 240)(46 233)(47 234)(48 235)(49 268)(50 269)(51 270)(52 271)(53 272)(54 265)(55 266)(56 267)(57 156)(58 157)(59 158)(60 159)(61 160)(62 153)(63 154)(64 155)(65 118)(66 119)(67 120)(68 113)(69 114)(70 115)(71 116)(72 117)(73 150)(74 151)(75 152)(76 145)(77 146)(78 147)(79 148)(80 149)(81 134)(82 135)(83 136)(84 129)(85 130)(86 131)(87 132)(88 133)(89 142)(90 143)(91 144)(92 137)(93 138)(94 139)(95 140)(96 141)(97 284)(98 285)(99 286)(100 287)(101 288)(102 281)(103 282)(104 283)(121 393)(122 394)(123 395)(124 396)(125 397)(126 398)(127 399)(128 400)(161 316)(162 317)(163 318)(164 319)(165 320)(166 313)(167 314)(168 315)(169 300)(170 301)(171 302)(172 303)(173 304)(174 297)(175 298)(176 299)(177 382)(178 383)(179 384)(180 377)(181 378)(182 379)(183 380)(184 381)(185 390)(186 391)(187 392)(188 385)(189 386)(190 387)(191 388)(192 389)(193 324)(194 325)(195 326)(196 327)(197 328)(198 321)(199 322)(200 323)(201 356)(202 357)(203 358)(204 359)(205 360)(206 353)(207 354)(208 355)(209 364)(210 365)(211 366)(212 367)(213 368)(214 361)(215 362)(216 363)(217 348)(218 349)(219 350)(220 351)(221 352)(222 345)(223 346)(224 347)(249 329)(250 330)(251 331)(252 332)(253 333)(254 334)(255 335)(256 336)(257 337)(258 338)(259 339)(260 340)(261 341)(262 342)(263 343)(264 344)(289 370)(290 371)(291 372)(292 373)(293 374)(294 375)(295 376)(296 369)
(1 211 159 97 132)(2 98 212 133 160)(3 134 99 153 213)(4 154 135 214 100)(5 215 155 101 136)(6 102 216 129 156)(7 130 103 157 209)(8 158 131 210 104)(9 38 315 191 272)(10 192 39 265 316)(11 266 185 317 40)(12 318 267 33 186)(13 34 319 187 268)(14 188 35 269 320)(15 270 189 313 36)(16 314 271 37 190)(17 353 199 170 235)(18 171 354 236 200)(19 237 172 193 355)(20 194 238 356 173)(21 357 195 174 239)(22 175 358 240 196)(23 233 176 197 359)(24 198 234 360 169)(25 397 182 75 343)(26 76 398 344 183)(27 337 77 184 399)(28 177 338 400 78)(29 393 178 79 339)(30 80 394 340 179)(31 341 73 180 395)(32 181 342 396 74)(41 323 280 302 207)(42 303 324 208 273)(43 201 304 274 325)(44 275 202 326 297)(45 327 276 298 203)(46 299 328 204 277)(47 205 300 278 321)(48 279 206 322 301)(49 110 229 164 392)(50 165 111 385 230)(51 386 166 231 112)(52 232 387 105 167)(53 106 225 168 388)(54 161 107 389 226)(55 390 162 227 108)(56 228 391 109 163)(57 241 281 363 84)(58 364 242 85 282)(59 86 365 283 243)(60 284 87 244 366)(61 245 285 367 88)(62 368 246 81 286)(63 82 361 287 247)(64 288 83 248 362)(65 137 222 293 330)(66 294 138 331 223)(67 332 295 224 139)(68 217 333 140 296)(69 141 218 289 334)(70 290 142 335 219)(71 336 291 220 143)(72 221 329 144 292)(89 255 350 115 371)(90 116 256 372 351)(91 373 117 352 249)(92 345 374 250 118)(93 251 346 119 375)(94 120 252 376 347)(95 369 113 348 253)(96 349 370 254 114)(121 383 148 259 306)(122 260 384 307 149)(123 308 261 150 377)(124 151 309 378 262)(125 379 152 263 310)(126 264 380 311 145)(127 312 257 146 381)(128 147 305 382 258)
(1 119 343 389 304)(2 390 120 297 344)(3 298 391 337 113)(4 338 299 114 392)(5 115 339 385 300)(6 386 116 301 340)(7 302 387 341 117)(8 342 303 118 388)(9 86 151 355 222)(10 356 87 223 152)(11 224 357 145 88)(12 146 217 81 358)(13 82 147 359 218)(14 360 83 219 148)(15 220 353 149 84)(16 150 221 85 354)(17 307 363 36 291)(18 37 308 292 364)(19 293 38 365 309)(20 366 294 310 39)(21 311 367 40 295)(22 33 312 296 368)(23 289 34 361 305)(24 362 290 306 35)(25 226 274 211 375)(26 212 227 376 275)(27 369 213 276 228)(28 277 370 229 214)(29 230 278 215 371)(30 216 231 372 279)(31 373 209 280 232)(32 273 374 225 210)(41 167 180 249 103)(42 250 168 104 181)(43 97 251 182 161)(44 183 98 162 252)(45 163 184 253 99)(46 254 164 100 177)(47 101 255 178 165)(48 179 102 166 256)(49 154 400 328 96)(50 321 155 89 393)(51 90 322 394 156)(52 395 91 157 323)(53 158 396 324 92)(54 325 159 93 397)(55 94 326 398 160)(56 399 95 153 327)(57 270 143 199 122)(58 200 271 123 144)(59 124 193 137 272)(60 138 125 265 194)(61 266 139 195 126)(62 196 267 127 140)(63 128 197 141 268)(64 142 121 269 198)(65 191 243 262 172)(66 263 192 173 244)(67 174 264 245 185)(68 246 175 186 257)(69 187 247 258 176)(70 259 188 169 248)(71 170 260 241 189)(72 242 171 190 261)(73 352 130 207 105)(74 208 345 106 131)(75 107 201 132 346)(76 133 108 347 202)(77 348 134 203 109)(78 204 349 110 135)(79 111 205 136 350)(80 129 112 351 206)(233 334 319 287 382)(234 288 335 383 320)(235 384 281 313 336)(236 314 377 329 282)(237 330 315 283 378)(238 284 331 379 316)(239 380 285 317 332)(240 318 381 333 286)
(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)| (1,244)(2,245)(3,246)(4,247)(5,248)(6,241)(7,242)(8,243)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,105)(17,279)(18,280)(19,273)(20,274)(21,275)(22,276)(23,277)(24,278)(25,310)(26,311)(27,312)(28,305)(29,306)(30,307)(31,308)(32,309)(33,228)(34,229)(35,230)(36,231)(37,232)(38,225)(39,226)(40,227)(41,236)(42,237)(43,238)(44,239)(45,240)(46,233)(47,234)(48,235)(49,268)(50,269)(51,270)(52,271)(53,272)(54,265)(55,266)(56,267)(57,156)(58,157)(59,158)(60,159)(61,160)(62,153)(63,154)(64,155)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(73,150)(74,151)(75,152)(76,145)(77,146)(78,147)(79,148)(80,149)(81,134)(82,135)(83,136)(84,129)(85,130)(86,131)(87,132)(88,133)(89,142)(90,143)(91,144)(92,137)(93,138)(94,139)(95,140)(96,141)(97,284)(98,285)(99,286)(100,287)(101,288)(102,281)(103,282)(104,283)(121,393)(122,394)(123,395)(124,396)(125,397)(126,398)(127,399)(128,400)(161,316)(162,317)(163,318)(164,319)(165,320)(166,313)(167,314)(168,315)(169,300)(170,301)(171,302)(172,303)(173,304)(174,297)(175,298)(176,299)(177,382)(178,383)(179,384)(180,377)(181,378)(182,379)(183,380)(184,381)(185,390)(186,391)(187,392)(188,385)(189,386)(190,387)(191,388)(192,389)(193,324)(194,325)(195,326)(196,327)(197,328)(198,321)(199,322)(200,323)(201,356)(202,357)(203,358)(204,359)(205,360)(206,353)(207,354)(208,355)(209,364)(210,365)(211,366)(212,367)(213,368)(214,361)(215,362)(216,363)(217,348)(218,349)(219,350)(220,351)(221,352)(222,345)(223,346)(224,347)(249,329)(250,330)(251,331)(252,332)(253,333)(254,334)(255,335)(256,336)(257,337)(258,338)(259,339)(260,340)(261,341)(262,342)(263,343)(264,344)(289,370)(290,371)(291,372)(292,373)(293,374)(294,375)(295,376)(296,369), (1,211,159,97,132)(2,98,212,133,160)(3,134,99,153,213)(4,154,135,214,100)(5,215,155,101,136)(6,102,216,129,156)(7,130,103,157,209)(8,158,131,210,104)(9,38,315,191,272)(10,192,39,265,316)(11,266,185,317,40)(12,318,267,33,186)(13,34,319,187,268)(14,188,35,269,320)(15,270,189,313,36)(16,314,271,37,190)(17,353,199,170,235)(18,171,354,236,200)(19,237,172,193,355)(20,194,238,356,173)(21,357,195,174,239)(22,175,358,240,196)(23,233,176,197,359)(24,198,234,360,169)(25,397,182,75,343)(26,76,398,344,183)(27,337,77,184,399)(28,177,338,400,78)(29,393,178,79,339)(30,80,394,340,179)(31,341,73,180,395)(32,181,342,396,74)(41,323,280,302,207)(42,303,324,208,273)(43,201,304,274,325)(44,275,202,326,297)(45,327,276,298,203)(46,299,328,204,277)(47,205,300,278,321)(48,279,206,322,301)(49,110,229,164,392)(50,165,111,385,230)(51,386,166,231,112)(52,232,387,105,167)(53,106,225,168,388)(54,161,107,389,226)(55,390,162,227,108)(56,228,391,109,163)(57,241,281,363,84)(58,364,242,85,282)(59,86,365,283,243)(60,284,87,244,366)(61,245,285,367,88)(62,368,246,81,286)(63,82,361,287,247)(64,288,83,248,362)(65,137,222,293,330)(66,294,138,331,223)(67,332,295,224,139)(68,217,333,140,296)(69,141,218,289,334)(70,290,142,335,219)(71,336,291,220,143)(72,221,329,144,292)(89,255,350,115,371)(90,116,256,372,351)(91,373,117,352,249)(92,345,374,250,118)(93,251,346,119,375)(94,120,252,376,347)(95,369,113,348,253)(96,349,370,254,114)(121,383,148,259,306)(122,260,384,307,149)(123,308,261,150,377)(124,151,309,378,262)(125,379,152,263,310)(126,264,380,311,145)(127,312,257,146,381)(128,147,305,382,258), (1,119,343,389,304)(2,390,120,297,344)(3,298,391,337,113)(4,338,299,114,392)(5,115,339,385,300)(6,386,116,301,340)(7,302,387,341,117)(8,342,303,118,388)(9,86,151,355,222)(10,356,87,223,152)(11,224,357,145,88)(12,146,217,81,358)(13,82,147,359,218)(14,360,83,219,148)(15,220,353,149,84)(16,150,221,85,354)(17,307,363,36,291)(18,37,308,292,364)(19,293,38,365,309)(20,366,294,310,39)(21,311,367,40,295)(22,33,312,296,368)(23,289,34,361,305)(24,362,290,306,35)(25,226,274,211,375)(26,212,227,376,275)(27,369,213,276,228)(28,277,370,229,214)(29,230,278,215,371)(30,216,231,372,279)(31,373,209,280,232)(32,273,374,225,210)(41,167,180,249,103)(42,250,168,104,181)(43,97,251,182,161)(44,183,98,162,252)(45,163,184,253,99)(46,254,164,100,177)(47,101,255,178,165)(48,179,102,166,256)(49,154,400,328,96)(50,321,155,89,393)(51,90,322,394,156)(52,395,91,157,323)(53,158,396,324,92)(54,325,159,93,397)(55,94,326,398,160)(56,399,95,153,327)(57,270,143,199,122)(58,200,271,123,144)(59,124,193,137,272)(60,138,125,265,194)(61,266,139,195,126)(62,196,267,127,140)(63,128,197,141,268)(64,142,121,269,198)(65,191,243,262,172)(66,263,192,173,244)(67,174,264,245,185)(68,246,175,186,257)(69,187,247,258,176)(70,259,188,169,248)(71,170,260,241,189)(72,242,171,190,261)(73,352,130,207,105)(74,208,345,106,131)(75,107,201,132,346)(76,133,108,347,202)(77,348,134,203,109)(78,204,349,110,135)(79,111,205,136,350)(80,129,112,351,206)(233,334,319,287,382)(234,288,335,383,320)(235,384,281,313,336)(236,314,377,329,282)(237,330,315,283,378)(238,284,331,379,316)(239,380,285,317,332)(240,318,381,333,286), (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( (1,244)(2,245)(3,246)(4,247)(5,248)(6,241)(7,242)(8,243)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,105)(17,279)(18,280)(19,273)(20,274)(21,275)(22,276)(23,277)(24,278)(25,310)(26,311)(27,312)(28,305)(29,306)(30,307)(31,308)(32,309)(33,228)(34,229)(35,230)(36,231)(37,232)(38,225)(39,226)(40,227)(41,236)(42,237)(43,238)(44,239)(45,240)(46,233)(47,234)(48,235)(49,268)(50,269)(51,270)(52,271)(53,272)(54,265)(55,266)(56,267)(57,156)(58,157)(59,158)(60,159)(61,160)(62,153)(63,154)(64,155)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(73,150)(74,151)(75,152)(76,145)(77,146)(78,147)(79,148)(80,149)(81,134)(82,135)(83,136)(84,129)(85,130)(86,131)(87,132)(88,133)(89,142)(90,143)(91,144)(92,137)(93,138)(94,139)(95,140)(96,141)(97,284)(98,285)(99,286)(100,287)(101,288)(102,281)(103,282)(104,283)(121,393)(122,394)(123,395)(124,396)(125,397)(126,398)(127,399)(128,400)(161,316)(162,317)(163,318)(164,319)(165,320)(166,313)(167,314)(168,315)(169,300)(170,301)(171,302)(172,303)(173,304)(174,297)(175,298)(176,299)(177,382)(178,383)(179,384)(180,377)(181,378)(182,379)(183,380)(184,381)(185,390)(186,391)(187,392)(188,385)(189,386)(190,387)(191,388)(192,389)(193,324)(194,325)(195,326)(196,327)(197,328)(198,321)(199,322)(200,323)(201,356)(202,357)(203,358)(204,359)(205,360)(206,353)(207,354)(208,355)(209,364)(210,365)(211,366)(212,367)(213,368)(214,361)(215,362)(216,363)(217,348)(218,349)(219,350)(220,351)(221,352)(222,345)(223,346)(224,347)(249,329)(250,330)(251,331)(252,332)(253,333)(254,334)(255,335)(256,336)(257,337)(258,338)(259,339)(260,340)(261,341)(262,342)(263,343)(264,344)(289,370)(290,371)(291,372)(292,373)(293,374)(294,375)(295,376)(296,369), (1,211,159,97,132)(2,98,212,133,160)(3,134,99,153,213)(4,154,135,214,100)(5,215,155,101,136)(6,102,216,129,156)(7,130,103,157,209)(8,158,131,210,104)(9,38,315,191,272)(10,192,39,265,316)(11,266,185,317,40)(12,318,267,33,186)(13,34,319,187,268)(14,188,35,269,320)(15,270,189,313,36)(16,314,271,37,190)(17,353,199,170,235)(18,171,354,236,200)(19,237,172,193,355)(20,194,238,356,173)(21,357,195,174,239)(22,175,358,240,196)(23,233,176,197,359)(24,198,234,360,169)(25,397,182,75,343)(26,76,398,344,183)(27,337,77,184,399)(28,177,338,400,78)(29,393,178,79,339)(30,80,394,340,179)(31,341,73,180,395)(32,181,342,396,74)(41,323,280,302,207)(42,303,324,208,273)(43,201,304,274,325)(44,275,202,326,297)(45,327,276,298,203)(46,299,328,204,277)(47,205,300,278,321)(48,279,206,322,301)(49,110,229,164,392)(50,165,111,385,230)(51,386,166,231,112)(52,232,387,105,167)(53,106,225,168,388)(54,161,107,389,226)(55,390,162,227,108)(56,228,391,109,163)(57,241,281,363,84)(58,364,242,85,282)(59,86,365,283,243)(60,284,87,244,366)(61,245,285,367,88)(62,368,246,81,286)(63,82,361,287,247)(64,288,83,248,362)(65,137,222,293,330)(66,294,138,331,223)(67,332,295,224,139)(68,217,333,140,296)(69,141,218,289,334)(70,290,142,335,219)(71,336,291,220,143)(72,221,329,144,292)(89,255,350,115,371)(90,116,256,372,351)(91,373,117,352,249)(92,345,374,250,118)(93,251,346,119,375)(94,120,252,376,347)(95,369,113,348,253)(96,349,370,254,114)(121,383,148,259,306)(122,260,384,307,149)(123,308,261,150,377)(124,151,309,378,262)(125,379,152,263,310)(126,264,380,311,145)(127,312,257,146,381)(128,147,305,382,258), (1,119,343,389,304)(2,390,120,297,344)(3,298,391,337,113)(4,338,299,114,392)(5,115,339,385,300)(6,386,116,301,340)(7,302,387,341,117)(8,342,303,118,388)(9,86,151,355,222)(10,356,87,223,152)(11,224,357,145,88)(12,146,217,81,358)(13,82,147,359,218)(14,360,83,219,148)(15,220,353,149,84)(16,150,221,85,354)(17,307,363,36,291)(18,37,308,292,364)(19,293,38,365,309)(20,366,294,310,39)(21,311,367,40,295)(22,33,312,296,368)(23,289,34,361,305)(24,362,290,306,35)(25,226,274,211,375)(26,212,227,376,275)(27,369,213,276,228)(28,277,370,229,214)(29,230,278,215,371)(30,216,231,372,279)(31,373,209,280,232)(32,273,374,225,210)(41,167,180,249,103)(42,250,168,104,181)(43,97,251,182,161)(44,183,98,162,252)(45,163,184,253,99)(46,254,164,100,177)(47,101,255,178,165)(48,179,102,166,256)(49,154,400,328,96)(50,321,155,89,393)(51,90,322,394,156)(52,395,91,157,323)(53,158,396,324,92)(54,325,159,93,397)(55,94,326,398,160)(56,399,95,153,327)(57,270,143,199,122)(58,200,271,123,144)(59,124,193,137,272)(60,138,125,265,194)(61,266,139,195,126)(62,196,267,127,140)(63,128,197,141,268)(64,142,121,269,198)(65,191,243,262,172)(66,263,192,173,244)(67,174,264,245,185)(68,246,175,186,257)(69,187,247,258,176)(70,259,188,169,248)(71,170,260,241,189)(72,242,171,190,261)(73,352,130,207,105)(74,208,345,106,131)(75,107,201,132,346)(76,133,108,347,202)(77,348,134,203,109)(78,204,349,110,135)(79,111,205,136,350)(80,129,112,351,206)(233,334,319,287,382)(234,288,335,383,320)(235,384,281,313,336)(236,314,377,329,282)(237,330,315,283,378)(238,284,331,379,316)(239,380,285,317,332)(240,318,381,333,286), (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([[(1,244),(2,245),(3,246),(4,247),(5,248),(6,241),(7,242),(8,243),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,112),(16,105),(17,279),(18,280),(19,273),(20,274),(21,275),(22,276),(23,277),(24,278),(25,310),(26,311),(27,312),(28,305),(29,306),(30,307),(31,308),(32,309),(33,228),(34,229),(35,230),(36,231),(37,232),(38,225),(39,226),(40,227),(41,236),(42,237),(43,238),(44,239),(45,240),(46,233),(47,234),(48,235),(49,268),(50,269),(51,270),(52,271),(53,272),(54,265),(55,266),(56,267),(57,156),(58,157),(59,158),(60,159),(61,160),(62,153),(63,154),(64,155),(65,118),(66,119),(67,120),(68,113),(69,114),(70,115),(71,116),(72,117),(73,150),(74,151),(75,152),(76,145),(77,146),(78,147),(79,148),(80,149),(81,134),(82,135),(83,136),(84,129),(85,130),(86,131),(87,132),(88,133),(89,142),(90,143),(91,144),(92,137),(93,138),(94,139),(95,140),(96,141),(97,284),(98,285),(99,286),(100,287),(101,288),(102,281),(103,282),(104,283),(121,393),(122,394),(123,395),(124,396),(125,397),(126,398),(127,399),(128,400),(161,316),(162,317),(163,318),(164,319),(165,320),(166,313),(167,314),(168,315),(169,300),(170,301),(171,302),(172,303),(173,304),(174,297),(175,298),(176,299),(177,382),(178,383),(179,384),(180,377),(181,378),(182,379),(183,380),(184,381),(185,390),(186,391),(187,392),(188,385),(189,386),(190,387),(191,388),(192,389),(193,324),(194,325),(195,326),(196,327),(197,328),(198,321),(199,322),(200,323),(201,356),(202,357),(203,358),(204,359),(205,360),(206,353),(207,354),(208,355),(209,364),(210,365),(211,366),(212,367),(213,368),(214,361),(215,362),(216,363),(217,348),(218,349),(219,350),(220,351),(221,352),(222,345),(223,346),(224,347),(249,329),(250,330),(251,331),(252,332),(253,333),(254,334),(255,335),(256,336),(257,337),(258,338),(259,339),(260,340),(261,341),(262,342),(263,343),(264,344),(289,370),(290,371),(291,372),(292,373),(293,374),(294,375),(295,376),(296,369)], [(1,211,159,97,132),(2,98,212,133,160),(3,134,99,153,213),(4,154,135,214,100),(5,215,155,101,136),(6,102,216,129,156),(7,130,103,157,209),(8,158,131,210,104),(9,38,315,191,272),(10,192,39,265,316),(11,266,185,317,40),(12,318,267,33,186),(13,34,319,187,268),(14,188,35,269,320),(15,270,189,313,36),(16,314,271,37,190),(17,353,199,170,235),(18,171,354,236,200),(19,237,172,193,355),(20,194,238,356,173),(21,357,195,174,239),(22,175,358,240,196),(23,233,176,197,359),(24,198,234,360,169),(25,397,182,75,343),(26,76,398,344,183),(27,337,77,184,399),(28,177,338,400,78),(29,393,178,79,339),(30,80,394,340,179),(31,341,73,180,395),(32,181,342,396,74),(41,323,280,302,207),(42,303,324,208,273),(43,201,304,274,325),(44,275,202,326,297),(45,327,276,298,203),(46,299,328,204,277),(47,205,300,278,321),(48,279,206,322,301),(49,110,229,164,392),(50,165,111,385,230),(51,386,166,231,112),(52,232,387,105,167),(53,106,225,168,388),(54,161,107,389,226),(55,390,162,227,108),(56,228,391,109,163),(57,241,281,363,84),(58,364,242,85,282),(59,86,365,283,243),(60,284,87,244,366),(61,245,285,367,88),(62,368,246,81,286),(63,82,361,287,247),(64,288,83,248,362),(65,137,222,293,330),(66,294,138,331,223),(67,332,295,224,139),(68,217,333,140,296),(69,141,218,289,334),(70,290,142,335,219),(71,336,291,220,143),(72,221,329,144,292),(89,255,350,115,371),(90,116,256,372,351),(91,373,117,352,249),(92,345,374,250,118),(93,251,346,119,375),(94,120,252,376,347),(95,369,113,348,253),(96,349,370,254,114),(121,383,148,259,306),(122,260,384,307,149),(123,308,261,150,377),(124,151,309,378,262),(125,379,152,263,310),(126,264,380,311,145),(127,312,257,146,381),(128,147,305,382,258)], [(1,119,343,389,304),(2,390,120,297,344),(3,298,391,337,113),(4,338,299,114,392),(5,115,339,385,300),(6,386,116,301,340),(7,302,387,341,117),(8,342,303,118,388),(9,86,151,355,222),(10,356,87,223,152),(11,224,357,145,88),(12,146,217,81,358),(13,82,147,359,218),(14,360,83,219,148),(15,220,353,149,84),(16,150,221,85,354),(17,307,363,36,291),(18,37,308,292,364),(19,293,38,365,309),(20,366,294,310,39),(21,311,367,40,295),(22,33,312,296,368),(23,289,34,361,305),(24,362,290,306,35),(25,226,274,211,375),(26,212,227,376,275),(27,369,213,276,228),(28,277,370,229,214),(29,230,278,215,371),(30,216,231,372,279),(31,373,209,280,232),(32,273,374,225,210),(41,167,180,249,103),(42,250,168,104,181),(43,97,251,182,161),(44,183,98,162,252),(45,163,184,253,99),(46,254,164,100,177),(47,101,255,178,165),(48,179,102,166,256),(49,154,400,328,96),(50,321,155,89,393),(51,90,322,394,156),(52,395,91,157,323),(53,158,396,324,92),(54,325,159,93,397),(55,94,326,398,160),(56,399,95,153,327),(57,270,143,199,122),(58,200,271,123,144),(59,124,193,137,272),(60,138,125,265,194),(61,266,139,195,126),(62,196,267,127,140),(63,128,197,141,268),(64,142,121,269,198),(65,191,243,262,172),(66,263,192,173,244),(67,174,264,245,185),(68,246,175,186,257),(69,187,247,258,176),(70,259,188,169,248),(71,170,260,241,189),(72,242,171,190,261),(73,352,130,207,105),(74,208,345,106,131),(75,107,201,132,346),(76,133,108,347,202),(77,348,134,203,109),(78,204,349,110,135),(79,111,205,136,350),(80,129,112,351,206),(233,334,319,287,382),(234,288,335,383,320),(235,384,281,313,336),(236,314,377,329,282),(237,330,315,283,378),(238,284,331,379,316),(239,380,285,317,332),(240,318,381,333,286)], [(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)]])

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A ··· 5F 8A ··· 8H 10A ··· 10R order 1 2 2 2 4 4 4 4 5 ··· 5 8 ··· 8 10 ··· 10 size 1 1 1 1 25 25 25 25 4 ··· 4 25 ··· 25 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 1 4 4 4 type + + + + - + image C1 C2 C2 C4 C4 C8 F5 C5⋊C8 C2×F5 kernel C2×C52⋊4C8 C52⋊4C8 C2×C52⋊6C4 C52⋊6C4 C102 C5×C10 C2×C10 C10 C10 # reps 1 2 1 2 2 8 6 12 6

Matrix representation of C2×C524C8 in GL9(𝔽41)

 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 40 40 40 40 0 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 40 40 40 40
,
 40 0 0 0 0 0 0 0 0 0 17 18 21 24 0 0 0 0 0 3 6 23 40 0 0 0 0 0 17 34 35 38 0 0 0 0 0 1 4 7 24 0 0 0 0 0 0 0 0 0 21 0 31 37 0 0 0 0 0 31 37 0 21 0 0 0 0 0 4 25 4 35 0 0 0 0 0 20 10 16 20

G:=sub<GL(9,GF(41))| [40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,1,40,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,1,40],[40,0,0,0,0,0,0,0,0,0,17,3,17,1,0,0,0,0,0,18,6,34,4,0,0,0,0,0,21,23,35,7,0,0,0,0,0,24,40,38,24,0,0,0,0,0,0,0,0,0,21,31,4,20,0,0,0,0,0,0,37,25,10,0,0,0,0,0,31,0,4,16,0,0,0,0,0,37,21,35,20] >;

C2×C524C8 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes_4C_8
% in TeX

G:=Group("C2xC5^2:4C8");
// GroupNames label

G:=SmallGroup(400,153);
// by ID

G=gap.SmallGroup(400,153);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,50,964,496,5765,2897]);
// 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^3,d*c*d^-1=c^3>;
// generators/relations

׿
×
𝔽