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## G = C4×C52⋊6C4order 400 = 24·52

### Direct product of C4 and C52⋊6C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C4×C52⋊6C4
 Chief series C1 — C5 — C52 — C5×C10 — C102 — C2×C52⋊6C4 — C4×C52⋊6C4
 Lower central C52 — C4×C52⋊6C4
 Upper central C1 — C2×C4

Generators and relations for C4×C526C4
G = < a,b,c,d | a4=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 456 in 120 conjugacy classes, 71 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, C2×C4, C10, C42, Dic5, C20, C2×C10, C52, C2×Dic5, C2×C20, C5×C10, C5×C10, C4×Dic5, C526C4, C5×C20, C102, C2×C526C4, C10×C20, C4×C526C4
Quotients: C1, C2, C4, C22, C2×C4, D5, C42, Dic5, D10, C4×D5, C2×Dic5, C5⋊D5, C4×Dic5, C526C4, C2×C5⋊D5, C4×C5⋊D5, C2×C526C4, C4×C526C4

Smallest permutation representation of C4×C526C4
Regular action on 400 points
Generators in S400
(1 148 48 123)(2 149 49 124)(3 150 50 125)(4 146 46 121)(5 147 47 122)(6 320 395 295)(7 316 391 291)(8 317 392 292)(9 318 393 293)(10 319 394 294)(11 315 390 290)(12 311 386 286)(13 312 387 287)(14 313 388 288)(15 314 389 289)(16 310 385 285)(17 306 381 281)(18 307 382 282)(19 308 383 283)(20 309 384 284)(21 305 380 280)(22 301 376 276)(23 302 377 277)(24 303 378 278)(25 304 379 279)(26 151 51 126)(27 152 52 127)(28 153 53 128)(29 154 54 129)(30 155 55 130)(31 156 56 131)(32 157 57 132)(33 158 58 133)(34 159 59 134)(35 160 60 135)(36 161 61 136)(37 162 62 137)(38 163 63 138)(39 164 64 139)(40 165 65 140)(41 166 66 141)(42 167 67 142)(43 168 68 143)(44 169 69 144)(45 170 70 145)(71 196 96 171)(72 197 97 172)(73 198 98 173)(74 199 99 174)(75 200 100 175)(76 216 101 176)(77 217 102 177)(78 218 103 178)(79 219 104 179)(80 220 105 180)(81 211 106 181)(82 212 107 182)(83 213 108 183)(84 214 109 184)(85 215 110 185)(86 206 111 186)(87 207 112 187)(88 208 113 188)(89 209 114 189)(90 210 115 190)(91 201 116 191)(92 202 117 192)(93 203 118 193)(94 204 119 194)(95 205 120 195)(221 346 246 321)(222 347 247 322)(223 348 248 323)(224 349 249 324)(225 350 250 325)(226 351 251 326)(227 352 252 327)(228 353 253 328)(229 354 254 329)(230 355 255 330)(231 356 256 331)(232 357 257 332)(233 358 258 333)(234 359 259 334)(235 360 260 335)(236 361 261 336)(237 362 262 337)(238 363 263 338)(239 364 264 339)(240 365 265 340)(241 366 266 341)(242 367 267 342)(243 368 268 343)(244 369 269 344)(245 370 270 345)(271 396 296 371)(272 397 297 372)(273 398 298 373)(274 399 299 374)(275 400 300 375)
(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 44 38 33 28)(2 45 39 34 29)(3 41 40 35 30)(4 42 36 31 26)(5 43 37 32 27)(6 400 25 16 11)(7 396 21 17 12)(8 397 22 18 13)(9 398 23 19 14)(10 399 24 20 15)(46 67 61 56 51)(47 68 62 57 52)(48 69 63 58 53)(49 70 64 59 54)(50 66 65 60 55)(71 92 86 81 76)(72 93 87 82 77)(73 94 88 83 78)(74 95 89 84 79)(75 91 90 85 80)(96 117 111 106 101)(97 118 112 107 102)(98 119 113 108 103)(99 120 114 109 104)(100 116 115 110 105)(121 142 136 131 126)(122 143 137 132 127)(123 144 138 133 128)(124 145 139 134 129)(125 141 140 135 130)(146 167 161 156 151)(147 168 162 157 152)(148 169 163 158 153)(149 170 164 159 154)(150 166 165 160 155)(171 192 186 181 176)(172 193 187 182 177)(173 194 188 183 178)(174 195 189 184 179)(175 191 190 185 180)(196 202 206 211 216)(197 203 207 212 217)(198 204 208 213 218)(199 205 209 214 219)(200 201 210 215 220)(221 230 231 236 241)(222 226 232 237 242)(223 227 233 238 243)(224 228 234 239 244)(225 229 235 240 245)(246 255 256 261 266)(247 251 257 262 267)(248 252 258 263 268)(249 253 259 264 269)(250 254 260 265 270)(271 280 281 286 291)(272 276 282 287 292)(273 277 283 288 293)(274 278 284 289 294)(275 279 285 290 295)(296 305 306 311 316)(297 301 307 312 317)(298 302 308 313 318)(299 303 309 314 319)(300 304 310 315 320)(321 330 331 336 341)(322 326 332 337 342)(323 327 333 338 343)(324 328 334 339 344)(325 329 335 340 345)(346 355 356 361 366)(347 351 357 362 367)(348 352 358 363 368)(349 353 359 364 369)(350 354 360 365 370)(371 380 381 386 391)(372 376 382 387 392)(373 377 383 388 393)(374 378 384 389 394)(375 379 385 390 395)
(1 282 73 232)(2 281 74 231)(3 285 75 235)(4 284 71 234)(5 283 72 233)(6 215 370 160)(7 214 366 159)(8 213 367 158)(9 212 368 157)(10 211 369 156)(11 220 365 155)(12 219 361 154)(13 218 362 153)(14 217 363 152)(15 216 364 151)(16 200 360 150)(17 199 356 149)(18 198 357 148)(19 197 358 147)(20 196 359 146)(21 205 355 170)(22 204 351 169)(23 203 352 168)(24 202 353 167)(25 201 354 166)(26 289 76 239)(27 288 77 238)(28 287 78 237)(29 286 79 236)(30 290 80 240)(31 294 81 244)(32 293 82 243)(33 292 83 242)(34 291 84 241)(35 295 85 245)(36 274 86 224)(37 273 87 223)(38 272 88 222)(39 271 89 221)(40 275 90 225)(41 279 91 229)(42 278 92 228)(43 277 93 227)(44 276 94 226)(45 280 95 230)(46 309 96 259)(47 308 97 258)(48 307 98 257)(49 306 99 256)(50 310 100 260)(51 314 101 264)(52 313 102 263)(53 312 103 262)(54 311 104 261)(55 315 105 265)(56 319 106 269)(57 318 107 268)(58 317 108 267)(59 316 109 266)(60 320 110 270)(61 299 111 249)(62 298 112 248)(63 297 113 247)(64 296 114 246)(65 300 115 250)(66 304 116 254)(67 303 117 253)(68 302 118 252)(69 301 119 251)(70 305 120 255)(121 384 171 334)(122 383 172 333)(123 382 173 332)(124 381 174 331)(125 385 175 335)(126 389 176 339)(127 388 177 338)(128 387 178 337)(129 386 179 336)(130 390 180 340)(131 394 181 344)(132 393 182 343)(133 392 183 342)(134 391 184 341)(135 395 185 345)(136 374 186 324)(137 373 187 323)(138 372 188 322)(139 371 189 321)(140 375 190 325)(141 379 191 329)(142 378 192 328)(143 377 193 327)(144 376 194 326)(145 380 195 330)(161 399 206 349)(162 398 207 348)(163 397 208 347)(164 396 209 346)(165 400 210 350)

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

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

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

112 conjugacy classes

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

112 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 D5 Dic5 D10 C4×D5 kernel C4×C52⋊6C4 C2×C52⋊6C4 C10×C20 C52⋊6C4 C5×C20 C2×C20 C20 C2×C10 C10 # reps 1 2 1 8 4 12 24 12 48

Matrix representation of C4×C526C4 in GL5(𝔽41)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 19 7 0 0 0 29 28
,
 1 0 0 0 0 0 10 0 0 0 0 0 37 0 0 0 0 0 31 1 0 0 0 10 3
,
 32 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 31 27 0 0 0 10 10

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,19,29,0,0,0,7,28],[1,0,0,0,0,0,10,0,0,0,0,0,37,0,0,0,0,0,31,10,0,0,0,1,3],[32,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,31,10,0,0,0,27,10] >;

C4×C526C4 in GAP, Magma, Sage, TeX

C_4\times C_5^2\rtimes_6C_4
% in TeX

G:=Group("C4xC5^2:6C4");
// GroupNames label

G:=SmallGroup(400,99);
// by ID

G=gap.SmallGroup(400,99);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,1924,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^5=c^5=d^4=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