Copied to
clipboard

## G = C4⋊C4×C52order 400 = 24·52

### Direct product of C52 and C4⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4⋊C4×C52
 Chief series C1 — C2 — C22 — C2×C10 — C102 — C10×C20 — C4⋊C4×C52
 Lower central C1 — C2 — C4⋊C4×C52
 Upper central C1 — C102 — C4⋊C4×C52

Generators and relations for C4⋊C4×C52
G = < a,b,c,d | a5=b5=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 120 in 104 conjugacy classes, 88 normal (14 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C10, C4⋊C4, C20, C20, C2×C10, C52, C2×C20, C5×C10, C5×C4⋊C4, C5×C20, C5×C20, C102, C10×C20, C10×C20, C4⋊C4×C52
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C10, C4⋊C4, C20, C2×C10, C52, C2×C20, C5×D4, C5×Q8, C5×C10, C5×C4⋊C4, C5×C20, C102, C10×C20, D4×C52, Q8×C52, C4⋊C4×C52

Smallest permutation representation of C4⋊C4×C52
Regular action on 400 points
Generators in S400
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)(136 137 138 139 140)(141 142 143 144 145)(146 147 148 149 150)(151 152 153 154 155)(156 157 158 159 160)(161 162 163 164 165)(166 167 168 169 170)(171 172 173 174 175)(176 177 178 179 180)(181 182 183 184 185)(186 187 188 189 190)(191 192 193 194 195)(196 197 198 199 200)(201 202 203 204 205)(206 207 208 209 210)(211 212 213 214 215)(216 217 218 219 220)(221 222 223 224 225)(226 227 228 229 230)(231 232 233 234 235)(236 237 238 239 240)(241 242 243 244 245)(246 247 248 249 250)(251 252 253 254 255)(256 257 258 259 260)(261 262 263 264 265)(266 267 268 269 270)(271 272 273 274 275)(276 277 278 279 280)(281 282 283 284 285)(286 287 288 289 290)(291 292 293 294 295)(296 297 298 299 300)(301 302 303 304 305)(306 307 308 309 310)(311 312 313 314 315)(316 317 318 319 320)(321 322 323 324 325)(326 327 328 329 330)(331 332 333 334 335)(336 337 338 339 340)(341 342 343 344 345)(346 347 348 349 350)(351 352 353 354 355)(356 357 358 359 360)(361 362 363 364 365)(366 367 368 369 370)(371 372 373 374 375)(376 377 378 379 380)(381 382 383 384 385)(386 387 388 389 390)(391 392 393 394 395)(396 397 398 399 400)
(1 44 93 108 7)(2 45 94 109 8)(3 41 95 110 9)(4 42 91 106 10)(5 43 92 107 6)(11 88 51 157 34)(12 89 52 158 35)(13 90 53 159 31)(14 86 54 160 32)(15 87 55 156 33)(16 79 165 57 39)(17 80 161 58 40)(18 76 162 59 36)(19 77 163 60 37)(20 78 164 56 38)(21 382 369 346 328)(22 383 370 347 329)(23 384 366 348 330)(24 385 367 349 326)(25 381 368 350 327)(26 132 63 81 68)(27 133 64 82 69)(28 134 65 83 70)(29 135 61 84 66)(30 131 62 85 67)(46 312 389 371 353)(47 313 390 372 354)(48 314 386 373 355)(49 315 387 374 351)(50 311 388 375 352)(71 337 319 391 378)(72 338 320 392 379)(73 339 316 393 380)(74 340 317 394 376)(75 336 318 395 377)(96 342 324 301 398)(97 343 325 302 399)(98 344 321 303 400)(99 345 322 304 396)(100 341 323 305 397)(101 198 265 142 124)(102 199 261 143 125)(103 200 262 144 121)(104 196 263 145 122)(105 197 264 141 123)(111 188 175 152 234)(112 189 171 153 235)(113 190 172 154 231)(114 186 173 155 232)(115 187 174 151 233)(116 193 180 257 139)(117 194 176 258 140)(118 195 177 259 136)(119 191 178 260 137)(120 192 179 256 138)(126 208 185 167 149)(127 209 181 168 150)(128 210 182 169 146)(129 206 183 170 147)(130 207 184 166 148)(201 298 365 242 224)(202 299 361 243 225)(203 300 362 244 221)(204 296 363 245 222)(205 297 364 241 223)(211 288 275 252 334)(212 289 271 253 335)(213 290 272 254 331)(214 286 273 255 332)(215 287 274 251 333)(216 293 280 357 239)(217 294 276 358 240)(218 295 277 359 236)(219 291 278 360 237)(220 292 279 356 238)(226 308 285 267 249)(227 309 281 268 250)(228 310 282 269 246)(229 306 283 270 247)(230 307 284 266 248)
(1 265 55 173)(2 261 51 174)(3 262 52 175)(4 263 53 171)(5 264 54 172)(6 197 86 190)(7 198 87 186)(8 199 88 187)(9 200 89 188)(10 196 90 189)(11 115 109 102)(12 111 110 103)(13 112 106 104)(14 113 107 105)(15 114 108 101)(16 208 63 120)(17 209 64 116)(18 210 65 117)(19 206 61 118)(20 207 62 119)(21 203 317 211)(22 204 318 212)(23 205 319 213)(24 201 320 214)(25 202 316 215)(26 256 57 149)(27 257 58 150)(28 258 59 146)(29 259 60 147)(30 260 56 148)(31 235 91 122)(32 231 92 123)(33 232 93 124)(34 233 94 125)(35 234 95 121)(36 128 134 140)(37 129 135 136)(38 130 131 137)(39 126 132 138)(40 127 133 139)(41 144 158 152)(42 145 159 153)(43 141 160 154)(44 142 156 155)(45 143 157 151)(46 236 322 229)(47 237 323 230)(48 238 324 226)(49 239 325 227)(50 240 321 228)(66 177 163 170)(67 178 164 166)(68 179 165 167)(69 180 161 168)(70 176 162 169)(71 254 348 241)(72 255 349 242)(73 251 350 243)(74 252 346 244)(75 253 347 245)(76 182 83 194)(77 183 84 195)(78 184 85 191)(79 185 81 192)(80 181 82 193)(96 267 373 279)(97 268 374 280)(98 269 375 276)(99 270 371 277)(100 266 372 278)(216 302 309 315)(217 303 310 311)(218 304 306 312)(219 305 307 313)(220 301 308 314)(221 340 334 328)(222 336 335 329)(223 337 331 330)(224 338 332 326)(225 339 333 327)(246 352 358 344)(247 353 359 345)(248 354 360 341)(249 355 356 342)(250 351 357 343)(271 370 363 377)(272 366 364 378)(273 367 365 379)(274 368 361 380)(275 369 362 376)(281 387 293 399)(282 388 294 400)(283 389 295 396)(284 390 291 397)(285 386 292 398)(286 385 298 392)(287 381 299 393)(288 382 300 394)(289 383 296 395)(290 384 297 391)
(1 273 68 267)(2 274 69 268)(3 275 70 269)(4 271 66 270)(5 272 67 266)(6 290 85 284)(7 286 81 285)(8 287 82 281)(9 288 83 282)(10 289 84 283)(11 202 17 216)(12 203 18 217)(13 204 19 218)(14 205 20 219)(15 201 16 220)(21 210 311 111)(22 206 312 112)(23 207 313 113)(24 208 314 114)(25 209 315 115)(26 249 44 255)(27 250 45 251)(28 246 41 252)(29 247 42 253)(30 248 43 254)(31 222 37 236)(32 223 38 237)(33 224 39 238)(34 225 40 239)(35 221 36 240)(46 235 329 129)(47 231 330 130)(48 232 326 126)(49 233 327 127)(50 234 328 128)(51 361 161 280)(52 362 162 276)(53 363 163 277)(54 364 164 278)(55 365 165 279)(56 360 160 241)(57 356 156 242)(58 357 157 243)(59 358 158 244)(60 359 159 245)(61 306 106 212)(62 307 107 213)(63 308 108 214)(64 309 109 215)(65 310 110 211)(71 260 341 141)(72 256 342 142)(73 257 343 143)(74 258 344 144)(75 259 345 145)(76 294 89 300)(77 295 90 296)(78 291 86 297)(79 292 87 298)(80 293 88 299)(91 335 135 229)(92 331 131 230)(93 332 132 226)(94 333 133 227)(95 334 134 228)(96 265 379 179)(97 261 380 180)(98 262 376 176)(99 263 377 177)(100 264 378 178)(101 320 120 301)(102 316 116 302)(103 317 117 303)(104 318 118 304)(105 319 119 305)(121 340 140 321)(122 336 136 322)(123 337 137 323)(124 338 138 324)(125 339 139 325)(146 352 152 346)(147 353 153 347)(148 354 154 348)(149 355 155 349)(150 351 151 350)(166 372 172 366)(167 373 173 367)(168 374 174 368)(169 375 175 369)(170 371 171 370)(181 387 187 381)(182 388 188 382)(183 389 189 383)(184 390 190 384)(185 386 186 385)(191 397 197 391)(192 398 198 392)(193 399 199 393)(194 400 200 394)(195 396 196 395)

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

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

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

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 2 2 type + + + - image C1 C2 C4 C5 C10 C20 D4 Q8 C5×D4 C5×Q8 kernel C4⋊C4×C52 C10×C20 C5×C20 C5×C4⋊C4 C2×C20 C20 C5×C10 C5×C10 C10 C10 # reps 1 3 4 24 72 96 1 1 24 24

Matrix representation of C4⋊C4×C52 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 10 0 0 0 0 10
,
 1 0 0 0 0 18 0 0 0 0 37 0 0 0 0 37
,
 40 0 0 0 0 40 0 0 0 0 0 1 0 0 40 0
,
 9 0 0 0 0 1 0 0 0 0 9 17 0 0 17 32
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,18,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[9,0,0,0,0,1,0,0,0,0,9,17,0,0,17,32] >;

C4⋊C4×C52 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times C_5^2
% in TeX

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

G:=SmallGroup(400,110);
// by ID

G=gap.SmallGroup(400,110);
# by ID

G:=PCGroup([6,-2,-2,-5,-5,-2,-2,1200,1225,607]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^4=d^4=1,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

׿
×
𝔽