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## G = C4⋊C4×C25order 400 = 24·52

### Direct product of C25 and C4⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4⋊C4×C25
 Chief series C1 — C5 — C10 — C2×C10 — C2×C50 — C2×C100 — C4⋊C4×C25
 Lower central C1 — C2 — C4⋊C4×C25
 Upper central C1 — C2×C50 — C4⋊C4×C25

Generators and relations for C4⋊C4×C25
G = < a,b,c | a25=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C4⋊C4×C25
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)
(1 186 302 203)(2 187 303 204)(3 188 304 205)(4 189 305 206)(5 190 306 207)(6 191 307 208)(7 192 308 209)(8 193 309 210)(9 194 310 211)(10 195 311 212)(11 196 312 213)(12 197 313 214)(13 198 314 215)(14 199 315 216)(15 200 316 217)(16 176 317 218)(17 177 318 219)(18 178 319 220)(19 179 320 221)(20 180 321 222)(21 181 322 223)(22 182 323 224)(23 183 324 225)(24 184 325 201)(25 185 301 202)(26 355 248 129)(27 356 249 130)(28 357 250 131)(29 358 226 132)(30 359 227 133)(31 360 228 134)(32 361 229 135)(33 362 230 136)(34 363 231 137)(35 364 232 138)(36 365 233 139)(37 366 234 140)(38 367 235 141)(39 368 236 142)(40 369 237 143)(41 370 238 144)(42 371 239 145)(43 372 240 146)(44 373 241 147)(45 374 242 148)(46 375 243 149)(47 351 244 150)(48 352 245 126)(49 353 246 127)(50 354 247 128)(51 101 92 329)(52 102 93 330)(53 103 94 331)(54 104 95 332)(55 105 96 333)(56 106 97 334)(57 107 98 335)(58 108 99 336)(59 109 100 337)(60 110 76 338)(61 111 77 339)(62 112 78 340)(63 113 79 341)(64 114 80 342)(65 115 81 343)(66 116 82 344)(67 117 83 345)(68 118 84 346)(69 119 85 347)(70 120 86 348)(71 121 87 349)(72 122 88 350)(73 123 89 326)(74 124 90 327)(75 125 91 328)(151 263 284 397)(152 264 285 398)(153 265 286 399)(154 266 287 400)(155 267 288 376)(156 268 289 377)(157 269 290 378)(158 270 291 379)(159 271 292 380)(160 272 293 381)(161 273 294 382)(162 274 295 383)(163 275 296 384)(164 251 297 385)(165 252 298 386)(166 253 299 387)(167 254 300 388)(168 255 276 389)(169 256 277 390)(170 257 278 391)(171 258 279 392)(172 259 280 393)(173 260 281 394)(174 261 282 395)(175 262 283 396)
(1 173 349 134)(2 174 350 135)(3 175 326 136)(4 151 327 137)(5 152 328 138)(6 153 329 139)(7 154 330 140)(8 155 331 141)(9 156 332 142)(10 157 333 143)(11 158 334 144)(12 159 335 145)(13 160 336 146)(14 161 337 147)(15 162 338 148)(16 163 339 149)(17 164 340 150)(18 165 341 126)(19 166 342 127)(20 167 343 128)(21 168 344 129)(22 169 345 130)(23 170 346 131)(24 171 347 132)(25 172 348 133)(26 223 255 82)(27 224 256 83)(28 225 257 84)(29 201 258 85)(30 202 259 86)(31 203 260 87)(32 204 261 88)(33 205 262 89)(34 206 263 90)(35 207 264 91)(36 208 265 92)(37 209 266 93)(38 210 267 94)(39 211 268 95)(40 212 269 96)(41 213 270 97)(42 214 271 98)(43 215 272 99)(44 216 273 100)(45 217 274 76)(46 218 275 77)(47 219 251 78)(48 220 252 79)(49 221 253 80)(50 222 254 81)(51 233 191 399)(52 234 192 400)(53 235 193 376)(54 236 194 377)(55 237 195 378)(56 238 196 379)(57 239 197 380)(58 240 198 381)(59 241 199 382)(60 242 200 383)(61 243 176 384)(62 244 177 385)(63 245 178 386)(64 246 179 387)(65 247 180 388)(66 248 181 389)(67 249 182 390)(68 250 183 391)(69 226 184 392)(70 227 185 393)(71 228 186 394)(72 229 187 395)(73 230 188 396)(74 231 189 397)(75 232 190 398)(101 365 307 286)(102 366 308 287)(103 367 309 288)(104 368 310 289)(105 369 311 290)(106 370 312 291)(107 371 313 292)(108 372 314 293)(109 373 315 294)(110 374 316 295)(111 375 317 296)(112 351 318 297)(113 352 319 298)(114 353 320 299)(115 354 321 300)(116 355 322 276)(117 356 323 277)(118 357 324 278)(119 358 325 279)(120 359 301 280)(121 360 302 281)(122 361 303 282)(123 362 304 283)(124 363 305 284)(125 364 306 285)

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

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

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

250 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 5A 5B 5C 5D 10A ··· 10L 20A ··· 20X 25A ··· 25T 50A ··· 50BH 100A ··· 100DP order 1 2 2 2 4 ··· 4 5 5 5 5 10 ··· 10 20 ··· 20 25 ··· 25 50 ··· 50 100 ··· 100 size 1 1 1 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

250 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C4 C5 C10 C20 C25 C50 C100 D4 Q8 C5×D4 C5×Q8 D4×C25 Q8×C25 kernel C4⋊C4×C25 C2×C100 C100 C5×C4⋊C4 C2×C20 C20 C4⋊C4 C2×C4 C4 C50 C50 C10 C10 C2 C2 # reps 1 3 4 4 12 16 20 60 80 1 1 4 4 20 20

Matrix representation of C4⋊C4×C25 in GL3(𝔽101) generated by

 1 0 0 0 56 0 0 0 56
,
 1 0 0 0 0 100 0 1 0
,
 10 0 0 0 74 52 0 52 27
G:=sub<GL(3,GF(101))| [1,0,0,0,56,0,0,0,56],[1,0,0,0,0,1,0,100,0],[10,0,0,0,74,52,0,52,27] >;

C4⋊C4×C25 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times C_{25}
% in TeX

G:=Group("C4:C4xC25");
// GroupNames label

G:=SmallGroup(400,22);
// by ID

G=gap.SmallGroup(400,22);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,240,265,127,374]);
// Polycyclic

G:=Group<a,b,c|a^25=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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