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G = C252C16order 400 = 24·52

The semidirect product of C25 and C16 acting via C16/C8=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C252C16, C50.2C8, C40.5D5, C8.2D25, C200.2C2, C100.5C4, C20.7Dic5, C4.2Dic25, C5.(C52C16), C2.(C252C8), C10.2(C52C8), SmallGroup(400,1)

Series: Derived Chief Lower central Upper central

C1C25 — C252C16
C1C5C25C50C100C200 — C252C16
C25 — C252C16
C1C8

Generators and relations for C252C16
 G = < a,b | a25=b16=1, bab-1=a-1 >

25C16
5C52C16

Smallest permutation representation of C252C16
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)
(1 380 197 299 85 333 135 232 39 367 164 258 60 318 110 207)(2 379 198 298 86 332 136 231 40 366 165 257 61 317 111 206)(3 378 199 297 87 331 137 230 41 365 166 256 62 316 112 205)(4 377 200 296 88 330 138 229 42 364 167 255 63 315 113 204)(5 376 176 295 89 329 139 228 43 363 168 254 64 314 114 203)(6 400 177 294 90 328 140 227 44 362 169 253 65 313 115 202)(7 399 178 293 91 327 141 226 45 361 170 252 66 312 116 201)(8 398 179 292 92 326 142 250 46 360 171 251 67 311 117 225)(9 397 180 291 93 350 143 249 47 359 172 275 68 310 118 224)(10 396 181 290 94 349 144 248 48 358 173 274 69 309 119 223)(11 395 182 289 95 348 145 247 49 357 174 273 70 308 120 222)(12 394 183 288 96 347 146 246 50 356 175 272 71 307 121 221)(13 393 184 287 97 346 147 245 26 355 151 271 72 306 122 220)(14 392 185 286 98 345 148 244 27 354 152 270 73 305 123 219)(15 391 186 285 99 344 149 243 28 353 153 269 74 304 124 218)(16 390 187 284 100 343 150 242 29 352 154 268 75 303 125 217)(17 389 188 283 76 342 126 241 30 351 155 267 51 302 101 216)(18 388 189 282 77 341 127 240 31 375 156 266 52 301 102 215)(19 387 190 281 78 340 128 239 32 374 157 265 53 325 103 214)(20 386 191 280 79 339 129 238 33 373 158 264 54 324 104 213)(21 385 192 279 80 338 130 237 34 372 159 263 55 323 105 212)(22 384 193 278 81 337 131 236 35 371 160 262 56 322 106 211)(23 383 194 277 82 336 132 235 36 370 161 261 57 321 107 210)(24 382 195 276 83 335 133 234 37 369 162 260 58 320 108 209)(25 381 196 300 84 334 134 233 38 368 163 259 59 319 109 208)

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

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

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

112 conjugacy classes

class 1  2 4A4B5A5B8A8B8C8D10A10B16A···16H20A20B20C20D25A···25J40A···40H50A···50J100A···100T200A···200AN
order1244558888101016···162020202025···2540···4050···50100···100200···200
size11112211112225···2522222···22···22···22···22···2

112 irreducible representations

dim1111122222222
type+++-+-
imageC1C2C4C8C16D5Dic5C52C8D25C52C16Dic25C252C8C252C16
kernelC252C16C200C100C50C25C40C20C10C8C5C4C2C1
# reps11248224108102040

Matrix representation of C252C16 in GL2(𝔽401) generated by

2669
39260
,
276242
207125
G:=sub<GL(2,GF(401))| [266,392,9,60],[276,207,242,125] >;

C252C16 in GAP, Magma, Sage, TeX

C_{25}\rtimes_2C_{16}
% in TeX

G:=Group("C25:2C16");
// GroupNames label

G:=SmallGroup(400,1);
// by ID

G=gap.SmallGroup(400,1);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,12,31,50,4324,628,11525]);
// Polycyclic

G:=Group<a,b|a^25=b^16=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C252C16 in TeX

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