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G = C202order 400 = 24·52

Abelian group of type [20,20]

direct product, abelian, monomial

Aliases: C202, SmallGroup(400,108)

Series: Derived Chief Lower central Upper central

C1 — C202
C1C2C22C2×C10C102C10×C20 — C202
C1 — C202
C1 — C202

Generators and relations for C202
 G = < a,b | a20=b20=1, ab=ba >

Subgroups: 120, all normal (6 characteristic)
C1, C2 [×3], C4 [×6], C22, C5 [×6], C2×C4 [×3], C10 [×18], C42, C20 [×36], C2×C10 [×6], C52, C2×C20 [×18], C5×C10 [×3], C4×C20 [×6], C5×C20 [×6], C102, C10×C20 [×3], C202
Quotients: C1, C2 [×3], C4 [×6], C22, C5 [×6], C2×C4 [×3], C10 [×18], C42, C20 [×36], C2×C10 [×6], C52, C2×C20 [×18], C5×C10 [×3], C4×C20 [×6], C5×C20 [×6], C102, C10×C20 [×3], C202

Smallest permutation representation of C202
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 87 327 150 130 195 370 218 46 173 250 271 233 392 307 32 344 63 281 106)(2 88 328 151 131 196 371 219 47 174 251 272 234 393 308 33 345 64 282 107)(3 89 329 152 132 197 372 220 48 175 252 273 235 394 309 34 346 65 283 108)(4 90 330 153 133 198 373 201 49 176 253 274 236 395 310 35 347 66 284 109)(5 91 331 154 134 199 374 202 50 177 254 275 237 396 311 36 348 67 285 110)(6 92 332 155 135 200 375 203 51 178 255 276 238 397 312 37 349 68 286 111)(7 93 333 156 136 181 376 204 52 179 256 277 239 398 313 38 350 69 287 112)(8 94 334 157 137 182 377 205 53 180 257 278 240 399 314 39 351 70 288 113)(9 95 335 158 138 183 378 206 54 161 258 279 221 400 315 40 352 71 289 114)(10 96 336 159 139 184 379 207 55 162 259 280 222 381 316 21 353 72 290 115)(11 97 337 160 140 185 380 208 56 163 260 261 223 382 317 22 354 73 291 116)(12 98 338 141 121 186 361 209 57 164 241 262 224 383 318 23 355 74 292 117)(13 99 339 142 122 187 362 210 58 165 242 263 225 384 319 24 356 75 293 118)(14 100 340 143 123 188 363 211 59 166 243 264 226 385 320 25 357 76 294 119)(15 81 321 144 124 189 364 212 60 167 244 265 227 386 301 26 358 77 295 120)(16 82 322 145 125 190 365 213 41 168 245 266 228 387 302 27 359 78 296 101)(17 83 323 146 126 191 366 214 42 169 246 267 229 388 303 28 360 79 297 102)(18 84 324 147 127 192 367 215 43 170 247 268 230 389 304 29 341 80 298 103)(19 85 325 148 128 193 368 216 44 171 248 269 231 390 305 30 342 61 299 104)(20 86 326 149 129 194 369 217 45 172 249 270 232 391 306 31 343 62 300 105)

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

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

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

400 conjugacy classes

class 1 2A2B2C4A···4L5A···5X10A···10BT20A···20KB
order12224···45···510···1020···20
size11111···11···11···11···1

400 irreducible representations

dim111111
type++
imageC1C2C4C5C10C20
kernelC202C10×C20C5×C20C4×C20C2×C20C20
# reps13122472288

Matrix representation of C202 in GL3(𝔽41) generated by

400
0180
0039
,
1800
050
005
G:=sub<GL(3,GF(41))| [4,0,0,0,18,0,0,0,39],[18,0,0,0,5,0,0,0,5] >;

C202 in GAP, Magma, Sage, TeX

C_{20}^2
% in TeX

G:=Group("C20^2");
// GroupNames label

G:=SmallGroup(400,108);
// by ID

G=gap.SmallGroup(400,108);
# by ID

G:=PCGroup([6,-2,-2,-5,-5,-2,-2,600,1207]);
// Polycyclic

G:=Group<a,b|a^20=b^20=1,a*b=b*a>;
// generators/relations

׿
×
𝔽