direct product, abelian, monomial
Aliases: C202, SmallGroup(400,108)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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, C4, C22, C5, C2×C4, C10, C42, C20, C2×C10, C52, C2×C20, C5×C10, C4×C20, C5×C20, C102, C10×C20, C202
Quotients: C1, C2, C4, C22, C5, C2×C4, C10, C42, C20, C2×C10, C52, C2×C20, C5×C10, C4×C20, C5×C20, C102, C10×C20, 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)
(1 342 304 226 244 206 173 190 393 334 149 271 292 23 375 57 103 91 123 65)(2 343 305 227 245 207 174 191 394 335 150 272 293 24 376 58 104 92 124 66)(3 344 306 228 246 208 175 192 395 336 151 273 294 25 377 59 105 93 125 67)(4 345 307 229 247 209 176 193 396 337 152 274 295 26 378 60 106 94 126 68)(5 346 308 230 248 210 177 194 397 338 153 275 296 27 379 41 107 95 127 69)(6 347 309 231 249 211 178 195 398 339 154 276 297 28 380 42 108 96 128 70)(7 348 310 232 250 212 179 196 399 340 155 277 298 29 361 43 109 97 129 71)(8 349 311 233 251 213 180 197 400 321 156 278 299 30 362 44 110 98 130 72)(9 350 312 234 252 214 161 198 381 322 157 279 300 31 363 45 111 99 131 73)(10 351 313 235 253 215 162 199 382 323 158 280 281 32 364 46 112 100 132 74)(11 352 314 236 254 216 163 200 383 324 159 261 282 33 365 47 113 81 133 75)(12 353 315 237 255 217 164 181 384 325 160 262 283 34 366 48 114 82 134 76)(13 354 316 238 256 218 165 182 385 326 141 263 284 35 367 49 115 83 135 77)(14 355 317 239 257 219 166 183 386 327 142 264 285 36 368 50 116 84 136 78)(15 356 318 240 258 220 167 184 387 328 143 265 286 37 369 51 117 85 137 79)(16 357 319 221 259 201 168 185 388 329 144 266 287 38 370 52 118 86 138 80)(17 358 320 222 260 202 169 186 389 330 145 267 288 39 371 53 119 87 139 61)(18 359 301 223 241 203 170 187 390 331 146 268 289 40 372 54 120 88 140 62)(19 360 302 224 242 204 171 188 391 332 147 269 290 21 373 55 101 89 121 63)(20 341 303 225 243 205 172 189 392 333 148 270 291 22 374 56 102 90 122 64)
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), (1,342,304,226,244,206,173,190,393,334,149,271,292,23,375,57,103,91,123,65)(2,343,305,227,245,207,174,191,394,335,150,272,293,24,376,58,104,92,124,66)(3,344,306,228,246,208,175,192,395,336,151,273,294,25,377,59,105,93,125,67)(4,345,307,229,247,209,176,193,396,337,152,274,295,26,378,60,106,94,126,68)(5,346,308,230,248,210,177,194,397,338,153,275,296,27,379,41,107,95,127,69)(6,347,309,231,249,211,178,195,398,339,154,276,297,28,380,42,108,96,128,70)(7,348,310,232,250,212,179,196,399,340,155,277,298,29,361,43,109,97,129,71)(8,349,311,233,251,213,180,197,400,321,156,278,299,30,362,44,110,98,130,72)(9,350,312,234,252,214,161,198,381,322,157,279,300,31,363,45,111,99,131,73)(10,351,313,235,253,215,162,199,382,323,158,280,281,32,364,46,112,100,132,74)(11,352,314,236,254,216,163,200,383,324,159,261,282,33,365,47,113,81,133,75)(12,353,315,237,255,217,164,181,384,325,160,262,283,34,366,48,114,82,134,76)(13,354,316,238,256,218,165,182,385,326,141,263,284,35,367,49,115,83,135,77)(14,355,317,239,257,219,166,183,386,327,142,264,285,36,368,50,116,84,136,78)(15,356,318,240,258,220,167,184,387,328,143,265,286,37,369,51,117,85,137,79)(16,357,319,221,259,201,168,185,388,329,144,266,287,38,370,52,118,86,138,80)(17,358,320,222,260,202,169,186,389,330,145,267,288,39,371,53,119,87,139,61)(18,359,301,223,241,203,170,187,390,331,146,268,289,40,372,54,120,88,140,62)(19,360,302,224,242,204,171,188,391,332,147,269,290,21,373,55,101,89,121,63)(20,341,303,225,243,205,172,189,392,333,148,270,291,22,374,56,102,90,122,64)>;
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), (1,342,304,226,244,206,173,190,393,334,149,271,292,23,375,57,103,91,123,65)(2,343,305,227,245,207,174,191,394,335,150,272,293,24,376,58,104,92,124,66)(3,344,306,228,246,208,175,192,395,336,151,273,294,25,377,59,105,93,125,67)(4,345,307,229,247,209,176,193,396,337,152,274,295,26,378,60,106,94,126,68)(5,346,308,230,248,210,177,194,397,338,153,275,296,27,379,41,107,95,127,69)(6,347,309,231,249,211,178,195,398,339,154,276,297,28,380,42,108,96,128,70)(7,348,310,232,250,212,179,196,399,340,155,277,298,29,361,43,109,97,129,71)(8,349,311,233,251,213,180,197,400,321,156,278,299,30,362,44,110,98,130,72)(9,350,312,234,252,214,161,198,381,322,157,279,300,31,363,45,111,99,131,73)(10,351,313,235,253,215,162,199,382,323,158,280,281,32,364,46,112,100,132,74)(11,352,314,236,254,216,163,200,383,324,159,261,282,33,365,47,113,81,133,75)(12,353,315,237,255,217,164,181,384,325,160,262,283,34,366,48,114,82,134,76)(13,354,316,238,256,218,165,182,385,326,141,263,284,35,367,49,115,83,135,77)(14,355,317,239,257,219,166,183,386,327,142,264,285,36,368,50,116,84,136,78)(15,356,318,240,258,220,167,184,387,328,143,265,286,37,369,51,117,85,137,79)(16,357,319,221,259,201,168,185,388,329,144,266,287,38,370,52,118,86,138,80)(17,358,320,222,260,202,169,186,389,330,145,267,288,39,371,53,119,87,139,61)(18,359,301,223,241,203,170,187,390,331,146,268,289,40,372,54,120,88,140,62)(19,360,302,224,242,204,171,188,391,332,147,269,290,21,373,55,101,89,121,63)(20,341,303,225,243,205,172,189,392,333,148,270,291,22,374,56,102,90,122,64) );
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)], [(1,342,304,226,244,206,173,190,393,334,149,271,292,23,375,57,103,91,123,65),(2,343,305,227,245,207,174,191,394,335,150,272,293,24,376,58,104,92,124,66),(3,344,306,228,246,208,175,192,395,336,151,273,294,25,377,59,105,93,125,67),(4,345,307,229,247,209,176,193,396,337,152,274,295,26,378,60,106,94,126,68),(5,346,308,230,248,210,177,194,397,338,153,275,296,27,379,41,107,95,127,69),(6,347,309,231,249,211,178,195,398,339,154,276,297,28,380,42,108,96,128,70),(7,348,310,232,250,212,179,196,399,340,155,277,298,29,361,43,109,97,129,71),(8,349,311,233,251,213,180,197,400,321,156,278,299,30,362,44,110,98,130,72),(9,350,312,234,252,214,161,198,381,322,157,279,300,31,363,45,111,99,131,73),(10,351,313,235,253,215,162,199,382,323,158,280,281,32,364,46,112,100,132,74),(11,352,314,236,254,216,163,200,383,324,159,261,282,33,365,47,113,81,133,75),(12,353,315,237,255,217,164,181,384,325,160,262,283,34,366,48,114,82,134,76),(13,354,316,238,256,218,165,182,385,326,141,263,284,35,367,49,115,83,135,77),(14,355,317,239,257,219,166,183,386,327,142,264,285,36,368,50,116,84,136,78),(15,356,318,240,258,220,167,184,387,328,143,265,286,37,369,51,117,85,137,79),(16,357,319,221,259,201,168,185,388,329,144,266,287,38,370,52,118,86,138,80),(17,358,320,222,260,202,169,186,389,330,145,267,288,39,371,53,119,87,139,61),(18,359,301,223,241,203,170,187,390,331,146,268,289,40,372,54,120,88,140,62),(19,360,302,224,242,204,171,188,391,332,147,269,290,21,373,55,101,89,121,63),(20,341,303,225,243,205,172,189,392,333,148,270,291,22,374,56,102,90,122,64)]])
400 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 5A | ··· | 5X | 10A | ··· | 10BT | 20A | ··· | 20KB |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
400 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C5 | C10 | C20 |
| kernel | C202 | C10×C20 | C5×C20 | C4×C20 | C2×C20 | C20 |
| # reps | 1 | 3 | 12 | 24 | 72 | 288 |
Matrix representation of C202 ►in GL3(𝔽41) generated by
| 4 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 39 |
| 18 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
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