direct product, abelian, monomial
Aliases: C10×C40, SmallGroup(400,111)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C10×C40 |
| C1 — C10×C40 |
| C1 — C10×C40 |
Generators and relations for C10×C40
G = < a,b | a10=b40=1, ab=ba >
Subgroups: 88, all normal (14 characteristic)
C1, C2, C2, C4, C22, C5, C8, C2×C4, C10, C2×C8, C20, C2×C10, C52, C40, C2×C20, C5×C10, C5×C10, C2×C40, C5×C20, C102, C5×C40, C10×C20, C10×C40
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, C10, C2×C8, C20, C2×C10, C52, C40, C2×C20, C5×C10, C2×C40, C5×C20, C102, C5×C40, C10×C20, C10×C40
►Regular action on 400 points
Generators in S400
(1 203 118 302 345 177 363 255 140 49)(2 204 119 303 346 178 364 256 141 50)(3 205 120 304 347 179 365 257 142 51)(4 206 81 305 348 180 366 258 143 52)(5 207 82 306 349 181 367 259 144 53)(6 208 83 307 350 182 368 260 145 54)(7 209 84 308 351 183 369 261 146 55)(8 210 85 309 352 184 370 262 147 56)(9 211 86 310 353 185 371 263 148 57)(10 212 87 311 354 186 372 264 149 58)(11 213 88 312 355 187 373 265 150 59)(12 214 89 313 356 188 374 266 151 60)(13 215 90 314 357 189 375 267 152 61)(14 216 91 315 358 190 376 268 153 62)(15 217 92 316 359 191 377 269 154 63)(16 218 93 317 360 192 378 270 155 64)(17 219 94 318 321 193 379 271 156 65)(18 220 95 319 322 194 380 272 157 66)(19 221 96 320 323 195 381 273 158 67)(20 222 97 281 324 196 382 274 159 68)(21 223 98 282 325 197 383 275 160 69)(22 224 99 283 326 198 384 276 121 70)(23 225 100 284 327 199 385 277 122 71)(24 226 101 285 328 200 386 278 123 72)(25 227 102 286 329 161 387 279 124 73)(26 228 103 287 330 162 388 280 125 74)(27 229 104 288 331 163 389 241 126 75)(28 230 105 289 332 164 390 242 127 76)(29 231 106 290 333 165 391 243 128 77)(30 232 107 291 334 166 392 244 129 78)(31 233 108 292 335 167 393 245 130 79)(32 234 109 293 336 168 394 246 131 80)(33 235 110 294 337 169 395 247 132 41)(34 236 111 295 338 170 396 248 133 42)(35 237 112 296 339 171 397 249 134 43)(36 238 113 297 340 172 398 250 135 44)(37 239 114 298 341 173 399 251 136 45)(38 240 115 299 342 174 400 252 137 46)(39 201 116 300 343 175 361 253 138 47)(40 202 117 301 344 176 362 254 139 48)
(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)
G:=sub<Sym(400)| (1,203,118,302,345,177,363,255,140,49)(2,204,119,303,346,178,364,256,141,50)(3,205,120,304,347,179,365,257,142,51)(4,206,81,305,348,180,366,258,143,52)(5,207,82,306,349,181,367,259,144,53)(6,208,83,307,350,182,368,260,145,54)(7,209,84,308,351,183,369,261,146,55)(8,210,85,309,352,184,370,262,147,56)(9,211,86,310,353,185,371,263,148,57)(10,212,87,311,354,186,372,264,149,58)(11,213,88,312,355,187,373,265,150,59)(12,214,89,313,356,188,374,266,151,60)(13,215,90,314,357,189,375,267,152,61)(14,216,91,315,358,190,376,268,153,62)(15,217,92,316,359,191,377,269,154,63)(16,218,93,317,360,192,378,270,155,64)(17,219,94,318,321,193,379,271,156,65)(18,220,95,319,322,194,380,272,157,66)(19,221,96,320,323,195,381,273,158,67)(20,222,97,281,324,196,382,274,159,68)(21,223,98,282,325,197,383,275,160,69)(22,224,99,283,326,198,384,276,121,70)(23,225,100,284,327,199,385,277,122,71)(24,226,101,285,328,200,386,278,123,72)(25,227,102,286,329,161,387,279,124,73)(26,228,103,287,330,162,388,280,125,74)(27,229,104,288,331,163,389,241,126,75)(28,230,105,289,332,164,390,242,127,76)(29,231,106,290,333,165,391,243,128,77)(30,232,107,291,334,166,392,244,129,78)(31,233,108,292,335,167,393,245,130,79)(32,234,109,293,336,168,394,246,131,80)(33,235,110,294,337,169,395,247,132,41)(34,236,111,295,338,170,396,248,133,42)(35,237,112,296,339,171,397,249,134,43)(36,238,113,297,340,172,398,250,135,44)(37,239,114,298,341,173,399,251,136,45)(38,240,115,299,342,174,400,252,137,46)(39,201,116,300,343,175,361,253,138,47)(40,202,117,301,344,176,362,254,139,48), (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)>;
G:=Group( (1,203,118,302,345,177,363,255,140,49)(2,204,119,303,346,178,364,256,141,50)(3,205,120,304,347,179,365,257,142,51)(4,206,81,305,348,180,366,258,143,52)(5,207,82,306,349,181,367,259,144,53)(6,208,83,307,350,182,368,260,145,54)(7,209,84,308,351,183,369,261,146,55)(8,210,85,309,352,184,370,262,147,56)(9,211,86,310,353,185,371,263,148,57)(10,212,87,311,354,186,372,264,149,58)(11,213,88,312,355,187,373,265,150,59)(12,214,89,313,356,188,374,266,151,60)(13,215,90,314,357,189,375,267,152,61)(14,216,91,315,358,190,376,268,153,62)(15,217,92,316,359,191,377,269,154,63)(16,218,93,317,360,192,378,270,155,64)(17,219,94,318,321,193,379,271,156,65)(18,220,95,319,322,194,380,272,157,66)(19,221,96,320,323,195,381,273,158,67)(20,222,97,281,324,196,382,274,159,68)(21,223,98,282,325,197,383,275,160,69)(22,224,99,283,326,198,384,276,121,70)(23,225,100,284,327,199,385,277,122,71)(24,226,101,285,328,200,386,278,123,72)(25,227,102,286,329,161,387,279,124,73)(26,228,103,287,330,162,388,280,125,74)(27,229,104,288,331,163,389,241,126,75)(28,230,105,289,332,164,390,242,127,76)(29,231,106,290,333,165,391,243,128,77)(30,232,107,291,334,166,392,244,129,78)(31,233,108,292,335,167,393,245,130,79)(32,234,109,293,336,168,394,246,131,80)(33,235,110,294,337,169,395,247,132,41)(34,236,111,295,338,170,396,248,133,42)(35,237,112,296,339,171,397,249,134,43)(36,238,113,297,340,172,398,250,135,44)(37,239,114,298,341,173,399,251,136,45)(38,240,115,299,342,174,400,252,137,46)(39,201,116,300,343,175,361,253,138,47)(40,202,117,301,344,176,362,254,139,48), (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) );
G=PermutationGroup([[(1,203,118,302,345,177,363,255,140,49),(2,204,119,303,346,178,364,256,141,50),(3,205,120,304,347,179,365,257,142,51),(4,206,81,305,348,180,366,258,143,52),(5,207,82,306,349,181,367,259,144,53),(6,208,83,307,350,182,368,260,145,54),(7,209,84,308,351,183,369,261,146,55),(8,210,85,309,352,184,370,262,147,56),(9,211,86,310,353,185,371,263,148,57),(10,212,87,311,354,186,372,264,149,58),(11,213,88,312,355,187,373,265,150,59),(12,214,89,313,356,188,374,266,151,60),(13,215,90,314,357,189,375,267,152,61),(14,216,91,315,358,190,376,268,153,62),(15,217,92,316,359,191,377,269,154,63),(16,218,93,317,360,192,378,270,155,64),(17,219,94,318,321,193,379,271,156,65),(18,220,95,319,322,194,380,272,157,66),(19,221,96,320,323,195,381,273,158,67),(20,222,97,281,324,196,382,274,159,68),(21,223,98,282,325,197,383,275,160,69),(22,224,99,283,326,198,384,276,121,70),(23,225,100,284,327,199,385,277,122,71),(24,226,101,285,328,200,386,278,123,72),(25,227,102,286,329,161,387,279,124,73),(26,228,103,287,330,162,388,280,125,74),(27,229,104,288,331,163,389,241,126,75),(28,230,105,289,332,164,390,242,127,76),(29,231,106,290,333,165,391,243,128,77),(30,232,107,291,334,166,392,244,129,78),(31,233,108,292,335,167,393,245,130,79),(32,234,109,293,336,168,394,246,131,80),(33,235,110,294,337,169,395,247,132,41),(34,236,111,295,338,170,396,248,133,42),(35,237,112,296,339,171,397,249,134,43),(36,238,113,297,340,172,398,250,135,44),(37,239,114,298,341,173,399,251,136,45),(38,240,115,299,342,174,400,252,137,46),(39,201,116,300,343,175,361,253,138,47),(40,202,117,301,344,176,362,254,139,48)], [(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)]])
400 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | ··· | 5X | 8A | ··· | 8H | 10A | ··· | 10BT | 20A | ··· | 20CR | 40A | ··· | 40GJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | ··· | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
400 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C20 | C20 | C40 |
| kernel | C10×C40 | C5×C40 | C10×C20 | C5×C20 | C102 | C2×C40 | C5×C10 | C40 | C2×C20 | C20 | C2×C10 | C10 |
| # reps | 1 | 2 | 1 | 2 | 2 | 24 | 8 | 48 | 24 | 48 | 48 | 192 |
Matrix representation of C10×C40 ►in GL2(𝔽41) generated by
| 31 | 0 |
| 0 | 10 |
| 5 | 0 |
| 0 | 34 |
G:=sub<GL(2,GF(41))| [31,0,0,10],[5,0,0,34] >;
C10×C40 in GAP, Magma, Sage, TeX
C_{10}\times C_{40} % in TeX
G:=Group("C10xC40"); // GroupNames label
G:=SmallGroup(400,111);
// by ID
G=gap.SmallGroup(400,111);
# by ID
G:=PCGroup([6,-2,-2,-5,-5,-2,-2,600,88]);
// Polycyclic
G:=Group<a,b|a^10=b^40=1,a*b=b*a>;
// generators/relations