direct product, abelian, monomial, 2-elementary
Aliases: C4×C100, SmallGroup(400,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C100 |
| C1 — C4×C100 |
| C1 — C4×C100 |
Generators and relations for C4×C100
G = < a,b | a4=b100=1, ab=ba >
Smallest permutation representation of C4×C100
►Regular action on 400 points
Generators in S400
(1 124 231 394)(2 125 232 395)(3 126 233 396)(4 127 234 397)(5 128 235 398)(6 129 236 399)(7 130 237 400)(8 131 238 301)(9 132 239 302)(10 133 240 303)(11 134 241 304)(12 135 242 305)(13 136 243 306)(14 137 244 307)(15 138 245 308)(16 139 246 309)(17 140 247 310)(18 141 248 311)(19 142 249 312)(20 143 250 313)(21 144 251 314)(22 145 252 315)(23 146 253 316)(24 147 254 317)(25 148 255 318)(26 149 256 319)(27 150 257 320)(28 151 258 321)(29 152 259 322)(30 153 260 323)(31 154 261 324)(32 155 262 325)(33 156 263 326)(34 157 264 327)(35 158 265 328)(36 159 266 329)(37 160 267 330)(38 161 268 331)(39 162 269 332)(40 163 270 333)(41 164 271 334)(42 165 272 335)(43 166 273 336)(44 167 274 337)(45 168 275 338)(46 169 276 339)(47 170 277 340)(48 171 278 341)(49 172 279 342)(50 173 280 343)(51 174 281 344)(52 175 282 345)(53 176 283 346)(54 177 284 347)(55 178 285 348)(56 179 286 349)(57 180 287 350)(58 181 288 351)(59 182 289 352)(60 183 290 353)(61 184 291 354)(62 185 292 355)(63 186 293 356)(64 187 294 357)(65 188 295 358)(66 189 296 359)(67 190 297 360)(68 191 298 361)(69 192 299 362)(70 193 300 363)(71 194 201 364)(72 195 202 365)(73 196 203 366)(74 197 204 367)(75 198 205 368)(76 199 206 369)(77 200 207 370)(78 101 208 371)(79 102 209 372)(80 103 210 373)(81 104 211 374)(82 105 212 375)(83 106 213 376)(84 107 214 377)(85 108 215 378)(86 109 216 379)(87 110 217 380)(88 111 218 381)(89 112 219 382)(90 113 220 383)(91 114 221 384)(92 115 222 385)(93 116 223 386)(94 117 224 387)(95 118 225 388)(96 119 226 389)(97 120 227 390)(98 121 228 391)(99 122 229 392)(100 123 230 393)
(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)| (1,124,231,394)(2,125,232,395)(3,126,233,396)(4,127,234,397)(5,128,235,398)(6,129,236,399)(7,130,237,400)(8,131,238,301)(9,132,239,302)(10,133,240,303)(11,134,241,304)(12,135,242,305)(13,136,243,306)(14,137,244,307)(15,138,245,308)(16,139,246,309)(17,140,247,310)(18,141,248,311)(19,142,249,312)(20,143,250,313)(21,144,251,314)(22,145,252,315)(23,146,253,316)(24,147,254,317)(25,148,255,318)(26,149,256,319)(27,150,257,320)(28,151,258,321)(29,152,259,322)(30,153,260,323)(31,154,261,324)(32,155,262,325)(33,156,263,326)(34,157,264,327)(35,158,265,328)(36,159,266,329)(37,160,267,330)(38,161,268,331)(39,162,269,332)(40,163,270,333)(41,164,271,334)(42,165,272,335)(43,166,273,336)(44,167,274,337)(45,168,275,338)(46,169,276,339)(47,170,277,340)(48,171,278,341)(49,172,279,342)(50,173,280,343)(51,174,281,344)(52,175,282,345)(53,176,283,346)(54,177,284,347)(55,178,285,348)(56,179,286,349)(57,180,287,350)(58,181,288,351)(59,182,289,352)(60,183,290,353)(61,184,291,354)(62,185,292,355)(63,186,293,356)(64,187,294,357)(65,188,295,358)(66,189,296,359)(67,190,297,360)(68,191,298,361)(69,192,299,362)(70,193,300,363)(71,194,201,364)(72,195,202,365)(73,196,203,366)(74,197,204,367)(75,198,205,368)(76,199,206,369)(77,200,207,370)(78,101,208,371)(79,102,209,372)(80,103,210,373)(81,104,211,374)(82,105,212,375)(83,106,213,376)(84,107,214,377)(85,108,215,378)(86,109,216,379)(87,110,217,380)(88,111,218,381)(89,112,219,382)(90,113,220,383)(91,114,221,384)(92,115,222,385)(93,116,223,386)(94,117,224,387)(95,118,225,388)(96,119,226,389)(97,120,227,390)(98,121,228,391)(99,122,229,392)(100,123,230,393), (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( (1,124,231,394)(2,125,232,395)(3,126,233,396)(4,127,234,397)(5,128,235,398)(6,129,236,399)(7,130,237,400)(8,131,238,301)(9,132,239,302)(10,133,240,303)(11,134,241,304)(12,135,242,305)(13,136,243,306)(14,137,244,307)(15,138,245,308)(16,139,246,309)(17,140,247,310)(18,141,248,311)(19,142,249,312)(20,143,250,313)(21,144,251,314)(22,145,252,315)(23,146,253,316)(24,147,254,317)(25,148,255,318)(26,149,256,319)(27,150,257,320)(28,151,258,321)(29,152,259,322)(30,153,260,323)(31,154,261,324)(32,155,262,325)(33,156,263,326)(34,157,264,327)(35,158,265,328)(36,159,266,329)(37,160,267,330)(38,161,268,331)(39,162,269,332)(40,163,270,333)(41,164,271,334)(42,165,272,335)(43,166,273,336)(44,167,274,337)(45,168,275,338)(46,169,276,339)(47,170,277,340)(48,171,278,341)(49,172,279,342)(50,173,280,343)(51,174,281,344)(52,175,282,345)(53,176,283,346)(54,177,284,347)(55,178,285,348)(56,179,286,349)(57,180,287,350)(58,181,288,351)(59,182,289,352)(60,183,290,353)(61,184,291,354)(62,185,292,355)(63,186,293,356)(64,187,294,357)(65,188,295,358)(66,189,296,359)(67,190,297,360)(68,191,298,361)(69,192,299,362)(70,193,300,363)(71,194,201,364)(72,195,202,365)(73,196,203,366)(74,197,204,367)(75,198,205,368)(76,199,206,369)(77,200,207,370)(78,101,208,371)(79,102,209,372)(80,103,210,373)(81,104,211,374)(82,105,212,375)(83,106,213,376)(84,107,214,377)(85,108,215,378)(86,109,216,379)(87,110,217,380)(88,111,218,381)(89,112,219,382)(90,113,220,383)(91,114,221,384)(92,115,222,385)(93,116,223,386)(94,117,224,387)(95,118,225,388)(96,119,226,389)(97,120,227,390)(98,121,228,391)(99,122,229,392)(100,123,230,393), (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([[(1,124,231,394),(2,125,232,395),(3,126,233,396),(4,127,234,397),(5,128,235,398),(6,129,236,399),(7,130,237,400),(8,131,238,301),(9,132,239,302),(10,133,240,303),(11,134,241,304),(12,135,242,305),(13,136,243,306),(14,137,244,307),(15,138,245,308),(16,139,246,309),(17,140,247,310),(18,141,248,311),(19,142,249,312),(20,143,250,313),(21,144,251,314),(22,145,252,315),(23,146,253,316),(24,147,254,317),(25,148,255,318),(26,149,256,319),(27,150,257,320),(28,151,258,321),(29,152,259,322),(30,153,260,323),(31,154,261,324),(32,155,262,325),(33,156,263,326),(34,157,264,327),(35,158,265,328),(36,159,266,329),(37,160,267,330),(38,161,268,331),(39,162,269,332),(40,163,270,333),(41,164,271,334),(42,165,272,335),(43,166,273,336),(44,167,274,337),(45,168,275,338),(46,169,276,339),(47,170,277,340),(48,171,278,341),(49,172,279,342),(50,173,280,343),(51,174,281,344),(52,175,282,345),(53,176,283,346),(54,177,284,347),(55,178,285,348),(56,179,286,349),(57,180,287,350),(58,181,288,351),(59,182,289,352),(60,183,290,353),(61,184,291,354),(62,185,292,355),(63,186,293,356),(64,187,294,357),(65,188,295,358),(66,189,296,359),(67,190,297,360),(68,191,298,361),(69,192,299,362),(70,193,300,363),(71,194,201,364),(72,195,202,365),(73,196,203,366),(74,197,204,367),(75,198,205,368),(76,199,206,369),(77,200,207,370),(78,101,208,371),(79,102,209,372),(80,103,210,373),(81,104,211,374),(82,105,212,375),(83,106,213,376),(84,107,214,377),(85,108,215,378),(86,109,216,379),(87,110,217,380),(88,111,218,381),(89,112,219,382),(90,113,220,383),(91,114,221,384),(92,115,222,385),(93,116,223,386),(94,117,224,387),(95,118,225,388),(96,119,226,389),(97,120,227,390),(98,121,228,391),(99,122,229,392),(100,123,230,393)], [(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 | ··· | 4L | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 20A | ··· | 20AV | 25A | ··· | 25T | 50A | ··· | 50BH | 100A | ··· | 100IF |
| 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 | 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 |
| type | + | + | |||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | C25 | C50 | C100 |
| kernel | C4×C100 | C2×C100 | C100 | C4×C20 | C2×C20 | C20 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 12 | 4 | 12 | 48 | 20 | 60 | 240 |
Matrix representation of C4×C100 ►in GL2(𝔽101) generated by
| 1 | 0 |
| 0 | 91 |
| 29 | 0 |
| 0 | 73 |
G:=sub<GL(2,GF(101))| [1,0,0,91],[29,0,0,73] >;
C4×C100 in GAP, Magma, Sage, TeX
C_4\times C_{100} % in TeX
G:=Group("C4xC100"); // GroupNames label
G:=SmallGroup(400,20);
// by ID
G=gap.SmallGroup(400,20);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,247,374]);
// Polycyclic
G:=Group<a,b|a^4=b^100=1,a*b=b*a>;
// generators/relations
Export