direct product, abelian, monomial, 5-elementary
Aliases: C5×C75, SmallGroup(375,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C75 |
| C1 — C5×C75 |
| C1 — C5×C75 |
Generators and relations for C5×C75
G = < a,b | a5=b75=1, ab=ba >
Smallest permutation representation of C5×C75
►Regular action on 375 points
Generators in S375
(1 261 111 160 307)(2 262 112 161 308)(3 263 113 162 309)(4 264 114 163 310)(5 265 115 164 311)(6 266 116 165 312)(7 267 117 166 313)(8 268 118 167 314)(9 269 119 168 315)(10 270 120 169 316)(11 271 121 170 317)(12 272 122 171 318)(13 273 123 172 319)(14 274 124 173 320)(15 275 125 174 321)(16 276 126 175 322)(17 277 127 176 323)(18 278 128 177 324)(19 279 129 178 325)(20 280 130 179 326)(21 281 131 180 327)(22 282 132 181 328)(23 283 133 182 329)(24 284 134 183 330)(25 285 135 184 331)(26 286 136 185 332)(27 287 137 186 333)(28 288 138 187 334)(29 289 139 188 335)(30 290 140 189 336)(31 291 141 190 337)(32 292 142 191 338)(33 293 143 192 339)(34 294 144 193 340)(35 295 145 194 341)(36 296 146 195 342)(37 297 147 196 343)(38 298 148 197 344)(39 299 149 198 345)(40 300 150 199 346)(41 226 76 200 347)(42 227 77 201 348)(43 228 78 202 349)(44 229 79 203 350)(45 230 80 204 351)(46 231 81 205 352)(47 232 82 206 353)(48 233 83 207 354)(49 234 84 208 355)(50 235 85 209 356)(51 236 86 210 357)(52 237 87 211 358)(53 238 88 212 359)(54 239 89 213 360)(55 240 90 214 361)(56 241 91 215 362)(57 242 92 216 363)(58 243 93 217 364)(59 244 94 218 365)(60 245 95 219 366)(61 246 96 220 367)(62 247 97 221 368)(63 248 98 222 369)(64 249 99 223 370)(65 250 100 224 371)(66 251 101 225 372)(67 252 102 151 373)(68 253 103 152 374)(69 254 104 153 375)(70 255 105 154 301)(71 256 106 155 302)(72 257 107 156 303)(73 258 108 157 304)(74 259 109 158 305)(75 260 110 159 306)
(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)
G:=sub<Sym(375)| (1,261,111,160,307)(2,262,112,161,308)(3,263,113,162,309)(4,264,114,163,310)(5,265,115,164,311)(6,266,116,165,312)(7,267,117,166,313)(8,268,118,167,314)(9,269,119,168,315)(10,270,120,169,316)(11,271,121,170,317)(12,272,122,171,318)(13,273,123,172,319)(14,274,124,173,320)(15,275,125,174,321)(16,276,126,175,322)(17,277,127,176,323)(18,278,128,177,324)(19,279,129,178,325)(20,280,130,179,326)(21,281,131,180,327)(22,282,132,181,328)(23,283,133,182,329)(24,284,134,183,330)(25,285,135,184,331)(26,286,136,185,332)(27,287,137,186,333)(28,288,138,187,334)(29,289,139,188,335)(30,290,140,189,336)(31,291,141,190,337)(32,292,142,191,338)(33,293,143,192,339)(34,294,144,193,340)(35,295,145,194,341)(36,296,146,195,342)(37,297,147,196,343)(38,298,148,197,344)(39,299,149,198,345)(40,300,150,199,346)(41,226,76,200,347)(42,227,77,201,348)(43,228,78,202,349)(44,229,79,203,350)(45,230,80,204,351)(46,231,81,205,352)(47,232,82,206,353)(48,233,83,207,354)(49,234,84,208,355)(50,235,85,209,356)(51,236,86,210,357)(52,237,87,211,358)(53,238,88,212,359)(54,239,89,213,360)(55,240,90,214,361)(56,241,91,215,362)(57,242,92,216,363)(58,243,93,217,364)(59,244,94,218,365)(60,245,95,219,366)(61,246,96,220,367)(62,247,97,221,368)(63,248,98,222,369)(64,249,99,223,370)(65,250,100,224,371)(66,251,101,225,372)(67,252,102,151,373)(68,253,103,152,374)(69,254,104,153,375)(70,255,105,154,301)(71,256,106,155,302)(72,257,107,156,303)(73,258,108,157,304)(74,259,109,158,305)(75,260,110,159,306), (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)>;
G:=Group( (1,261,111,160,307)(2,262,112,161,308)(3,263,113,162,309)(4,264,114,163,310)(5,265,115,164,311)(6,266,116,165,312)(7,267,117,166,313)(8,268,118,167,314)(9,269,119,168,315)(10,270,120,169,316)(11,271,121,170,317)(12,272,122,171,318)(13,273,123,172,319)(14,274,124,173,320)(15,275,125,174,321)(16,276,126,175,322)(17,277,127,176,323)(18,278,128,177,324)(19,279,129,178,325)(20,280,130,179,326)(21,281,131,180,327)(22,282,132,181,328)(23,283,133,182,329)(24,284,134,183,330)(25,285,135,184,331)(26,286,136,185,332)(27,287,137,186,333)(28,288,138,187,334)(29,289,139,188,335)(30,290,140,189,336)(31,291,141,190,337)(32,292,142,191,338)(33,293,143,192,339)(34,294,144,193,340)(35,295,145,194,341)(36,296,146,195,342)(37,297,147,196,343)(38,298,148,197,344)(39,299,149,198,345)(40,300,150,199,346)(41,226,76,200,347)(42,227,77,201,348)(43,228,78,202,349)(44,229,79,203,350)(45,230,80,204,351)(46,231,81,205,352)(47,232,82,206,353)(48,233,83,207,354)(49,234,84,208,355)(50,235,85,209,356)(51,236,86,210,357)(52,237,87,211,358)(53,238,88,212,359)(54,239,89,213,360)(55,240,90,214,361)(56,241,91,215,362)(57,242,92,216,363)(58,243,93,217,364)(59,244,94,218,365)(60,245,95,219,366)(61,246,96,220,367)(62,247,97,221,368)(63,248,98,222,369)(64,249,99,223,370)(65,250,100,224,371)(66,251,101,225,372)(67,252,102,151,373)(68,253,103,152,374)(69,254,104,153,375)(70,255,105,154,301)(71,256,106,155,302)(72,257,107,156,303)(73,258,108,157,304)(74,259,109,158,305)(75,260,110,159,306), (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) );
G=PermutationGroup([[(1,261,111,160,307),(2,262,112,161,308),(3,263,113,162,309),(4,264,114,163,310),(5,265,115,164,311),(6,266,116,165,312),(7,267,117,166,313),(8,268,118,167,314),(9,269,119,168,315),(10,270,120,169,316),(11,271,121,170,317),(12,272,122,171,318),(13,273,123,172,319),(14,274,124,173,320),(15,275,125,174,321),(16,276,126,175,322),(17,277,127,176,323),(18,278,128,177,324),(19,279,129,178,325),(20,280,130,179,326),(21,281,131,180,327),(22,282,132,181,328),(23,283,133,182,329),(24,284,134,183,330),(25,285,135,184,331),(26,286,136,185,332),(27,287,137,186,333),(28,288,138,187,334),(29,289,139,188,335),(30,290,140,189,336),(31,291,141,190,337),(32,292,142,191,338),(33,293,143,192,339),(34,294,144,193,340),(35,295,145,194,341),(36,296,146,195,342),(37,297,147,196,343),(38,298,148,197,344),(39,299,149,198,345),(40,300,150,199,346),(41,226,76,200,347),(42,227,77,201,348),(43,228,78,202,349),(44,229,79,203,350),(45,230,80,204,351),(46,231,81,205,352),(47,232,82,206,353),(48,233,83,207,354),(49,234,84,208,355),(50,235,85,209,356),(51,236,86,210,357),(52,237,87,211,358),(53,238,88,212,359),(54,239,89,213,360),(55,240,90,214,361),(56,241,91,215,362),(57,242,92,216,363),(58,243,93,217,364),(59,244,94,218,365),(60,245,95,219,366),(61,246,96,220,367),(62,247,97,221,368),(63,248,98,222,369),(64,249,99,223,370),(65,250,100,224,371),(66,251,101,225,372),(67,252,102,151,373),(68,253,103,152,374),(69,254,104,153,375),(70,255,105,154,301),(71,256,106,155,302),(72,257,107,156,303),(73,258,108,157,304),(74,259,109,158,305),(75,260,110,159,306)], [(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)]])
375 conjugacy classes
| class | 1 | 3A | 3B | 5A | ··· | 5X | 15A | ··· | 15AV | 25A | ··· | 25CV | 75A | ··· | 75GR |
| order | 1 | 3 | 3 | 5 | ··· | 5 | 15 | ··· | 15 | 25 | ··· | 25 | 75 | ··· | 75 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
375 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||||
| image | C1 | C3 | C5 | C5 | C15 | C15 | C25 | C75 |
| kernel | C5×C75 | C5×C25 | C75 | C5×C15 | C25 | C52 | C15 | C5 |
| # reps | 1 | 2 | 20 | 4 | 40 | 8 | 100 | 200 |
Matrix representation of C5×C75 ►in GL2(𝔽151) generated by
| 59 | 0 |
| 0 | 1 |
| 21 | 0 |
| 0 | 137 |
G:=sub<GL(2,GF(151))| [59,0,0,1],[21,0,0,137] >;
C5×C75 in GAP, Magma, Sage, TeX
C_5\times C_{75} % in TeX
G:=Group("C5xC75"); // GroupNames label
G:=SmallGroup(375,3);
// by ID
G=gap.SmallGroup(375,3);
# by ID
G:=PCGroup([4,-3,-5,-5,-5,205]);
// Polycyclic
G:=Group<a,b|a^5=b^75=1,a*b=b*a>;
// generators/relations
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