direct product, abelian, monomial, 5-elementary
Aliases: C5×C70, SmallGroup(350,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C70 |
| C1 — C5×C70 |
| C1 — C5×C70 |
Generators and relations for C5×C70
G = < a,b | a5=b70=1, ab=ba >
Smallest permutation representation of C5×C70
►Regular action on 350 points
Generators in S350
(1 218 320 152 111)(2 219 321 153 112)(3 220 322 154 113)(4 221 323 155 114)(5 222 324 156 115)(6 223 325 157 116)(7 224 326 158 117)(8 225 327 159 118)(9 226 328 160 119)(10 227 329 161 120)(11 228 330 162 121)(12 229 331 163 122)(13 230 332 164 123)(14 231 333 165 124)(15 232 334 166 125)(16 233 335 167 126)(17 234 336 168 127)(18 235 337 169 128)(19 236 338 170 129)(20 237 339 171 130)(21 238 340 172 131)(22 239 341 173 132)(23 240 342 174 133)(24 241 343 175 134)(25 242 344 176 135)(26 243 345 177 136)(27 244 346 178 137)(28 245 347 179 138)(29 246 348 180 139)(30 247 349 181 140)(31 248 350 182 71)(32 249 281 183 72)(33 250 282 184 73)(34 251 283 185 74)(35 252 284 186 75)(36 253 285 187 76)(37 254 286 188 77)(38 255 287 189 78)(39 256 288 190 79)(40 257 289 191 80)(41 258 290 192 81)(42 259 291 193 82)(43 260 292 194 83)(44 261 293 195 84)(45 262 294 196 85)(46 263 295 197 86)(47 264 296 198 87)(48 265 297 199 88)(49 266 298 200 89)(50 267 299 201 90)(51 268 300 202 91)(52 269 301 203 92)(53 270 302 204 93)(54 271 303 205 94)(55 272 304 206 95)(56 273 305 207 96)(57 274 306 208 97)(58 275 307 209 98)(59 276 308 210 99)(60 277 309 141 100)(61 278 310 142 101)(62 279 311 143 102)(63 280 312 144 103)(64 211 313 145 104)(65 212 314 146 105)(66 213 315 147 106)(67 214 316 148 107)(68 215 317 149 108)(69 216 318 150 109)(70 217 319 151 110)
(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)
G:=sub<Sym(350)| (1,218,320,152,111)(2,219,321,153,112)(3,220,322,154,113)(4,221,323,155,114)(5,222,324,156,115)(6,223,325,157,116)(7,224,326,158,117)(8,225,327,159,118)(9,226,328,160,119)(10,227,329,161,120)(11,228,330,162,121)(12,229,331,163,122)(13,230,332,164,123)(14,231,333,165,124)(15,232,334,166,125)(16,233,335,167,126)(17,234,336,168,127)(18,235,337,169,128)(19,236,338,170,129)(20,237,339,171,130)(21,238,340,172,131)(22,239,341,173,132)(23,240,342,174,133)(24,241,343,175,134)(25,242,344,176,135)(26,243,345,177,136)(27,244,346,178,137)(28,245,347,179,138)(29,246,348,180,139)(30,247,349,181,140)(31,248,350,182,71)(32,249,281,183,72)(33,250,282,184,73)(34,251,283,185,74)(35,252,284,186,75)(36,253,285,187,76)(37,254,286,188,77)(38,255,287,189,78)(39,256,288,190,79)(40,257,289,191,80)(41,258,290,192,81)(42,259,291,193,82)(43,260,292,194,83)(44,261,293,195,84)(45,262,294,196,85)(46,263,295,197,86)(47,264,296,198,87)(48,265,297,199,88)(49,266,298,200,89)(50,267,299,201,90)(51,268,300,202,91)(52,269,301,203,92)(53,270,302,204,93)(54,271,303,205,94)(55,272,304,206,95)(56,273,305,207,96)(57,274,306,208,97)(58,275,307,209,98)(59,276,308,210,99)(60,277,309,141,100)(61,278,310,142,101)(62,279,311,143,102)(63,280,312,144,103)(64,211,313,145,104)(65,212,314,146,105)(66,213,315,147,106)(67,214,316,148,107)(68,215,317,149,108)(69,216,318,150,109)(70,217,319,151,110), (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)>;
G:=Group( (1,218,320,152,111)(2,219,321,153,112)(3,220,322,154,113)(4,221,323,155,114)(5,222,324,156,115)(6,223,325,157,116)(7,224,326,158,117)(8,225,327,159,118)(9,226,328,160,119)(10,227,329,161,120)(11,228,330,162,121)(12,229,331,163,122)(13,230,332,164,123)(14,231,333,165,124)(15,232,334,166,125)(16,233,335,167,126)(17,234,336,168,127)(18,235,337,169,128)(19,236,338,170,129)(20,237,339,171,130)(21,238,340,172,131)(22,239,341,173,132)(23,240,342,174,133)(24,241,343,175,134)(25,242,344,176,135)(26,243,345,177,136)(27,244,346,178,137)(28,245,347,179,138)(29,246,348,180,139)(30,247,349,181,140)(31,248,350,182,71)(32,249,281,183,72)(33,250,282,184,73)(34,251,283,185,74)(35,252,284,186,75)(36,253,285,187,76)(37,254,286,188,77)(38,255,287,189,78)(39,256,288,190,79)(40,257,289,191,80)(41,258,290,192,81)(42,259,291,193,82)(43,260,292,194,83)(44,261,293,195,84)(45,262,294,196,85)(46,263,295,197,86)(47,264,296,198,87)(48,265,297,199,88)(49,266,298,200,89)(50,267,299,201,90)(51,268,300,202,91)(52,269,301,203,92)(53,270,302,204,93)(54,271,303,205,94)(55,272,304,206,95)(56,273,305,207,96)(57,274,306,208,97)(58,275,307,209,98)(59,276,308,210,99)(60,277,309,141,100)(61,278,310,142,101)(62,279,311,143,102)(63,280,312,144,103)(64,211,313,145,104)(65,212,314,146,105)(66,213,315,147,106)(67,214,316,148,107)(68,215,317,149,108)(69,216,318,150,109)(70,217,319,151,110), (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) );
G=PermutationGroup([[(1,218,320,152,111),(2,219,321,153,112),(3,220,322,154,113),(4,221,323,155,114),(5,222,324,156,115),(6,223,325,157,116),(7,224,326,158,117),(8,225,327,159,118),(9,226,328,160,119),(10,227,329,161,120),(11,228,330,162,121),(12,229,331,163,122),(13,230,332,164,123),(14,231,333,165,124),(15,232,334,166,125),(16,233,335,167,126),(17,234,336,168,127),(18,235,337,169,128),(19,236,338,170,129),(20,237,339,171,130),(21,238,340,172,131),(22,239,341,173,132),(23,240,342,174,133),(24,241,343,175,134),(25,242,344,176,135),(26,243,345,177,136),(27,244,346,178,137),(28,245,347,179,138),(29,246,348,180,139),(30,247,349,181,140),(31,248,350,182,71),(32,249,281,183,72),(33,250,282,184,73),(34,251,283,185,74),(35,252,284,186,75),(36,253,285,187,76),(37,254,286,188,77),(38,255,287,189,78),(39,256,288,190,79),(40,257,289,191,80),(41,258,290,192,81),(42,259,291,193,82),(43,260,292,194,83),(44,261,293,195,84),(45,262,294,196,85),(46,263,295,197,86),(47,264,296,198,87),(48,265,297,199,88),(49,266,298,200,89),(50,267,299,201,90),(51,268,300,202,91),(52,269,301,203,92),(53,270,302,204,93),(54,271,303,205,94),(55,272,304,206,95),(56,273,305,207,96),(57,274,306,208,97),(58,275,307,209,98),(59,276,308,210,99),(60,277,309,141,100),(61,278,310,142,101),(62,279,311,143,102),(63,280,312,144,103),(64,211,313,145,104),(65,212,314,146,105),(66,213,315,147,106),(67,214,316,148,107),(68,215,317,149,108),(69,216,318,150,109),(70,217,319,151,110)], [(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)]])
350 conjugacy classes
| class | 1 | 2 | 5A | ··· | 5X | 7A | ··· | 7F | 10A | ··· | 10X | 14A | ··· | 14F | 35A | ··· | 35EN | 70A | ··· | 70EN |
| order | 1 | 2 | 5 | ··· | 5 | 7 | ··· | 7 | 10 | ··· | 10 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
350 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C5 | C7 | C10 | C14 | C35 | C70 |
| kernel | C5×C70 | C5×C35 | C70 | C5×C10 | C35 | C52 | C10 | C5 |
| # reps | 1 | 1 | 24 | 6 | 24 | 6 | 144 | 144 |
Matrix representation of C5×C70 ►in GL2(𝔽71) generated by
| 1 | 0 |
| 0 | 25 |
| 46 | 0 |
| 0 | 51 |
G:=sub<GL(2,GF(71))| [1,0,0,25],[46,0,0,51] >;
C5×C70 in GAP, Magma, Sage, TeX
C_5\times C_{70} % in TeX
G:=Group("C5xC70"); // GroupNames label
G:=SmallGroup(350,10);
// by ID
G=gap.SmallGroup(350,10);
# by ID
G:=PCGroup([4,-2,-5,-5,-7]);
// Polycyclic
G:=Group<a,b|a^5=b^70=1,a*b=b*a>;
// generators/relations
Export