direct product, abelian, monomial, 5-elementary
Aliases: C5×C65, SmallGroup(325,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C65 |
| C1 — C5×C65 |
| C1 — C5×C65 |
Generators and relations for C5×C65
G = < a,b | a5=b65=1, ab=ba >
Smallest permutation representation of C5×C65
►Regular action on 325 points
Generators in S325
(1 280 244 161 88)(2 281 245 162 89)(3 282 246 163 90)(4 283 247 164 91)(5 284 248 165 92)(6 285 249 166 93)(7 286 250 167 94)(8 287 251 168 95)(9 288 252 169 96)(10 289 253 170 97)(11 290 254 171 98)(12 291 255 172 99)(13 292 256 173 100)(14 293 257 174 101)(15 294 258 175 102)(16 295 259 176 103)(17 296 260 177 104)(18 297 196 178 105)(19 298 197 179 106)(20 299 198 180 107)(21 300 199 181 108)(22 301 200 182 109)(23 302 201 183 110)(24 303 202 184 111)(25 304 203 185 112)(26 305 204 186 113)(27 306 205 187 114)(28 307 206 188 115)(29 308 207 189 116)(30 309 208 190 117)(31 310 209 191 118)(32 311 210 192 119)(33 312 211 193 120)(34 313 212 194 121)(35 314 213 195 122)(36 315 214 131 123)(37 316 215 132 124)(38 317 216 133 125)(39 318 217 134 126)(40 319 218 135 127)(41 320 219 136 128)(42 321 220 137 129)(43 322 221 138 130)(44 323 222 139 66)(45 324 223 140 67)(46 325 224 141 68)(47 261 225 142 69)(48 262 226 143 70)(49 263 227 144 71)(50 264 228 145 72)(51 265 229 146 73)(52 266 230 147 74)(53 267 231 148 75)(54 268 232 149 76)(55 269 233 150 77)(56 270 234 151 78)(57 271 235 152 79)(58 272 236 153 80)(59 273 237 154 81)(60 274 238 155 82)(61 275 239 156 83)(62 276 240 157 84)(63 277 241 158 85)(64 278 242 159 86)(65 279 243 160 87)
(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)
G:=sub<Sym(325)| (1,280,244,161,88)(2,281,245,162,89)(3,282,246,163,90)(4,283,247,164,91)(5,284,248,165,92)(6,285,249,166,93)(7,286,250,167,94)(8,287,251,168,95)(9,288,252,169,96)(10,289,253,170,97)(11,290,254,171,98)(12,291,255,172,99)(13,292,256,173,100)(14,293,257,174,101)(15,294,258,175,102)(16,295,259,176,103)(17,296,260,177,104)(18,297,196,178,105)(19,298,197,179,106)(20,299,198,180,107)(21,300,199,181,108)(22,301,200,182,109)(23,302,201,183,110)(24,303,202,184,111)(25,304,203,185,112)(26,305,204,186,113)(27,306,205,187,114)(28,307,206,188,115)(29,308,207,189,116)(30,309,208,190,117)(31,310,209,191,118)(32,311,210,192,119)(33,312,211,193,120)(34,313,212,194,121)(35,314,213,195,122)(36,315,214,131,123)(37,316,215,132,124)(38,317,216,133,125)(39,318,217,134,126)(40,319,218,135,127)(41,320,219,136,128)(42,321,220,137,129)(43,322,221,138,130)(44,323,222,139,66)(45,324,223,140,67)(46,325,224,141,68)(47,261,225,142,69)(48,262,226,143,70)(49,263,227,144,71)(50,264,228,145,72)(51,265,229,146,73)(52,266,230,147,74)(53,267,231,148,75)(54,268,232,149,76)(55,269,233,150,77)(56,270,234,151,78)(57,271,235,152,79)(58,272,236,153,80)(59,273,237,154,81)(60,274,238,155,82)(61,275,239,156,83)(62,276,240,157,84)(63,277,241,158,85)(64,278,242,159,86)(65,279,243,160,87), (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)>;
G:=Group( (1,280,244,161,88)(2,281,245,162,89)(3,282,246,163,90)(4,283,247,164,91)(5,284,248,165,92)(6,285,249,166,93)(7,286,250,167,94)(8,287,251,168,95)(9,288,252,169,96)(10,289,253,170,97)(11,290,254,171,98)(12,291,255,172,99)(13,292,256,173,100)(14,293,257,174,101)(15,294,258,175,102)(16,295,259,176,103)(17,296,260,177,104)(18,297,196,178,105)(19,298,197,179,106)(20,299,198,180,107)(21,300,199,181,108)(22,301,200,182,109)(23,302,201,183,110)(24,303,202,184,111)(25,304,203,185,112)(26,305,204,186,113)(27,306,205,187,114)(28,307,206,188,115)(29,308,207,189,116)(30,309,208,190,117)(31,310,209,191,118)(32,311,210,192,119)(33,312,211,193,120)(34,313,212,194,121)(35,314,213,195,122)(36,315,214,131,123)(37,316,215,132,124)(38,317,216,133,125)(39,318,217,134,126)(40,319,218,135,127)(41,320,219,136,128)(42,321,220,137,129)(43,322,221,138,130)(44,323,222,139,66)(45,324,223,140,67)(46,325,224,141,68)(47,261,225,142,69)(48,262,226,143,70)(49,263,227,144,71)(50,264,228,145,72)(51,265,229,146,73)(52,266,230,147,74)(53,267,231,148,75)(54,268,232,149,76)(55,269,233,150,77)(56,270,234,151,78)(57,271,235,152,79)(58,272,236,153,80)(59,273,237,154,81)(60,274,238,155,82)(61,275,239,156,83)(62,276,240,157,84)(63,277,241,158,85)(64,278,242,159,86)(65,279,243,160,87), (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) );
G=PermutationGroup([[(1,280,244,161,88),(2,281,245,162,89),(3,282,246,163,90),(4,283,247,164,91),(5,284,248,165,92),(6,285,249,166,93),(7,286,250,167,94),(8,287,251,168,95),(9,288,252,169,96),(10,289,253,170,97),(11,290,254,171,98),(12,291,255,172,99),(13,292,256,173,100),(14,293,257,174,101),(15,294,258,175,102),(16,295,259,176,103),(17,296,260,177,104),(18,297,196,178,105),(19,298,197,179,106),(20,299,198,180,107),(21,300,199,181,108),(22,301,200,182,109),(23,302,201,183,110),(24,303,202,184,111),(25,304,203,185,112),(26,305,204,186,113),(27,306,205,187,114),(28,307,206,188,115),(29,308,207,189,116),(30,309,208,190,117),(31,310,209,191,118),(32,311,210,192,119),(33,312,211,193,120),(34,313,212,194,121),(35,314,213,195,122),(36,315,214,131,123),(37,316,215,132,124),(38,317,216,133,125),(39,318,217,134,126),(40,319,218,135,127),(41,320,219,136,128),(42,321,220,137,129),(43,322,221,138,130),(44,323,222,139,66),(45,324,223,140,67),(46,325,224,141,68),(47,261,225,142,69),(48,262,226,143,70),(49,263,227,144,71),(50,264,228,145,72),(51,265,229,146,73),(52,266,230,147,74),(53,267,231,148,75),(54,268,232,149,76),(55,269,233,150,77),(56,270,234,151,78),(57,271,235,152,79),(58,272,236,153,80),(59,273,237,154,81),(60,274,238,155,82),(61,275,239,156,83),(62,276,240,157,84),(63,277,241,158,85),(64,278,242,159,86),(65,279,243,160,87)], [(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)]])
325 conjugacy classes
| class | 1 | 5A | ··· | 5X | 13A | ··· | 13L | 65A | ··· | 65KB |
| order | 1 | 5 | ··· | 5 | 13 | ··· | 13 | 65 | ··· | 65 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
325 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C5 | C13 | C65 |
| kernel | C5×C65 | C65 | C52 | C5 |
| # reps | 1 | 24 | 12 | 288 |
Matrix representation of C5×C65 ►in GL2(𝔽131) generated by
| 89 | 0 |
| 0 | 1 |
| 13 | 0 |
| 0 | 43 |
G:=sub<GL(2,GF(131))| [89,0,0,1],[13,0,0,43] >;
C5×C65 in GAP, Magma, Sage, TeX
C_5\times C_{65} % in TeX
G:=Group("C5xC65"); // GroupNames label
G:=SmallGroup(325,2);
// by ID
G=gap.SmallGroup(325,2);
# by ID
G:=PCGroup([3,-5,-5,-13]);
// Polycyclic
G:=Group<a,b|a^5=b^65=1,a*b=b*a>;
// generators/relations
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