direct product, abelian, monomial, 5-elementary
Aliases: C5×C60, SmallGroup(300,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C60 |
| C1 — C5×C60 |
| C1 — C5×C60 |
Generators and relations for C5×C60
G = < a,b | a5=b60=1, ab=ba >
Smallest permutation representation of C5×C60
►Regular action on 300 points
Generators in S300
(1 103 178 279 183)(2 104 179 280 184)(3 105 180 281 185)(4 106 121 282 186)(5 107 122 283 187)(6 108 123 284 188)(7 109 124 285 189)(8 110 125 286 190)(9 111 126 287 191)(10 112 127 288 192)(11 113 128 289 193)(12 114 129 290 194)(13 115 130 291 195)(14 116 131 292 196)(15 117 132 293 197)(16 118 133 294 198)(17 119 134 295 199)(18 120 135 296 200)(19 61 136 297 201)(20 62 137 298 202)(21 63 138 299 203)(22 64 139 300 204)(23 65 140 241 205)(24 66 141 242 206)(25 67 142 243 207)(26 68 143 244 208)(27 69 144 245 209)(28 70 145 246 210)(29 71 146 247 211)(30 72 147 248 212)(31 73 148 249 213)(32 74 149 250 214)(33 75 150 251 215)(34 76 151 252 216)(35 77 152 253 217)(36 78 153 254 218)(37 79 154 255 219)(38 80 155 256 220)(39 81 156 257 221)(40 82 157 258 222)(41 83 158 259 223)(42 84 159 260 224)(43 85 160 261 225)(44 86 161 262 226)(45 87 162 263 227)(46 88 163 264 228)(47 89 164 265 229)(48 90 165 266 230)(49 91 166 267 231)(50 92 167 268 232)(51 93 168 269 233)(52 94 169 270 234)(53 95 170 271 235)(54 96 171 272 236)(55 97 172 273 237)(56 98 173 274 238)(57 99 174 275 239)(58 100 175 276 240)(59 101 176 277 181)(60 102 177 278 182)
(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)
G:=sub<Sym(300)| (1,103,178,279,183)(2,104,179,280,184)(3,105,180,281,185)(4,106,121,282,186)(5,107,122,283,187)(6,108,123,284,188)(7,109,124,285,189)(8,110,125,286,190)(9,111,126,287,191)(10,112,127,288,192)(11,113,128,289,193)(12,114,129,290,194)(13,115,130,291,195)(14,116,131,292,196)(15,117,132,293,197)(16,118,133,294,198)(17,119,134,295,199)(18,120,135,296,200)(19,61,136,297,201)(20,62,137,298,202)(21,63,138,299,203)(22,64,139,300,204)(23,65,140,241,205)(24,66,141,242,206)(25,67,142,243,207)(26,68,143,244,208)(27,69,144,245,209)(28,70,145,246,210)(29,71,146,247,211)(30,72,147,248,212)(31,73,148,249,213)(32,74,149,250,214)(33,75,150,251,215)(34,76,151,252,216)(35,77,152,253,217)(36,78,153,254,218)(37,79,154,255,219)(38,80,155,256,220)(39,81,156,257,221)(40,82,157,258,222)(41,83,158,259,223)(42,84,159,260,224)(43,85,160,261,225)(44,86,161,262,226)(45,87,162,263,227)(46,88,163,264,228)(47,89,164,265,229)(48,90,165,266,230)(49,91,166,267,231)(50,92,167,268,232)(51,93,168,269,233)(52,94,169,270,234)(53,95,170,271,235)(54,96,171,272,236)(55,97,172,273,237)(56,98,173,274,238)(57,99,174,275,239)(58,100,175,276,240)(59,101,176,277,181)(60,102,177,278,182), (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)>;
G:=Group( (1,103,178,279,183)(2,104,179,280,184)(3,105,180,281,185)(4,106,121,282,186)(5,107,122,283,187)(6,108,123,284,188)(7,109,124,285,189)(8,110,125,286,190)(9,111,126,287,191)(10,112,127,288,192)(11,113,128,289,193)(12,114,129,290,194)(13,115,130,291,195)(14,116,131,292,196)(15,117,132,293,197)(16,118,133,294,198)(17,119,134,295,199)(18,120,135,296,200)(19,61,136,297,201)(20,62,137,298,202)(21,63,138,299,203)(22,64,139,300,204)(23,65,140,241,205)(24,66,141,242,206)(25,67,142,243,207)(26,68,143,244,208)(27,69,144,245,209)(28,70,145,246,210)(29,71,146,247,211)(30,72,147,248,212)(31,73,148,249,213)(32,74,149,250,214)(33,75,150,251,215)(34,76,151,252,216)(35,77,152,253,217)(36,78,153,254,218)(37,79,154,255,219)(38,80,155,256,220)(39,81,156,257,221)(40,82,157,258,222)(41,83,158,259,223)(42,84,159,260,224)(43,85,160,261,225)(44,86,161,262,226)(45,87,162,263,227)(46,88,163,264,228)(47,89,164,265,229)(48,90,165,266,230)(49,91,166,267,231)(50,92,167,268,232)(51,93,168,269,233)(52,94,169,270,234)(53,95,170,271,235)(54,96,171,272,236)(55,97,172,273,237)(56,98,173,274,238)(57,99,174,275,239)(58,100,175,276,240)(59,101,176,277,181)(60,102,177,278,182), (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) );
G=PermutationGroup([[(1,103,178,279,183),(2,104,179,280,184),(3,105,180,281,185),(4,106,121,282,186),(5,107,122,283,187),(6,108,123,284,188),(7,109,124,285,189),(8,110,125,286,190),(9,111,126,287,191),(10,112,127,288,192),(11,113,128,289,193),(12,114,129,290,194),(13,115,130,291,195),(14,116,131,292,196),(15,117,132,293,197),(16,118,133,294,198),(17,119,134,295,199),(18,120,135,296,200),(19,61,136,297,201),(20,62,137,298,202),(21,63,138,299,203),(22,64,139,300,204),(23,65,140,241,205),(24,66,141,242,206),(25,67,142,243,207),(26,68,143,244,208),(27,69,144,245,209),(28,70,145,246,210),(29,71,146,247,211),(30,72,147,248,212),(31,73,148,249,213),(32,74,149,250,214),(33,75,150,251,215),(34,76,151,252,216),(35,77,152,253,217),(36,78,153,254,218),(37,79,154,255,219),(38,80,155,256,220),(39,81,156,257,221),(40,82,157,258,222),(41,83,158,259,223),(42,84,159,260,224),(43,85,160,261,225),(44,86,161,262,226),(45,87,162,263,227),(46,88,163,264,228),(47,89,164,265,229),(48,90,165,266,230),(49,91,166,267,231),(50,92,167,268,232),(51,93,168,269,233),(52,94,169,270,234),(53,95,170,271,235),(54,96,171,272,236),(55,97,172,273,237),(56,98,173,274,238),(57,99,174,275,239),(58,100,175,276,240),(59,101,176,277,181),(60,102,177,278,182)], [(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)]])
300 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | ··· | 5X | 6A | 6B | 10A | ··· | 10X | 12A | 12B | 12C | 12D | 15A | ··· | 15AV | 20A | ··· | 20AV | 30A | ··· | 30AV | 60A | ··· | 60CR |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | ··· | 5 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
300 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C10 | C12 | C15 | C20 | C30 | C60 |
| kernel | C5×C60 | C5×C30 | C5×C20 | C5×C15 | C60 | C5×C10 | C30 | C52 | C20 | C15 | C10 | C5 |
| # reps | 1 | 1 | 2 | 2 | 24 | 2 | 24 | 4 | 48 | 48 | 48 | 96 |
Matrix representation of C5×C60 ►in GL2(𝔽61) generated by
| 34 | 0 |
| 0 | 20 |
| 39 | 0 |
| 0 | 50 |
G:=sub<GL(2,GF(61))| [34,0,0,20],[39,0,0,50] >;
C5×C60 in GAP, Magma, Sage, TeX
C_5\times C_{60} % in TeX
G:=Group("C5xC60"); // GroupNames label
G:=SmallGroup(300,21);
// by ID
G=gap.SmallGroup(300,21);
# by ID
G:=PCGroup([5,-2,-3,-5,-5,-2,750]);
// Polycyclic
G:=Group<a,b|a^5=b^60=1,a*b=b*a>;
// generators/relations
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