direct product, abelian, monomial
Aliases: C10×C30, SmallGroup(300,49)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C10×C30 |
| C1 — C10×C30 |
| C1 — C10×C30 |
Generators and relations for C10×C30
G = < a,b | a10=b30=1, ab=ba >
Subgroups: 80, all normal (8 characteristic)
C1, C2, C3, C22, C5, C6, C10, C2×C6, C15, C2×C10, C52, C30, C5×C10, C2×C30, C5×C15, C102, C5×C30, C10×C30
Quotients: C1, C2, C3, C22, C5, C6, C10, C2×C6, C15, C2×C10, C52, C30, C5×C10, C2×C30, C5×C15, C102, C5×C30, C10×C30
►Regular action on 300 points
Generators in S300
(1 124 299 114 227 79 261 205 31 152)(2 125 300 115 228 80 262 206 32 153)(3 126 271 116 229 81 263 207 33 154)(4 127 272 117 230 82 264 208 34 155)(5 128 273 118 231 83 265 209 35 156)(6 129 274 119 232 84 266 210 36 157)(7 130 275 120 233 85 267 181 37 158)(8 131 276 91 234 86 268 182 38 159)(9 132 277 92 235 87 269 183 39 160)(10 133 278 93 236 88 270 184 40 161)(11 134 279 94 237 89 241 185 41 162)(12 135 280 95 238 90 242 186 42 163)(13 136 281 96 239 61 243 187 43 164)(14 137 282 97 240 62 244 188 44 165)(15 138 283 98 211 63 245 189 45 166)(16 139 284 99 212 64 246 190 46 167)(17 140 285 100 213 65 247 191 47 168)(18 141 286 101 214 66 248 192 48 169)(19 142 287 102 215 67 249 193 49 170)(20 143 288 103 216 68 250 194 50 171)(21 144 289 104 217 69 251 195 51 172)(22 145 290 105 218 70 252 196 52 173)(23 146 291 106 219 71 253 197 53 174)(24 147 292 107 220 72 254 198 54 175)(25 148 293 108 221 73 255 199 55 176)(26 149 294 109 222 74 256 200 56 177)(27 150 295 110 223 75 257 201 57 178)(28 121 296 111 224 76 258 202 58 179)(29 122 297 112 225 77 259 203 59 180)(30 123 298 113 226 78 260 204 60 151)
(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)| (1,124,299,114,227,79,261,205,31,152)(2,125,300,115,228,80,262,206,32,153)(3,126,271,116,229,81,263,207,33,154)(4,127,272,117,230,82,264,208,34,155)(5,128,273,118,231,83,265,209,35,156)(6,129,274,119,232,84,266,210,36,157)(7,130,275,120,233,85,267,181,37,158)(8,131,276,91,234,86,268,182,38,159)(9,132,277,92,235,87,269,183,39,160)(10,133,278,93,236,88,270,184,40,161)(11,134,279,94,237,89,241,185,41,162)(12,135,280,95,238,90,242,186,42,163)(13,136,281,96,239,61,243,187,43,164)(14,137,282,97,240,62,244,188,44,165)(15,138,283,98,211,63,245,189,45,166)(16,139,284,99,212,64,246,190,46,167)(17,140,285,100,213,65,247,191,47,168)(18,141,286,101,214,66,248,192,48,169)(19,142,287,102,215,67,249,193,49,170)(20,143,288,103,216,68,250,194,50,171)(21,144,289,104,217,69,251,195,51,172)(22,145,290,105,218,70,252,196,52,173)(23,146,291,106,219,71,253,197,53,174)(24,147,292,107,220,72,254,198,54,175)(25,148,293,108,221,73,255,199,55,176)(26,149,294,109,222,74,256,200,56,177)(27,150,295,110,223,75,257,201,57,178)(28,121,296,111,224,76,258,202,58,179)(29,122,297,112,225,77,259,203,59,180)(30,123,298,113,226,78,260,204,60,151), (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( (1,124,299,114,227,79,261,205,31,152)(2,125,300,115,228,80,262,206,32,153)(3,126,271,116,229,81,263,207,33,154)(4,127,272,117,230,82,264,208,34,155)(5,128,273,118,231,83,265,209,35,156)(6,129,274,119,232,84,266,210,36,157)(7,130,275,120,233,85,267,181,37,158)(8,131,276,91,234,86,268,182,38,159)(9,132,277,92,235,87,269,183,39,160)(10,133,278,93,236,88,270,184,40,161)(11,134,279,94,237,89,241,185,41,162)(12,135,280,95,238,90,242,186,42,163)(13,136,281,96,239,61,243,187,43,164)(14,137,282,97,240,62,244,188,44,165)(15,138,283,98,211,63,245,189,45,166)(16,139,284,99,212,64,246,190,46,167)(17,140,285,100,213,65,247,191,47,168)(18,141,286,101,214,66,248,192,48,169)(19,142,287,102,215,67,249,193,49,170)(20,143,288,103,216,68,250,194,50,171)(21,144,289,104,217,69,251,195,51,172)(22,145,290,105,218,70,252,196,52,173)(23,146,291,106,219,71,253,197,53,174)(24,147,292,107,220,72,254,198,54,175)(25,148,293,108,221,73,255,199,55,176)(26,149,294,109,222,74,256,200,56,177)(27,150,295,110,223,75,257,201,57,178)(28,121,296,111,224,76,258,202,58,179)(29,122,297,112,225,77,259,203,59,180)(30,123,298,113,226,78,260,204,60,151), (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([[(1,124,299,114,227,79,261,205,31,152),(2,125,300,115,228,80,262,206,32,153),(3,126,271,116,229,81,263,207,33,154),(4,127,272,117,230,82,264,208,34,155),(5,128,273,118,231,83,265,209,35,156),(6,129,274,119,232,84,266,210,36,157),(7,130,275,120,233,85,267,181,37,158),(8,131,276,91,234,86,268,182,38,159),(9,132,277,92,235,87,269,183,39,160),(10,133,278,93,236,88,270,184,40,161),(11,134,279,94,237,89,241,185,41,162),(12,135,280,95,238,90,242,186,42,163),(13,136,281,96,239,61,243,187,43,164),(14,137,282,97,240,62,244,188,44,165),(15,138,283,98,211,63,245,189,45,166),(16,139,284,99,212,64,246,190,46,167),(17,140,285,100,213,65,247,191,47,168),(18,141,286,101,214,66,248,192,48,169),(19,142,287,102,215,67,249,193,49,170),(20,143,288,103,216,68,250,194,50,171),(21,144,289,104,217,69,251,195,51,172),(22,145,290,105,218,70,252,196,52,173),(23,146,291,106,219,71,253,197,53,174),(24,147,292,107,220,72,254,198,54,175),(25,148,293,108,221,73,255,199,55,176),(26,149,294,109,222,74,256,200,56,177),(27,150,295,110,223,75,257,201,57,178),(28,121,296,111,224,76,258,202,58,179),(29,122,297,112,225,77,259,203,59,180),(30,123,298,113,226,78,260,204,60,151)], [(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 | 2A | 2B | 2C | 3A | 3B | 5A | ··· | 5X | 6A | ··· | 6F | 10A | ··· | 10BT | 15A | ··· | 15AV | 30A | ··· | 30EN |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 5 | ··· | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 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 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 |
| kernel | C10×C30 | C5×C30 | C102 | C2×C30 | C5×C10 | C30 | C2×C10 | C10 |
| # reps | 1 | 3 | 2 | 24 | 6 | 72 | 48 | 144 |
Matrix representation of C10×C30 ►in GL2(𝔽31) generated by
| 2 | 0 |
| 0 | 27 |
| 23 | 0 |
| 0 | 13 |
G:=sub<GL(2,GF(31))| [2,0,0,27],[23,0,0,13] >;
C10×C30 in GAP, Magma, Sage, TeX
C_{10}\times C_{30} % in TeX
G:=Group("C10xC30"); // GroupNames label
G:=SmallGroup(300,49);
// by ID
G=gap.SmallGroup(300,49);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5]);
// Polycyclic
G:=Group<a,b|a^10=b^30=1,a*b=b*a>;
// generators/relations