metabelian, supersoluble, monomial, A-group
Aliases: C30.3D5, C15⋊3Dic5, C5⋊2Dic15, C10.3D15, C52⋊8Dic3, (C5×C15)⋊9C4, C6.(C5⋊D5), C2.(C5⋊D15), C3⋊(C52⋊6C4), (C5×C10).3S3, (C5×C30).1C2, SmallGroup(300,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C15 — C30.D5 |
Generators and relations for C30.D5
G = < a,b,c | a30=b5=1, c2=a15, ab=ba, cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C30.D5
►Regular action on 300 points
Generators in S300
(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)
(1 98 263 236 158)(2 99 264 237 159)(3 100 265 238 160)(4 101 266 239 161)(5 102 267 240 162)(6 103 268 211 163)(7 104 269 212 164)(8 105 270 213 165)(9 106 241 214 166)(10 107 242 215 167)(11 108 243 216 168)(12 109 244 217 169)(13 110 245 218 170)(14 111 246 219 171)(15 112 247 220 172)(16 113 248 221 173)(17 114 249 222 174)(18 115 250 223 175)(19 116 251 224 176)(20 117 252 225 177)(21 118 253 226 178)(22 119 254 227 179)(23 120 255 228 180)(24 91 256 229 151)(25 92 257 230 152)(26 93 258 231 153)(27 94 259 232 154)(28 95 260 233 155)(29 96 261 234 156)(30 97 262 235 157)(31 86 199 275 136)(32 87 200 276 137)(33 88 201 277 138)(34 89 202 278 139)(35 90 203 279 140)(36 61 204 280 141)(37 62 205 281 142)(38 63 206 282 143)(39 64 207 283 144)(40 65 208 284 145)(41 66 209 285 146)(42 67 210 286 147)(43 68 181 287 148)(44 69 182 288 149)(45 70 183 289 150)(46 71 184 290 121)(47 72 185 291 122)(48 73 186 292 123)(49 74 187 293 124)(50 75 188 294 125)(51 76 189 295 126)(52 77 190 296 127)(53 78 191 297 128)(54 79 192 298 129)(55 80 193 299 130)(56 81 194 300 131)(57 82 195 271 132)(58 83 196 272 133)(59 84 197 273 134)(60 85 198 274 135)
(1 40 16 55)(2 39 17 54)(3 38 18 53)(4 37 19 52)(5 36 20 51)(6 35 21 50)(7 34 22 49)(8 33 23 48)(9 32 24 47)(10 31 25 46)(11 60 26 45)(12 59 27 44)(13 58 28 43)(14 57 29 42)(15 56 30 41)(61 177 76 162)(62 176 77 161)(63 175 78 160)(64 174 79 159)(65 173 80 158)(66 172 81 157)(67 171 82 156)(68 170 83 155)(69 169 84 154)(70 168 85 153)(71 167 86 152)(72 166 87 151)(73 165 88 180)(74 164 89 179)(75 163 90 178)(91 122 106 137)(92 121 107 136)(93 150 108 135)(94 149 109 134)(95 148 110 133)(96 147 111 132)(97 146 112 131)(98 145 113 130)(99 144 114 129)(100 143 115 128)(101 142 116 127)(102 141 117 126)(103 140 118 125)(104 139 119 124)(105 138 120 123)(181 218 196 233)(182 217 197 232)(183 216 198 231)(184 215 199 230)(185 214 200 229)(186 213 201 228)(187 212 202 227)(188 211 203 226)(189 240 204 225)(190 239 205 224)(191 238 206 223)(192 237 207 222)(193 236 208 221)(194 235 209 220)(195 234 210 219)(241 276 256 291)(242 275 257 290)(243 274 258 289)(244 273 259 288)(245 272 260 287)(246 271 261 286)(247 300 262 285)(248 299 263 284)(249 298 264 283)(250 297 265 282)(251 296 266 281)(252 295 267 280)(253 294 268 279)(254 293 269 278)(255 292 270 277)
G:=sub<Sym(300)| (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), (1,98,263,236,158)(2,99,264,237,159)(3,100,265,238,160)(4,101,266,239,161)(5,102,267,240,162)(6,103,268,211,163)(7,104,269,212,164)(8,105,270,213,165)(9,106,241,214,166)(10,107,242,215,167)(11,108,243,216,168)(12,109,244,217,169)(13,110,245,218,170)(14,111,246,219,171)(15,112,247,220,172)(16,113,248,221,173)(17,114,249,222,174)(18,115,250,223,175)(19,116,251,224,176)(20,117,252,225,177)(21,118,253,226,178)(22,119,254,227,179)(23,120,255,228,180)(24,91,256,229,151)(25,92,257,230,152)(26,93,258,231,153)(27,94,259,232,154)(28,95,260,233,155)(29,96,261,234,156)(30,97,262,235,157)(31,86,199,275,136)(32,87,200,276,137)(33,88,201,277,138)(34,89,202,278,139)(35,90,203,279,140)(36,61,204,280,141)(37,62,205,281,142)(38,63,206,282,143)(39,64,207,283,144)(40,65,208,284,145)(41,66,209,285,146)(42,67,210,286,147)(43,68,181,287,148)(44,69,182,288,149)(45,70,183,289,150)(46,71,184,290,121)(47,72,185,291,122)(48,73,186,292,123)(49,74,187,293,124)(50,75,188,294,125)(51,76,189,295,126)(52,77,190,296,127)(53,78,191,297,128)(54,79,192,298,129)(55,80,193,299,130)(56,81,194,300,131)(57,82,195,271,132)(58,83,196,272,133)(59,84,197,273,134)(60,85,198,274,135), (1,40,16,55)(2,39,17,54)(3,38,18,53)(4,37,19,52)(5,36,20,51)(6,35,21,50)(7,34,22,49)(8,33,23,48)(9,32,24,47)(10,31,25,46)(11,60,26,45)(12,59,27,44)(13,58,28,43)(14,57,29,42)(15,56,30,41)(61,177,76,162)(62,176,77,161)(63,175,78,160)(64,174,79,159)(65,173,80,158)(66,172,81,157)(67,171,82,156)(68,170,83,155)(69,169,84,154)(70,168,85,153)(71,167,86,152)(72,166,87,151)(73,165,88,180)(74,164,89,179)(75,163,90,178)(91,122,106,137)(92,121,107,136)(93,150,108,135)(94,149,109,134)(95,148,110,133)(96,147,111,132)(97,146,112,131)(98,145,113,130)(99,144,114,129)(100,143,115,128)(101,142,116,127)(102,141,117,126)(103,140,118,125)(104,139,119,124)(105,138,120,123)(181,218,196,233)(182,217,197,232)(183,216,198,231)(184,215,199,230)(185,214,200,229)(186,213,201,228)(187,212,202,227)(188,211,203,226)(189,240,204,225)(190,239,205,224)(191,238,206,223)(192,237,207,222)(193,236,208,221)(194,235,209,220)(195,234,210,219)(241,276,256,291)(242,275,257,290)(243,274,258,289)(244,273,259,288)(245,272,260,287)(246,271,261,286)(247,300,262,285)(248,299,263,284)(249,298,264,283)(250,297,265,282)(251,296,266,281)(252,295,267,280)(253,294,268,279)(254,293,269,278)(255,292,270,277)>;
G:=Group( (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), (1,98,263,236,158)(2,99,264,237,159)(3,100,265,238,160)(4,101,266,239,161)(5,102,267,240,162)(6,103,268,211,163)(7,104,269,212,164)(8,105,270,213,165)(9,106,241,214,166)(10,107,242,215,167)(11,108,243,216,168)(12,109,244,217,169)(13,110,245,218,170)(14,111,246,219,171)(15,112,247,220,172)(16,113,248,221,173)(17,114,249,222,174)(18,115,250,223,175)(19,116,251,224,176)(20,117,252,225,177)(21,118,253,226,178)(22,119,254,227,179)(23,120,255,228,180)(24,91,256,229,151)(25,92,257,230,152)(26,93,258,231,153)(27,94,259,232,154)(28,95,260,233,155)(29,96,261,234,156)(30,97,262,235,157)(31,86,199,275,136)(32,87,200,276,137)(33,88,201,277,138)(34,89,202,278,139)(35,90,203,279,140)(36,61,204,280,141)(37,62,205,281,142)(38,63,206,282,143)(39,64,207,283,144)(40,65,208,284,145)(41,66,209,285,146)(42,67,210,286,147)(43,68,181,287,148)(44,69,182,288,149)(45,70,183,289,150)(46,71,184,290,121)(47,72,185,291,122)(48,73,186,292,123)(49,74,187,293,124)(50,75,188,294,125)(51,76,189,295,126)(52,77,190,296,127)(53,78,191,297,128)(54,79,192,298,129)(55,80,193,299,130)(56,81,194,300,131)(57,82,195,271,132)(58,83,196,272,133)(59,84,197,273,134)(60,85,198,274,135), (1,40,16,55)(2,39,17,54)(3,38,18,53)(4,37,19,52)(5,36,20,51)(6,35,21,50)(7,34,22,49)(8,33,23,48)(9,32,24,47)(10,31,25,46)(11,60,26,45)(12,59,27,44)(13,58,28,43)(14,57,29,42)(15,56,30,41)(61,177,76,162)(62,176,77,161)(63,175,78,160)(64,174,79,159)(65,173,80,158)(66,172,81,157)(67,171,82,156)(68,170,83,155)(69,169,84,154)(70,168,85,153)(71,167,86,152)(72,166,87,151)(73,165,88,180)(74,164,89,179)(75,163,90,178)(91,122,106,137)(92,121,107,136)(93,150,108,135)(94,149,109,134)(95,148,110,133)(96,147,111,132)(97,146,112,131)(98,145,113,130)(99,144,114,129)(100,143,115,128)(101,142,116,127)(102,141,117,126)(103,140,118,125)(104,139,119,124)(105,138,120,123)(181,218,196,233)(182,217,197,232)(183,216,198,231)(184,215,199,230)(185,214,200,229)(186,213,201,228)(187,212,202,227)(188,211,203,226)(189,240,204,225)(190,239,205,224)(191,238,206,223)(192,237,207,222)(193,236,208,221)(194,235,209,220)(195,234,210,219)(241,276,256,291)(242,275,257,290)(243,274,258,289)(244,273,259,288)(245,272,260,287)(246,271,261,286)(247,300,262,285)(248,299,263,284)(249,298,264,283)(250,297,265,282)(251,296,266,281)(252,295,267,280)(253,294,268,279)(254,293,269,278)(255,292,270,277) );
G=PermutationGroup([[(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)], [(1,98,263,236,158),(2,99,264,237,159),(3,100,265,238,160),(4,101,266,239,161),(5,102,267,240,162),(6,103,268,211,163),(7,104,269,212,164),(8,105,270,213,165),(9,106,241,214,166),(10,107,242,215,167),(11,108,243,216,168),(12,109,244,217,169),(13,110,245,218,170),(14,111,246,219,171),(15,112,247,220,172),(16,113,248,221,173),(17,114,249,222,174),(18,115,250,223,175),(19,116,251,224,176),(20,117,252,225,177),(21,118,253,226,178),(22,119,254,227,179),(23,120,255,228,180),(24,91,256,229,151),(25,92,257,230,152),(26,93,258,231,153),(27,94,259,232,154),(28,95,260,233,155),(29,96,261,234,156),(30,97,262,235,157),(31,86,199,275,136),(32,87,200,276,137),(33,88,201,277,138),(34,89,202,278,139),(35,90,203,279,140),(36,61,204,280,141),(37,62,205,281,142),(38,63,206,282,143),(39,64,207,283,144),(40,65,208,284,145),(41,66,209,285,146),(42,67,210,286,147),(43,68,181,287,148),(44,69,182,288,149),(45,70,183,289,150),(46,71,184,290,121),(47,72,185,291,122),(48,73,186,292,123),(49,74,187,293,124),(50,75,188,294,125),(51,76,189,295,126),(52,77,190,296,127),(53,78,191,297,128),(54,79,192,298,129),(55,80,193,299,130),(56,81,194,300,131),(57,82,195,271,132),(58,83,196,272,133),(59,84,197,273,134),(60,85,198,274,135)], [(1,40,16,55),(2,39,17,54),(3,38,18,53),(4,37,19,52),(5,36,20,51),(6,35,21,50),(7,34,22,49),(8,33,23,48),(9,32,24,47),(10,31,25,46),(11,60,26,45),(12,59,27,44),(13,58,28,43),(14,57,29,42),(15,56,30,41),(61,177,76,162),(62,176,77,161),(63,175,78,160),(64,174,79,159),(65,173,80,158),(66,172,81,157),(67,171,82,156),(68,170,83,155),(69,169,84,154),(70,168,85,153),(71,167,86,152),(72,166,87,151),(73,165,88,180),(74,164,89,179),(75,163,90,178),(91,122,106,137),(92,121,107,136),(93,150,108,135),(94,149,109,134),(95,148,110,133),(96,147,111,132),(97,146,112,131),(98,145,113,130),(99,144,114,129),(100,143,115,128),(101,142,116,127),(102,141,117,126),(103,140,118,125),(104,139,119,124),(105,138,120,123),(181,218,196,233),(182,217,197,232),(183,216,198,231),(184,215,199,230),(185,214,200,229),(186,213,201,228),(187,212,202,227),(188,211,203,226),(189,240,204,225),(190,239,205,224),(191,238,206,223),(192,237,207,222),(193,236,208,221),(194,235,209,220),(195,234,210,219),(241,276,256,291),(242,275,257,290),(243,274,258,289),(244,273,259,288),(245,272,260,287),(246,271,261,286),(247,300,262,285),(248,299,263,284),(249,298,264,283),(250,297,265,282),(251,296,266,281),(252,295,267,280),(253,294,268,279),(254,293,269,278),(255,292,270,277)]])
78 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | ··· | 5L | 6 | 10A | ··· | 10L | 15A | ··· | 15X | 30A | ··· | 30X |
| order | 1 | 2 | 3 | 4 | 4 | 5 | ··· | 5 | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 75 | 75 | 2 | ··· | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | |
| image | C1 | C2 | C4 | S3 | D5 | Dic3 | Dic5 | D15 | Dic15 |
| kernel | C30.D5 | C5×C30 | C5×C15 | C5×C10 | C30 | C52 | C15 | C10 | C5 |
| # reps | 1 | 1 | 2 | 1 | 12 | 1 | 12 | 24 | 24 |
Matrix representation of C30.D5 ►in GL4(𝔽61) generated by
| 44 | 1 | 0 | 0 |
| 16 | 60 | 0 | 0 |
| 0 | 0 | 14 | 30 |
| 0 | 0 | 31 | 36 |
| 44 | 1 | 0 | 0 |
| 16 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 17 |
| 20 | 59 | 0 | 0 |
| 47 | 41 | 0 | 0 |
| 0 | 0 | 24 | 53 |
| 0 | 0 | 34 | 37 |
G:=sub<GL(4,GF(61))| [44,16,0,0,1,60,0,0,0,0,14,31,0,0,30,36],[44,16,0,0,1,60,0,0,0,0,0,60,0,0,1,17],[20,47,0,0,59,41,0,0,0,0,24,34,0,0,53,37] >;
C30.D5 in GAP, Magma, Sage, TeX
C_{30}.D_5 % in TeX
G:=Group("C30.D5"); // GroupNames label
G:=SmallGroup(300,20);
// by ID
G=gap.SmallGroup(300,20);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5,10,122,963,6004]);
// Polycyclic
G:=Group<a,b,c|a^30=b^5=1,c^2=a^15,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
Export