metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.47D4, C12.20D20, (C4×Dic3)⋊4D5, C6.57(C2×D20), D30⋊4C4⋊8C2, (Dic3×C20)⋊4C2, (C6×Dic10)⋊6C2, (C2×Dic10)⋊4S3, (C2×D60).14C2, C6.7(C4○D20), C30.116(C2×D4), (C2×C20).295D6, C15⋊8(C4.4D4), C3⋊3(C4.D20), C30.40(C4○D4), (C2×C12).117D10, C4.10(C3⋊D20), C20.58(C3⋊D4), C5⋊1(C12.23D4), (C2×C30).64C23, (C2×Dic5).18D6, (C2×C60).114C22, C10.9(Q8⋊3S3), C2.12(D60⋊C2), (C2×Dic3).143D10, (C6×Dic5).37C22, (C22×D15).23C22, (C10×Dic3).167C22, (C2×C4).105(S3×D5), C10.12(C2×C3⋊D4), C2.16(C2×C3⋊D20), C22.150(C2×S3×D5), (C2×C6).76(C22×D5), (C2×C10).76(C22×S3), SmallGroup(480,450)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.47D4
G = < a,b,c | a60=b4=c2=1, bab-1=a41, cac=a-1, cbc=a30b-1 >
Subgroups: 988 in 152 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, D15, C30, C30, C4.4D4, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, D6⋊C4, C2×D12, C6×Q8, C5×Dic3, C3×Dic5, C60, D30, C2×C30, D10⋊C4, C4×C20, C2×Dic10, C2×D20, C12.23D4, C3×Dic10, C6×Dic5, C10×Dic3, D60, C2×C60, C22×D15, C4.D20, D30⋊4C4, Dic3×C20, C6×Dic10, C2×D60, C60.47D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4.4D4, D20, C22×D5, Q8⋊3S3, C2×C3⋊D4, S3×D5, C2×D20, C4○D20, C12.23D4, C3⋊D20, C2×S3×D5, C4.D20, D60⋊C2, C2×C3⋊D20, C60.47D4
►On 240 points
Generators in S240
(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)
(1 104 218 162)(2 85 219 143)(3 66 220 124)(4 107 221 165)(5 88 222 146)(6 69 223 127)(7 110 224 168)(8 91 225 149)(9 72 226 130)(10 113 227 171)(11 94 228 152)(12 75 229 133)(13 116 230 174)(14 97 231 155)(15 78 232 136)(16 119 233 177)(17 100 234 158)(18 81 235 139)(19 62 236 180)(20 103 237 161)(21 84 238 142)(22 65 239 123)(23 106 240 164)(24 87 181 145)(25 68 182 126)(26 109 183 167)(27 90 184 148)(28 71 185 129)(29 112 186 170)(30 93 187 151)(31 74 188 132)(32 115 189 173)(33 96 190 154)(34 77 191 135)(35 118 192 176)(36 99 193 157)(37 80 194 138)(38 61 195 179)(39 102 196 160)(40 83 197 141)(41 64 198 122)(42 105 199 163)(43 86 200 144)(44 67 201 125)(45 108 202 166)(46 89 203 147)(47 70 204 128)(48 111 205 169)(49 92 206 150)(50 73 207 131)(51 114 208 172)(52 95 209 153)(53 76 210 134)(54 117 211 175)(55 98 212 156)(56 79 213 137)(57 120 214 178)(58 101 215 159)(59 82 216 140)(60 63 217 121)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(61 130)(62 129)(63 128)(64 127)(65 126)(66 125)(67 124)(68 123)(69 122)(70 121)(71 180)(72 179)(73 178)(74 177)(75 176)(76 175)(77 174)(78 173)(79 172)(80 171)(81 170)(82 169)(83 168)(84 167)(85 166)(86 165)(87 164)(88 163)(89 162)(90 161)(91 160)(92 159)(93 158)(94 157)(95 156)(96 155)(97 154)(98 153)(99 152)(100 151)(101 150)(102 149)(103 148)(104 147)(105 146)(106 145)(107 144)(108 143)(109 142)(110 141)(111 140)(112 139)(113 138)(114 137)(115 136)(116 135)(117 134)(118 133)(119 132)(120 131)(181 210)(182 209)(183 208)(184 207)(185 206)(186 205)(187 204)(188 203)(189 202)(190 201)(191 200)(192 199)(193 198)(194 197)(195 196)(211 240)(212 239)(213 238)(214 237)(215 236)(216 235)(217 234)(218 233)(219 232)(220 231)(221 230)(222 229)(223 228)(224 227)(225 226)
G:=sub<Sym(240)| (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), (1,104,218,162)(2,85,219,143)(3,66,220,124)(4,107,221,165)(5,88,222,146)(6,69,223,127)(7,110,224,168)(8,91,225,149)(9,72,226,130)(10,113,227,171)(11,94,228,152)(12,75,229,133)(13,116,230,174)(14,97,231,155)(15,78,232,136)(16,119,233,177)(17,100,234,158)(18,81,235,139)(19,62,236,180)(20,103,237,161)(21,84,238,142)(22,65,239,123)(23,106,240,164)(24,87,181,145)(25,68,182,126)(26,109,183,167)(27,90,184,148)(28,71,185,129)(29,112,186,170)(30,93,187,151)(31,74,188,132)(32,115,189,173)(33,96,190,154)(34,77,191,135)(35,118,192,176)(36,99,193,157)(37,80,194,138)(38,61,195,179)(39,102,196,160)(40,83,197,141)(41,64,198,122)(42,105,199,163)(43,86,200,144)(44,67,201,125)(45,108,202,166)(46,89,203,147)(47,70,204,128)(48,111,205,169)(49,92,206,150)(50,73,207,131)(51,114,208,172)(52,95,209,153)(53,76,210,134)(54,117,211,175)(55,98,212,156)(56,79,213,137)(57,120,214,178)(58,101,215,159)(59,82,216,140)(60,63,217,121), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(61,130)(62,129)(63,128)(64,127)(65,126)(66,125)(67,124)(68,123)(69,122)(70,121)(71,180)(72,179)(73,178)(74,177)(75,176)(76,175)(77,174)(78,173)(79,172)(80,171)(81,170)(82,169)(83,168)(84,167)(85,166)(86,165)(87,164)(88,163)(89,162)(90,161)(91,160)(92,159)(93,158)(94,157)(95,156)(96,155)(97,154)(98,153)(99,152)(100,151)(101,150)(102,149)(103,148)(104,147)(105,146)(106,145)(107,144)(108,143)(109,142)(110,141)(111,140)(112,139)(113,138)(114,137)(115,136)(116,135)(117,134)(118,133)(119,132)(120,131)(181,210)(182,209)(183,208)(184,207)(185,206)(186,205)(187,204)(188,203)(189,202)(190,201)(191,200)(192,199)(193,198)(194,197)(195,196)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(217,234)(218,233)(219,232)(220,231)(221,230)(222,229)(223,228)(224,227)(225,226)>;
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), (1,104,218,162)(2,85,219,143)(3,66,220,124)(4,107,221,165)(5,88,222,146)(6,69,223,127)(7,110,224,168)(8,91,225,149)(9,72,226,130)(10,113,227,171)(11,94,228,152)(12,75,229,133)(13,116,230,174)(14,97,231,155)(15,78,232,136)(16,119,233,177)(17,100,234,158)(18,81,235,139)(19,62,236,180)(20,103,237,161)(21,84,238,142)(22,65,239,123)(23,106,240,164)(24,87,181,145)(25,68,182,126)(26,109,183,167)(27,90,184,148)(28,71,185,129)(29,112,186,170)(30,93,187,151)(31,74,188,132)(32,115,189,173)(33,96,190,154)(34,77,191,135)(35,118,192,176)(36,99,193,157)(37,80,194,138)(38,61,195,179)(39,102,196,160)(40,83,197,141)(41,64,198,122)(42,105,199,163)(43,86,200,144)(44,67,201,125)(45,108,202,166)(46,89,203,147)(47,70,204,128)(48,111,205,169)(49,92,206,150)(50,73,207,131)(51,114,208,172)(52,95,209,153)(53,76,210,134)(54,117,211,175)(55,98,212,156)(56,79,213,137)(57,120,214,178)(58,101,215,159)(59,82,216,140)(60,63,217,121), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(61,130)(62,129)(63,128)(64,127)(65,126)(66,125)(67,124)(68,123)(69,122)(70,121)(71,180)(72,179)(73,178)(74,177)(75,176)(76,175)(77,174)(78,173)(79,172)(80,171)(81,170)(82,169)(83,168)(84,167)(85,166)(86,165)(87,164)(88,163)(89,162)(90,161)(91,160)(92,159)(93,158)(94,157)(95,156)(96,155)(97,154)(98,153)(99,152)(100,151)(101,150)(102,149)(103,148)(104,147)(105,146)(106,145)(107,144)(108,143)(109,142)(110,141)(111,140)(112,139)(113,138)(114,137)(115,136)(116,135)(117,134)(118,133)(119,132)(120,131)(181,210)(182,209)(183,208)(184,207)(185,206)(186,205)(187,204)(188,203)(189,202)(190,201)(191,200)(192,199)(193,198)(194,197)(195,196)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(217,234)(218,233)(219,232)(220,231)(221,230)(222,229)(223,228)(224,227)(225,226) );
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)], [(1,104,218,162),(2,85,219,143),(3,66,220,124),(4,107,221,165),(5,88,222,146),(6,69,223,127),(7,110,224,168),(8,91,225,149),(9,72,226,130),(10,113,227,171),(11,94,228,152),(12,75,229,133),(13,116,230,174),(14,97,231,155),(15,78,232,136),(16,119,233,177),(17,100,234,158),(18,81,235,139),(19,62,236,180),(20,103,237,161),(21,84,238,142),(22,65,239,123),(23,106,240,164),(24,87,181,145),(25,68,182,126),(26,109,183,167),(27,90,184,148),(28,71,185,129),(29,112,186,170),(30,93,187,151),(31,74,188,132),(32,115,189,173),(33,96,190,154),(34,77,191,135),(35,118,192,176),(36,99,193,157),(37,80,194,138),(38,61,195,179),(39,102,196,160),(40,83,197,141),(41,64,198,122),(42,105,199,163),(43,86,200,144),(44,67,201,125),(45,108,202,166),(46,89,203,147),(47,70,204,128),(48,111,205,169),(49,92,206,150),(50,73,207,131),(51,114,208,172),(52,95,209,153),(53,76,210,134),(54,117,211,175),(55,98,212,156),(56,79,213,137),(57,120,214,178),(58,101,215,159),(59,82,216,140),(60,63,217,121)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(61,130),(62,129),(63,128),(64,127),(65,126),(66,125),(67,124),(68,123),(69,122),(70,121),(71,180),(72,179),(73,178),(74,177),(75,176),(76,175),(77,174),(78,173),(79,172),(80,171),(81,170),(82,169),(83,168),(84,167),(85,166),(86,165),(87,164),(88,163),(89,162),(90,161),(91,160),(92,159),(93,158),(94,157),(95,156),(96,155),(97,154),(98,153),(99,152),(100,151),(101,150),(102,149),(103,148),(104,147),(105,146),(106,145),(107,144),(108,143),(109,142),(110,141),(111,140),(112,139),(113,138),(114,137),(115,136),(116,135),(117,134),(118,133),(119,132),(120,131),(181,210),(182,209),(183,208),(184,207),(185,206),(186,205),(187,204),(188,203),(189,202),(190,201),(191,200),(192,199),(193,198),(194,197),(195,196),(211,240),(212,239),(213,238),(214,237),(215,236),(216,235),(217,234),(218,233),(219,232),(220,231),(221,230),(222,229),(223,228),(224,227),(225,226)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 60 | 60 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | D20 | C4○D20 | Q8⋊3S3 | S3×D5 | C3⋊D20 | C2×S3×D5 | D60⋊C2 |
| kernel | C60.47D4 | D30⋊4C4 | Dic3×C20 | C6×Dic10 | C2×D60 | C2×Dic10 | C60 | C4×Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C20 | C12 | C6 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 8 | 16 | 2 | 2 | 4 | 2 | 8 |
Matrix representation of C60.47D4 ►in GL4(𝔽61) generated by
| 29 | 59 | 0 | 0 |
| 2 | 2 | 0 | 0 |
| 0 | 0 | 2 | 49 |
| 0 | 0 | 46 | 60 |
| 47 | 45 | 0 | 0 |
| 16 | 14 | 0 | 0 |
| 0 | 0 | 27 | 14 |
| 0 | 0 | 35 | 34 |
| 29 | 59 | 0 | 0 |
| 54 | 32 | 0 | 0 |
| 0 | 0 | 60 | 12 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(61))| [29,2,0,0,59,2,0,0,0,0,2,46,0,0,49,60],[47,16,0,0,45,14,0,0,0,0,27,35,0,0,14,34],[29,54,0,0,59,32,0,0,0,0,60,0,0,0,12,1] >;
C60.47D4 in GAP, Magma, Sage, TeX
C_{60}._{47}D_4 % in TeX
G:=Group("C60.47D4"); // GroupNames label
G:=SmallGroup(480,450);
// by ID
G=gap.SmallGroup(480,450);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,176,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^41,c*a*c=a^-1,c*b*c=a^30*b^-1>;
// generators/relations