metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.46D4, D6⋊7Dic10, (S3×C10)⋊7Q8, C60⋊5C4⋊24C2, C5⋊4(D6⋊3Q8), C15⋊6(C22⋊Q8), C30.24(C2×Q8), C10.28(S3×Q8), (C2×Dic10)⋊3S3, (C6×Dic10)⋊5C2, C6.6(C4○D20), C30.113(C2×D4), (C2×C20).294D6, D6⋊Dic5.7C2, Dic15⋊5C4⋊8C2, C30.35(C4○D4), (C2×C12).115D10, C20.83(C3⋊D4), C3⋊4(C20.48D4), C4.25(C15⋊D4), C12.37(C5⋊D4), (C2×C30).59C23, (C2×Dic5).16D6, C2.12(S3×Dic10), C6.10(C2×Dic10), (C2×C60).113C22, C10.8(Q8⋊3S3), (C22×S3).67D10, C2.11(D60⋊C2), (C2×Dic3).141D10, (C6×Dic5).35C22, (C2×Dic15).58C22, (C10×Dic3).165C22, (S3×C2×C4).4D5, (S3×C2×C20).4C2, C6.82(C2×C5⋊D4), (C2×C4).104(S3×D5), C2.16(C2×C15⋊D4), C10.83(C2×C3⋊D4), C22.146(C2×S3×D5), (S3×C2×C10).79C22, (C2×C6).71(C22×D5), (C2×C10).71(C22×S3), SmallGroup(480,445)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.46D4
G = < a,b,c | a60=b4=1, c2=a30, bab-1=a-1, cac-1=a19, cbc-1=a30b-1 >
Subgroups: 604 in 148 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, Q8, C23, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C5×S3, C30, C22⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, Dic3⋊C4, C4⋊Dic3, D6⋊C4, S3×C2×C4, C6×Q8, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, S3×C10, C2×C30, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C22×C20, D6⋊3Q8, C3×Dic10, C6×Dic5, S3×C20, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C20.48D4, D6⋊Dic5, Dic15⋊5C4, C60⋊5C4, C6×Dic10, S3×C2×C20, C60.46D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, C3⋊D4, C22×S3, C22⋊Q8, Dic10, C5⋊D4, C22×D5, S3×Q8, Q8⋊3S3, C2×C3⋊D4, S3×D5, C2×Dic10, C4○D20, C2×C5⋊D4, D6⋊3Q8, C15⋊D4, C2×S3×D5, C20.48D4, S3×Dic10, D60⋊C2, C2×C15⋊D4, C60.46D4
►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)
(1 212 109 152)(2 211 110 151)(3 210 111 150)(4 209 112 149)(5 208 113 148)(6 207 114 147)(7 206 115 146)(8 205 116 145)(9 204 117 144)(10 203 118 143)(11 202 119 142)(12 201 120 141)(13 200 61 140)(14 199 62 139)(15 198 63 138)(16 197 64 137)(17 196 65 136)(18 195 66 135)(19 194 67 134)(20 193 68 133)(21 192 69 132)(22 191 70 131)(23 190 71 130)(24 189 72 129)(25 188 73 128)(26 187 74 127)(27 186 75 126)(28 185 76 125)(29 184 77 124)(30 183 78 123)(31 182 79 122)(32 181 80 121)(33 240 81 180)(34 239 82 179)(35 238 83 178)(36 237 84 177)(37 236 85 176)(38 235 86 175)(39 234 87 174)(40 233 88 173)(41 232 89 172)(42 231 90 171)(43 230 91 170)(44 229 92 169)(45 228 93 168)(46 227 94 167)(47 226 95 166)(48 225 96 165)(49 224 97 164)(50 223 98 163)(51 222 99 162)(52 221 100 161)(53 220 101 160)(54 219 102 159)(55 218 103 158)(56 217 104 157)(57 216 105 156)(58 215 106 155)(59 214 107 154)(60 213 108 153)
(1 122 31 152)(2 141 32 171)(3 160 33 130)(4 179 34 149)(5 138 35 168)(6 157 36 127)(7 176 37 146)(8 135 38 165)(9 154 39 124)(10 173 40 143)(11 132 41 162)(12 151 42 121)(13 170 43 140)(14 129 44 159)(15 148 45 178)(16 167 46 137)(17 126 47 156)(18 145 48 175)(19 164 49 134)(20 123 50 153)(21 142 51 172)(22 161 52 131)(23 180 53 150)(24 139 54 169)(25 158 55 128)(26 177 56 147)(27 136 57 166)(28 155 58 125)(29 174 59 144)(30 133 60 163)(61 230 91 200)(62 189 92 219)(63 208 93 238)(64 227 94 197)(65 186 95 216)(66 205 96 235)(67 224 97 194)(68 183 98 213)(69 202 99 232)(70 221 100 191)(71 240 101 210)(72 199 102 229)(73 218 103 188)(74 237 104 207)(75 196 105 226)(76 215 106 185)(77 234 107 204)(78 193 108 223)(79 212 109 182)(80 231 110 201)(81 190 111 220)(82 209 112 239)(83 228 113 198)(84 187 114 217)(85 206 115 236)(86 225 116 195)(87 184 117 214)(88 203 118 233)(89 222 119 192)(90 181 120 211)
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), (1,212,109,152)(2,211,110,151)(3,210,111,150)(4,209,112,149)(5,208,113,148)(6,207,114,147)(7,206,115,146)(8,205,116,145)(9,204,117,144)(10,203,118,143)(11,202,119,142)(12,201,120,141)(13,200,61,140)(14,199,62,139)(15,198,63,138)(16,197,64,137)(17,196,65,136)(18,195,66,135)(19,194,67,134)(20,193,68,133)(21,192,69,132)(22,191,70,131)(23,190,71,130)(24,189,72,129)(25,188,73,128)(26,187,74,127)(27,186,75,126)(28,185,76,125)(29,184,77,124)(30,183,78,123)(31,182,79,122)(32,181,80,121)(33,240,81,180)(34,239,82,179)(35,238,83,178)(36,237,84,177)(37,236,85,176)(38,235,86,175)(39,234,87,174)(40,233,88,173)(41,232,89,172)(42,231,90,171)(43,230,91,170)(44,229,92,169)(45,228,93,168)(46,227,94,167)(47,226,95,166)(48,225,96,165)(49,224,97,164)(50,223,98,163)(51,222,99,162)(52,221,100,161)(53,220,101,160)(54,219,102,159)(55,218,103,158)(56,217,104,157)(57,216,105,156)(58,215,106,155)(59,214,107,154)(60,213,108,153), (1,122,31,152)(2,141,32,171)(3,160,33,130)(4,179,34,149)(5,138,35,168)(6,157,36,127)(7,176,37,146)(8,135,38,165)(9,154,39,124)(10,173,40,143)(11,132,41,162)(12,151,42,121)(13,170,43,140)(14,129,44,159)(15,148,45,178)(16,167,46,137)(17,126,47,156)(18,145,48,175)(19,164,49,134)(20,123,50,153)(21,142,51,172)(22,161,52,131)(23,180,53,150)(24,139,54,169)(25,158,55,128)(26,177,56,147)(27,136,57,166)(28,155,58,125)(29,174,59,144)(30,133,60,163)(61,230,91,200)(62,189,92,219)(63,208,93,238)(64,227,94,197)(65,186,95,216)(66,205,96,235)(67,224,97,194)(68,183,98,213)(69,202,99,232)(70,221,100,191)(71,240,101,210)(72,199,102,229)(73,218,103,188)(74,237,104,207)(75,196,105,226)(76,215,106,185)(77,234,107,204)(78,193,108,223)(79,212,109,182)(80,231,110,201)(81,190,111,220)(82,209,112,239)(83,228,113,198)(84,187,114,217)(85,206,115,236)(86,225,116,195)(87,184,117,214)(88,203,118,233)(89,222,119,192)(90,181,120,211)>;
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), (1,212,109,152)(2,211,110,151)(3,210,111,150)(4,209,112,149)(5,208,113,148)(6,207,114,147)(7,206,115,146)(8,205,116,145)(9,204,117,144)(10,203,118,143)(11,202,119,142)(12,201,120,141)(13,200,61,140)(14,199,62,139)(15,198,63,138)(16,197,64,137)(17,196,65,136)(18,195,66,135)(19,194,67,134)(20,193,68,133)(21,192,69,132)(22,191,70,131)(23,190,71,130)(24,189,72,129)(25,188,73,128)(26,187,74,127)(27,186,75,126)(28,185,76,125)(29,184,77,124)(30,183,78,123)(31,182,79,122)(32,181,80,121)(33,240,81,180)(34,239,82,179)(35,238,83,178)(36,237,84,177)(37,236,85,176)(38,235,86,175)(39,234,87,174)(40,233,88,173)(41,232,89,172)(42,231,90,171)(43,230,91,170)(44,229,92,169)(45,228,93,168)(46,227,94,167)(47,226,95,166)(48,225,96,165)(49,224,97,164)(50,223,98,163)(51,222,99,162)(52,221,100,161)(53,220,101,160)(54,219,102,159)(55,218,103,158)(56,217,104,157)(57,216,105,156)(58,215,106,155)(59,214,107,154)(60,213,108,153), (1,122,31,152)(2,141,32,171)(3,160,33,130)(4,179,34,149)(5,138,35,168)(6,157,36,127)(7,176,37,146)(8,135,38,165)(9,154,39,124)(10,173,40,143)(11,132,41,162)(12,151,42,121)(13,170,43,140)(14,129,44,159)(15,148,45,178)(16,167,46,137)(17,126,47,156)(18,145,48,175)(19,164,49,134)(20,123,50,153)(21,142,51,172)(22,161,52,131)(23,180,53,150)(24,139,54,169)(25,158,55,128)(26,177,56,147)(27,136,57,166)(28,155,58,125)(29,174,59,144)(30,133,60,163)(61,230,91,200)(62,189,92,219)(63,208,93,238)(64,227,94,197)(65,186,95,216)(66,205,96,235)(67,224,97,194)(68,183,98,213)(69,202,99,232)(70,221,100,191)(71,240,101,210)(72,199,102,229)(73,218,103,188)(74,237,104,207)(75,196,105,226)(76,215,106,185)(77,234,107,204)(78,193,108,223)(79,212,109,182)(80,231,110,201)(81,190,111,220)(82,209,112,239)(83,228,113,198)(84,187,114,217)(85,206,115,236)(86,225,116,195)(87,184,117,214)(88,203,118,233)(89,222,119,192)(90,181,120,211) );
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)], [(1,212,109,152),(2,211,110,151),(3,210,111,150),(4,209,112,149),(5,208,113,148),(6,207,114,147),(7,206,115,146),(8,205,116,145),(9,204,117,144),(10,203,118,143),(11,202,119,142),(12,201,120,141),(13,200,61,140),(14,199,62,139),(15,198,63,138),(16,197,64,137),(17,196,65,136),(18,195,66,135),(19,194,67,134),(20,193,68,133),(21,192,69,132),(22,191,70,131),(23,190,71,130),(24,189,72,129),(25,188,73,128),(26,187,74,127),(27,186,75,126),(28,185,76,125),(29,184,77,124),(30,183,78,123),(31,182,79,122),(32,181,80,121),(33,240,81,180),(34,239,82,179),(35,238,83,178),(36,237,84,177),(37,236,85,176),(38,235,86,175),(39,234,87,174),(40,233,88,173),(41,232,89,172),(42,231,90,171),(43,230,91,170),(44,229,92,169),(45,228,93,168),(46,227,94,167),(47,226,95,166),(48,225,96,165),(49,224,97,164),(50,223,98,163),(51,222,99,162),(52,221,100,161),(53,220,101,160),(54,219,102,159),(55,218,103,158),(56,217,104,157),(57,216,105,156),(58,215,106,155),(59,214,107,154),(60,213,108,153)], [(1,122,31,152),(2,141,32,171),(3,160,33,130),(4,179,34,149),(5,138,35,168),(6,157,36,127),(7,176,37,146),(8,135,38,165),(9,154,39,124),(10,173,40,143),(11,132,41,162),(12,151,42,121),(13,170,43,140),(14,129,44,159),(15,148,45,178),(16,167,46,137),(17,126,47,156),(18,145,48,175),(19,164,49,134),(20,123,50,153),(21,142,51,172),(22,161,52,131),(23,180,53,150),(24,139,54,169),(25,158,55,128),(26,177,56,147),(27,136,57,166),(28,155,58,125),(29,174,59,144),(30,133,60,163),(61,230,91,200),(62,189,92,219),(63,208,93,238),(64,227,94,197),(65,186,95,216),(66,205,96,235),(67,224,97,194),(68,183,98,213),(69,202,99,232),(70,221,100,191),(71,240,101,210),(72,199,102,229),(73,218,103,188),(74,237,104,207),(75,196,105,226),(76,215,106,185),(77,234,107,204),(78,193,108,223),(79,212,109,182),(80,231,110,201),(81,190,111,220),(82,209,112,239),(83,228,113,198),(84,187,114,217),(85,206,115,236),(86,225,116,195),(87,184,117,214),(88,203,118,233),(89,222,119,192),(90,181,120,211)]])
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 | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 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 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 6 | 6 | 20 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | + | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C3⋊D4 | C5⋊D4 | Dic10 | C4○D20 | S3×Q8 | Q8⋊3S3 | S3×D5 | C15⋊D4 | C2×S3×D5 | S3×Dic10 | D60⋊C2 |
| kernel | C60.46D4 | D6⋊Dic5 | Dic15⋊5C4 | C60⋊5C4 | C6×Dic10 | S3×C2×C20 | C2×Dic10 | C60 | S3×C10 | S3×C2×C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | C20 | C12 | D6 | C6 | C10 | C10 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C60.46D4 ►in GL4(𝔽61) generated by
| 0 | 1 | 0 | 0 |
| 60 | 60 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 22 | 8 |
| 52 | 43 | 0 | 0 |
| 52 | 9 | 0 | 0 |
| 0 | 0 | 24 | 28 |
| 0 | 0 | 60 | 37 |
| 52 | 43 | 0 | 0 |
| 18 | 9 | 0 | 0 |
| 0 | 0 | 24 | 28 |
| 0 | 0 | 60 | 37 |
G:=sub<GL(4,GF(61))| [0,60,0,0,1,60,0,0,0,0,23,22,0,0,0,8],[52,52,0,0,43,9,0,0,0,0,24,60,0,0,28,37],[52,18,0,0,43,9,0,0,0,0,24,60,0,0,28,37] >;
C60.46D4 in GAP, Magma, Sage, TeX
C_{60}._{46}D_4 % in TeX
G:=Group("C60.46D4"); // GroupNames label
G:=SmallGroup(480,445);
// by ID
G=gap.SmallGroup(480,445);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^30*b^-1>;
// generators/relations