metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊15C4, C60.56D4, C30.18D8, C10.11D24, C12.49D20, C30.6SD16, C4⋊Dic3⋊1D5, C20.35(C4×S3), C6.6(D4⋊D5), (C2×C20).49D6, (C2×C30).16D4, C12.31(C4×D5), C5⋊2(C2.D24), C6.1(Q8⋊D5), C15⋊8(D4⋊C4), C3⋊1(D20⋊6C4), C60.106(C2×C4), (C2×D60).16C2, (C2×C10).30D12, C2.2(C5⋊D24), C10.4(C24⋊C2), C10.20(D6⋊C4), (C2×C12).284D10, C20.11(C3⋊D4), C4.14(C3⋊D20), C4.6(D30.C2), C6.5(D10⋊C4), C2.6(D30⋊4C4), C30.52(C22⋊C4), (C2×C60).128C22, C2.1(Dic6⋊D5), C22.13(C5⋊D12), (C6×C5⋊2C8)⋊5C2, (C2×C5⋊2C8)⋊2S3, (C5×C4⋊Dic3)⋊1C2, (C2×C4).134(S3×D5), (C2×C6).25(C5⋊D4), SmallGroup(480,45)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊15C4
G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a25b >
Subgroups: 796 in 100 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C8, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, D6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, C20, C20, D10, C2×C10, C24, D12, C2×Dic3, C2×C12, C22×S3, D15, C30, D4⋊C4, C5⋊2C8, D20, C2×C20, C2×C20, C22×D5, C4⋊Dic3, C2×C24, C2×D12, C5×Dic3, C60, D30, C2×C30, C2×C5⋊2C8, C5×C4⋊C4, C2×D20, C2.D24, C3×C5⋊2C8, C10×Dic3, D60, D60, C2×C60, C22×D15, D20⋊6C4, C6×C5⋊2C8, C5×C4⋊Dic3, C2×D60, D60⋊15C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D8, SD16, D10, C4×S3, D12, C3⋊D4, D4⋊C4, C4×D5, D20, C5⋊D4, C24⋊C2, D24, D6⋊C4, S3×D5, D10⋊C4, D4⋊D5, Q8⋊D5, C2.D24, D30.C2, C3⋊D20, C5⋊D12, D20⋊6C4, C5⋊D24, Dic6⋊D5, D30⋊4C4, D60⋊15C4
►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)
(1 165)(2 164)(3 163)(4 162)(5 161)(6 160)(7 159)(8 158)(9 157)(10 156)(11 155)(12 154)(13 153)(14 152)(15 151)(16 150)(17 149)(18 148)(19 147)(20 146)(21 145)(22 144)(23 143)(24 142)(25 141)(26 140)(27 139)(28 138)(29 137)(30 136)(31 135)(32 134)(33 133)(34 132)(35 131)(36 130)(37 129)(38 128)(39 127)(40 126)(41 125)(42 124)(43 123)(44 122)(45 121)(46 180)(47 179)(48 178)(49 177)(50 176)(51 175)(52 174)(53 173)(54 172)(55 171)(56 170)(57 169)(58 168)(59 167)(60 166)(61 185)(62 184)(63 183)(64 182)(65 181)(66 240)(67 239)(68 238)(69 237)(70 236)(71 235)(72 234)(73 233)(74 232)(75 231)(76 230)(77 229)(78 228)(79 227)(80 226)(81 225)(82 224)(83 223)(84 222)(85 221)(86 220)(87 219)(88 218)(89 217)(90 216)(91 215)(92 214)(93 213)(94 212)(95 211)(96 210)(97 209)(98 208)(99 207)(100 206)(101 205)(102 204)(103 203)(104 202)(105 201)(106 200)(107 199)(108 198)(109 197)(110 196)(111 195)(112 194)(113 193)(114 192)(115 191)(116 190)(117 189)(118 188)(119 187)(120 186)
(1 73 166 219)(2 84 167 230)(3 95 168 181)(4 106 169 192)(5 117 170 203)(6 68 171 214)(7 79 172 225)(8 90 173 236)(9 101 174 187)(10 112 175 198)(11 63 176 209)(12 74 177 220)(13 85 178 231)(14 96 179 182)(15 107 180 193)(16 118 121 204)(17 69 122 215)(18 80 123 226)(19 91 124 237)(20 102 125 188)(21 113 126 199)(22 64 127 210)(23 75 128 221)(24 86 129 232)(25 97 130 183)(26 108 131 194)(27 119 132 205)(28 70 133 216)(29 81 134 227)(30 92 135 238)(31 103 136 189)(32 114 137 200)(33 65 138 211)(34 76 139 222)(35 87 140 233)(36 98 141 184)(37 109 142 195)(38 120 143 206)(39 71 144 217)(40 82 145 228)(41 93 146 239)(42 104 147 190)(43 115 148 201)(44 66 149 212)(45 77 150 223)(46 88 151 234)(47 99 152 185)(48 110 153 196)(49 61 154 207)(50 72 155 218)(51 83 156 229)(52 94 157 240)(53 105 158 191)(54 116 159 202)(55 67 160 213)(56 78 161 224)(57 89 162 235)(58 100 163 186)(59 111 164 197)(60 62 165 208)
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), (1,165)(2,164)(3,163)(4,162)(5,161)(6,160)(7,159)(8,158)(9,157)(10,156)(11,155)(12,154)(13,153)(14,152)(15,151)(16,150)(17,149)(18,148)(19,147)(20,146)(21,145)(22,144)(23,143)(24,142)(25,141)(26,140)(27,139)(28,138)(29,137)(30,136)(31,135)(32,134)(33,133)(34,132)(35,131)(36,130)(37,129)(38,128)(39,127)(40,126)(41,125)(42,124)(43,123)(44,122)(45,121)(46,180)(47,179)(48,178)(49,177)(50,176)(51,175)(52,174)(53,173)(54,172)(55,171)(56,170)(57,169)(58,168)(59,167)(60,166)(61,185)(62,184)(63,183)(64,182)(65,181)(66,240)(67,239)(68,238)(69,237)(70,236)(71,235)(72,234)(73,233)(74,232)(75,231)(76,230)(77,229)(78,228)(79,227)(80,226)(81,225)(82,224)(83,223)(84,222)(85,221)(86,220)(87,219)(88,218)(89,217)(90,216)(91,215)(92,214)(93,213)(94,212)(95,211)(96,210)(97,209)(98,208)(99,207)(100,206)(101,205)(102,204)(103,203)(104,202)(105,201)(106,200)(107,199)(108,198)(109,197)(110,196)(111,195)(112,194)(113,193)(114,192)(115,191)(116,190)(117,189)(118,188)(119,187)(120,186), (1,73,166,219)(2,84,167,230)(3,95,168,181)(4,106,169,192)(5,117,170,203)(6,68,171,214)(7,79,172,225)(8,90,173,236)(9,101,174,187)(10,112,175,198)(11,63,176,209)(12,74,177,220)(13,85,178,231)(14,96,179,182)(15,107,180,193)(16,118,121,204)(17,69,122,215)(18,80,123,226)(19,91,124,237)(20,102,125,188)(21,113,126,199)(22,64,127,210)(23,75,128,221)(24,86,129,232)(25,97,130,183)(26,108,131,194)(27,119,132,205)(28,70,133,216)(29,81,134,227)(30,92,135,238)(31,103,136,189)(32,114,137,200)(33,65,138,211)(34,76,139,222)(35,87,140,233)(36,98,141,184)(37,109,142,195)(38,120,143,206)(39,71,144,217)(40,82,145,228)(41,93,146,239)(42,104,147,190)(43,115,148,201)(44,66,149,212)(45,77,150,223)(46,88,151,234)(47,99,152,185)(48,110,153,196)(49,61,154,207)(50,72,155,218)(51,83,156,229)(52,94,157,240)(53,105,158,191)(54,116,159,202)(55,67,160,213)(56,78,161,224)(57,89,162,235)(58,100,163,186)(59,111,164,197)(60,62,165,208)>;
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), (1,165)(2,164)(3,163)(4,162)(5,161)(6,160)(7,159)(8,158)(9,157)(10,156)(11,155)(12,154)(13,153)(14,152)(15,151)(16,150)(17,149)(18,148)(19,147)(20,146)(21,145)(22,144)(23,143)(24,142)(25,141)(26,140)(27,139)(28,138)(29,137)(30,136)(31,135)(32,134)(33,133)(34,132)(35,131)(36,130)(37,129)(38,128)(39,127)(40,126)(41,125)(42,124)(43,123)(44,122)(45,121)(46,180)(47,179)(48,178)(49,177)(50,176)(51,175)(52,174)(53,173)(54,172)(55,171)(56,170)(57,169)(58,168)(59,167)(60,166)(61,185)(62,184)(63,183)(64,182)(65,181)(66,240)(67,239)(68,238)(69,237)(70,236)(71,235)(72,234)(73,233)(74,232)(75,231)(76,230)(77,229)(78,228)(79,227)(80,226)(81,225)(82,224)(83,223)(84,222)(85,221)(86,220)(87,219)(88,218)(89,217)(90,216)(91,215)(92,214)(93,213)(94,212)(95,211)(96,210)(97,209)(98,208)(99,207)(100,206)(101,205)(102,204)(103,203)(104,202)(105,201)(106,200)(107,199)(108,198)(109,197)(110,196)(111,195)(112,194)(113,193)(114,192)(115,191)(116,190)(117,189)(118,188)(119,187)(120,186), (1,73,166,219)(2,84,167,230)(3,95,168,181)(4,106,169,192)(5,117,170,203)(6,68,171,214)(7,79,172,225)(8,90,173,236)(9,101,174,187)(10,112,175,198)(11,63,176,209)(12,74,177,220)(13,85,178,231)(14,96,179,182)(15,107,180,193)(16,118,121,204)(17,69,122,215)(18,80,123,226)(19,91,124,237)(20,102,125,188)(21,113,126,199)(22,64,127,210)(23,75,128,221)(24,86,129,232)(25,97,130,183)(26,108,131,194)(27,119,132,205)(28,70,133,216)(29,81,134,227)(30,92,135,238)(31,103,136,189)(32,114,137,200)(33,65,138,211)(34,76,139,222)(35,87,140,233)(36,98,141,184)(37,109,142,195)(38,120,143,206)(39,71,144,217)(40,82,145,228)(41,93,146,239)(42,104,147,190)(43,115,148,201)(44,66,149,212)(45,77,150,223)(46,88,151,234)(47,99,152,185)(48,110,153,196)(49,61,154,207)(50,72,155,218)(51,83,156,229)(52,94,157,240)(53,105,158,191)(54,116,159,202)(55,67,160,213)(56,78,161,224)(57,89,162,235)(58,100,163,186)(59,111,164,197)(60,62,165,208) );
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)], [(1,165),(2,164),(3,163),(4,162),(5,161),(6,160),(7,159),(8,158),(9,157),(10,156),(11,155),(12,154),(13,153),(14,152),(15,151),(16,150),(17,149),(18,148),(19,147),(20,146),(21,145),(22,144),(23,143),(24,142),(25,141),(26,140),(27,139),(28,138),(29,137),(30,136),(31,135),(32,134),(33,133),(34,132),(35,131),(36,130),(37,129),(38,128),(39,127),(40,126),(41,125),(42,124),(43,123),(44,122),(45,121),(46,180),(47,179),(48,178),(49,177),(50,176),(51,175),(52,174),(53,173),(54,172),(55,171),(56,170),(57,169),(58,168),(59,167),(60,166),(61,185),(62,184),(63,183),(64,182),(65,181),(66,240),(67,239),(68,238),(69,237),(70,236),(71,235),(72,234),(73,233),(74,232),(75,231),(76,230),(77,229),(78,228),(79,227),(80,226),(81,225),(82,224),(83,223),(84,222),(85,221),(86,220),(87,219),(88,218),(89,217),(90,216),(91,215),(92,214),(93,213),(94,212),(95,211),(96,210),(97,209),(98,208),(99,207),(100,206),(101,205),(102,204),(103,203),(104,202),(105,201),(106,200),(107,199),(108,198),(109,197),(110,196),(111,195),(112,194),(113,193),(114,192),(115,191),(116,190),(117,189),(118,188),(119,187),(120,186)], [(1,73,166,219),(2,84,167,230),(3,95,168,181),(4,106,169,192),(5,117,170,203),(6,68,171,214),(7,79,172,225),(8,90,173,236),(9,101,174,187),(10,112,175,198),(11,63,176,209),(12,74,177,220),(13,85,178,231),(14,96,179,182),(15,107,180,193),(16,118,121,204),(17,69,122,215),(18,80,123,226),(19,91,124,237),(20,102,125,188),(21,113,126,199),(22,64,127,210),(23,75,128,221),(24,86,129,232),(25,97,130,183),(26,108,131,194),(27,119,132,205),(28,70,133,216),(29,81,134,227),(30,92,135,238),(31,103,136,189),(32,114,137,200),(33,65,138,211),(34,76,139,222),(35,87,140,233),(36,98,141,184),(37,109,142,195),(38,120,143,206),(39,71,144,217),(40,82,145,228),(41,93,146,239),(42,104,147,190),(43,115,148,201),(44,66,149,212),(45,77,150,223),(46,88,151,234),(47,99,152,185),(48,110,153,196),(49,61,154,207),(50,72,155,218),(51,83,156,229),(52,94,157,240),(53,105,158,191),(54,116,159,202),(55,67,160,213),(56,78,161,224),(57,89,162,235),(58,100,163,186),(59,111,164,197),(60,62,165,208)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 60 | 60 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D5 | D6 | D8 | SD16 | D10 | C4×S3 | C3⋊D4 | D12 | C4×D5 | D20 | C5⋊D4 | C24⋊C2 | D24 | S3×D5 | D4⋊D5 | Q8⋊D5 | D30.C2 | C3⋊D20 | C5⋊D12 | C5⋊D24 | Dic6⋊D5 |
| kernel | D60⋊15C4 | C6×C5⋊2C8 | C5×C4⋊Dic3 | C2×D60 | D60 | C2×C5⋊2C8 | C60 | C2×C30 | C4⋊Dic3 | C2×C20 | C30 | C30 | C2×C12 | C20 | C20 | C2×C10 | C12 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of D60⋊15C4 ►in GL5(𝔽241)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 99 | 142 | 0 | 0 |
| 0 | 99 | 198 | 0 | 0 |
| 0 | 0 | 0 | 51 | 190 |
| 0 | 0 | 0 | 51 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 142 | 99 | 0 | 0 |
| 0 | 198 | 99 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 240 | 0 |
| 64 | 0 | 0 | 0 | 0 |
| 0 | 66 | 94 | 0 | 0 |
| 0 | 28 | 175 | 0 | 0 |
| 0 | 0 | 0 | 76 | 192 |
| 0 | 0 | 0 | 49 | 165 |
G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,99,99,0,0,0,142,198,0,0,0,0,0,51,51,0,0,0,190,1],[1,0,0,0,0,0,142,198,0,0,0,99,99,0,0,0,0,0,0,240,0,0,0,240,0],[64,0,0,0,0,0,66,28,0,0,0,94,175,0,0,0,0,0,76,49,0,0,0,192,165] >;
D60⋊15C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes_{15}C_4 % in TeX
G:=Group("D60:15C4"); // GroupNames label
G:=SmallGroup(480,45);
// by ID
G=gap.SmallGroup(480,45);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,197,64,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^25*b>;
// generators/relations