metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊11C4, Dic15⋊20D4, C4⋊C4⋊8D15, C4⋊1(C4×D15), C12⋊6(C4×D5), C15⋊37(C4×D4), C20⋊10(C4×S3), C60⋊12(C2×C4), C2.4(D4×D15), D30⋊20(C2×C4), (C2×D60).8C2, (C2×C4).30D30, C6.104(D4×D5), C3⋊3(D20⋊8C4), C5⋊4(Dic3⋊5D4), (C4×Dic15)⋊3C2, (C2×C20).211D6, C10.106(S3×D4), C30.312(C2×D4), D30⋊3C4⋊12C2, (C2×C12).209D10, C30.258(C4○D4), C2.2(Q8⋊3D15), (C2×C60).178C22, C30.162(C22×C4), (C2×C30).290C23, C6.40(Q8⋊2D5), C10.40(Q8⋊3S3), C22.18(C22×D15), (C22×D15).82C22, (C2×Dic15).239C22, (C5×C4⋊C4)⋊4S3, (C3×C4⋊C4)⋊4D5, (C15×C4⋊C4)⋊4C2, C6.67(C2×C4×D5), (C2×C4×D15)⋊18C2, C10.99(S3×C2×C4), C2.13(C2×C4×D15), (C2×C6).286(C22×D5), (C2×C10).285(C22×S3), SmallGroup(480,858)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊11C4
G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a31, cbc-1=a30b >
Subgroups: 1188 in 188 conjugacy classes, 65 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C4×D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, Dic15, Dic15, C60, C60, D30, D30, C2×C30, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, Dic3⋊5D4, C4×D15, D60, C2×Dic15, C2×C60, C2×C60, C22×D15, D20⋊8C4, C4×Dic15, D30⋊3C4, C15×C4⋊C4, C2×C4×D15, C2×D60, D60⋊11C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, C22×S3, D15, C4×D4, C4×D5, C22×D5, S3×C2×C4, S3×D4, Q8⋊3S3, D30, C2×C4×D5, D4×D5, Q8⋊2D5, Dic3⋊5D4, C4×D15, C22×D15, D20⋊8C4, C2×C4×D15, D4×D15, Q8⋊3D15, D60⋊11C4
►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)
(1 121)(2 180)(3 179)(4 178)(5 177)(6 176)(7 175)(8 174)(9 173)(10 172)(11 171)(12 170)(13 169)(14 168)(15 167)(16 166)(17 165)(18 164)(19 163)(20 162)(21 161)(22 160)(23 159)(24 158)(25 157)(26 156)(27 155)(28 154)(29 153)(30 152)(31 151)(32 150)(33 149)(34 148)(35 147)(36 146)(37 145)(38 144)(39 143)(40 142)(41 141)(42 140)(43 139)(44 138)(45 137)(46 136)(47 135)(48 134)(49 133)(50 132)(51 131)(52 130)(53 129)(54 128)(55 127)(56 126)(57 125)(58 124)(59 123)(60 122)(61 235)(62 234)(63 233)(64 232)(65 231)(66 230)(67 229)(68 228)(69 227)(70 226)(71 225)(72 224)(73 223)(74 222)(75 221)(76 220)(77 219)(78 218)(79 217)(80 216)(81 215)(82 214)(83 213)(84 212)(85 211)(86 210)(87 209)(88 208)(89 207)(90 206)(91 205)(92 204)(93 203)(94 202)(95 201)(96 200)(97 199)(98 198)(99 197)(100 196)(101 195)(102 194)(103 193)(104 192)(105 191)(106 190)(107 189)(108 188)(109 187)(110 186)(111 185)(112 184)(113 183)(114 182)(115 181)(116 240)(117 239)(118 238)(119 237)(120 236)
(1 80 122 217)(2 111 123 188)(3 82 124 219)(4 113 125 190)(5 84 126 221)(6 115 127 192)(7 86 128 223)(8 117 129 194)(9 88 130 225)(10 119 131 196)(11 90 132 227)(12 61 133 198)(13 92 134 229)(14 63 135 200)(15 94 136 231)(16 65 137 202)(17 96 138 233)(18 67 139 204)(19 98 140 235)(20 69 141 206)(21 100 142 237)(22 71 143 208)(23 102 144 239)(24 73 145 210)(25 104 146 181)(26 75 147 212)(27 106 148 183)(28 77 149 214)(29 108 150 185)(30 79 151 216)(31 110 152 187)(32 81 153 218)(33 112 154 189)(34 83 155 220)(35 114 156 191)(36 85 157 222)(37 116 158 193)(38 87 159 224)(39 118 160 195)(40 89 161 226)(41 120 162 197)(42 91 163 228)(43 62 164 199)(44 93 165 230)(45 64 166 201)(46 95 167 232)(47 66 168 203)(48 97 169 234)(49 68 170 205)(50 99 171 236)(51 70 172 207)(52 101 173 238)(53 72 174 209)(54 103 175 240)(55 74 176 211)(56 105 177 182)(57 76 178 213)(58 107 179 184)(59 78 180 215)(60 109 121 186)
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), (1,121)(2,180)(3,179)(4,178)(5,177)(6,176)(7,175)(8,174)(9,173)(10,172)(11,171)(12,170)(13,169)(14,168)(15,167)(16,166)(17,165)(18,164)(19,163)(20,162)(21,161)(22,160)(23,159)(24,158)(25,157)(26,156)(27,155)(28,154)(29,153)(30,152)(31,151)(32,150)(33,149)(34,148)(35,147)(36,146)(37,145)(38,144)(39,143)(40,142)(41,141)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,133)(50,132)(51,131)(52,130)(53,129)(54,128)(55,127)(56,126)(57,125)(58,124)(59,123)(60,122)(61,235)(62,234)(63,233)(64,232)(65,231)(66,230)(67,229)(68,228)(69,227)(70,226)(71,225)(72,224)(73,223)(74,222)(75,221)(76,220)(77,219)(78,218)(79,217)(80,216)(81,215)(82,214)(83,213)(84,212)(85,211)(86,210)(87,209)(88,208)(89,207)(90,206)(91,205)(92,204)(93,203)(94,202)(95,201)(96,200)(97,199)(98,198)(99,197)(100,196)(101,195)(102,194)(103,193)(104,192)(105,191)(106,190)(107,189)(108,188)(109,187)(110,186)(111,185)(112,184)(113,183)(114,182)(115,181)(116,240)(117,239)(118,238)(119,237)(120,236), (1,80,122,217)(2,111,123,188)(3,82,124,219)(4,113,125,190)(5,84,126,221)(6,115,127,192)(7,86,128,223)(8,117,129,194)(9,88,130,225)(10,119,131,196)(11,90,132,227)(12,61,133,198)(13,92,134,229)(14,63,135,200)(15,94,136,231)(16,65,137,202)(17,96,138,233)(18,67,139,204)(19,98,140,235)(20,69,141,206)(21,100,142,237)(22,71,143,208)(23,102,144,239)(24,73,145,210)(25,104,146,181)(26,75,147,212)(27,106,148,183)(28,77,149,214)(29,108,150,185)(30,79,151,216)(31,110,152,187)(32,81,153,218)(33,112,154,189)(34,83,155,220)(35,114,156,191)(36,85,157,222)(37,116,158,193)(38,87,159,224)(39,118,160,195)(40,89,161,226)(41,120,162,197)(42,91,163,228)(43,62,164,199)(44,93,165,230)(45,64,166,201)(46,95,167,232)(47,66,168,203)(48,97,169,234)(49,68,170,205)(50,99,171,236)(51,70,172,207)(52,101,173,238)(53,72,174,209)(54,103,175,240)(55,74,176,211)(56,105,177,182)(57,76,178,213)(58,107,179,184)(59,78,180,215)(60,109,121,186)>;
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), (1,121)(2,180)(3,179)(4,178)(5,177)(6,176)(7,175)(8,174)(9,173)(10,172)(11,171)(12,170)(13,169)(14,168)(15,167)(16,166)(17,165)(18,164)(19,163)(20,162)(21,161)(22,160)(23,159)(24,158)(25,157)(26,156)(27,155)(28,154)(29,153)(30,152)(31,151)(32,150)(33,149)(34,148)(35,147)(36,146)(37,145)(38,144)(39,143)(40,142)(41,141)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,133)(50,132)(51,131)(52,130)(53,129)(54,128)(55,127)(56,126)(57,125)(58,124)(59,123)(60,122)(61,235)(62,234)(63,233)(64,232)(65,231)(66,230)(67,229)(68,228)(69,227)(70,226)(71,225)(72,224)(73,223)(74,222)(75,221)(76,220)(77,219)(78,218)(79,217)(80,216)(81,215)(82,214)(83,213)(84,212)(85,211)(86,210)(87,209)(88,208)(89,207)(90,206)(91,205)(92,204)(93,203)(94,202)(95,201)(96,200)(97,199)(98,198)(99,197)(100,196)(101,195)(102,194)(103,193)(104,192)(105,191)(106,190)(107,189)(108,188)(109,187)(110,186)(111,185)(112,184)(113,183)(114,182)(115,181)(116,240)(117,239)(118,238)(119,237)(120,236), (1,80,122,217)(2,111,123,188)(3,82,124,219)(4,113,125,190)(5,84,126,221)(6,115,127,192)(7,86,128,223)(8,117,129,194)(9,88,130,225)(10,119,131,196)(11,90,132,227)(12,61,133,198)(13,92,134,229)(14,63,135,200)(15,94,136,231)(16,65,137,202)(17,96,138,233)(18,67,139,204)(19,98,140,235)(20,69,141,206)(21,100,142,237)(22,71,143,208)(23,102,144,239)(24,73,145,210)(25,104,146,181)(26,75,147,212)(27,106,148,183)(28,77,149,214)(29,108,150,185)(30,79,151,216)(31,110,152,187)(32,81,153,218)(33,112,154,189)(34,83,155,220)(35,114,156,191)(36,85,157,222)(37,116,158,193)(38,87,159,224)(39,118,160,195)(40,89,161,226)(41,120,162,197)(42,91,163,228)(43,62,164,199)(44,93,165,230)(45,64,166,201)(46,95,167,232)(47,66,168,203)(48,97,169,234)(49,68,170,205)(50,99,171,236)(51,70,172,207)(52,101,173,238)(53,72,174,209)(54,103,175,240)(55,74,176,211)(56,105,177,182)(57,76,178,213)(58,107,179,184)(59,78,180,215)(60,109,121,186) );
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)], [(1,121),(2,180),(3,179),(4,178),(5,177),(6,176),(7,175),(8,174),(9,173),(10,172),(11,171),(12,170),(13,169),(14,168),(15,167),(16,166),(17,165),(18,164),(19,163),(20,162),(21,161),(22,160),(23,159),(24,158),(25,157),(26,156),(27,155),(28,154),(29,153),(30,152),(31,151),(32,150),(33,149),(34,148),(35,147),(36,146),(37,145),(38,144),(39,143),(40,142),(41,141),(42,140),(43,139),(44,138),(45,137),(46,136),(47,135),(48,134),(49,133),(50,132),(51,131),(52,130),(53,129),(54,128),(55,127),(56,126),(57,125),(58,124),(59,123),(60,122),(61,235),(62,234),(63,233),(64,232),(65,231),(66,230),(67,229),(68,228),(69,227),(70,226),(71,225),(72,224),(73,223),(74,222),(75,221),(76,220),(77,219),(78,218),(79,217),(80,216),(81,215),(82,214),(83,213),(84,212),(85,211),(86,210),(87,209),(88,208),(89,207),(90,206),(91,205),(92,204),(93,203),(94,202),(95,201),(96,200),(97,199),(98,198),(99,197),(100,196),(101,195),(102,194),(103,193),(104,192),(105,191),(106,190),(107,189),(108,188),(109,187),(110,186),(111,185),(112,184),(113,183),(114,182),(115,181),(116,240),(117,239),(118,238),(119,237),(120,236)], [(1,80,122,217),(2,111,123,188),(3,82,124,219),(4,113,125,190),(5,84,126,221),(6,115,127,192),(7,86,128,223),(8,117,129,194),(9,88,130,225),(10,119,131,196),(11,90,132,227),(12,61,133,198),(13,92,134,229),(14,63,135,200),(15,94,136,231),(16,65,137,202),(17,96,138,233),(18,67,139,204),(19,98,140,235),(20,69,141,206),(21,100,142,237),(22,71,143,208),(23,102,144,239),(24,73,145,210),(25,104,146,181),(26,75,147,212),(27,106,148,183),(28,77,149,214),(29,108,150,185),(30,79,151,216),(31,110,152,187),(32,81,153,218),(33,112,154,189),(34,83,155,220),(35,114,156,191),(36,85,157,222),(37,116,158,193),(38,87,159,224),(39,118,160,195),(40,89,161,226),(41,120,162,197),(42,91,163,228),(43,62,164,199),(44,93,165,230),(45,64,166,201),(46,95,167,232),(47,66,168,203),(48,97,169,234),(49,68,170,205),(50,99,171,236),(51,70,172,207),(52,101,173,238),(53,72,174,209),(54,103,175,240),(55,74,176,211),(56,105,177,182),(57,76,178,213),(58,107,179,184),(59,78,180,215),(60,109,121,186)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 30 | 30 | 2 | 2 | ··· | 2 | 15 | 15 | 15 | 15 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | C4○D4 | D10 | C4×S3 | D15 | C4×D5 | D30 | C4×D15 | S3×D4 | Q8⋊3S3 | D4×D5 | Q8⋊2D5 | D4×D15 | Q8⋊3D15 |
| kernel | D60⋊11C4 | C4×Dic15 | D30⋊3C4 | C15×C4⋊C4 | C2×C4×D15 | C2×D60 | D60 | C5×C4⋊C4 | Dic15 | C3×C4⋊C4 | C2×C20 | C30 | C2×C12 | C20 | C4⋊C4 | C12 | C2×C4 | C4 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 3 | 2 | 6 | 4 | 4 | 8 | 12 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D60⋊11C4 ►in GL6(𝔽61)
| 31 | 14 | 0 | 0 | 0 | 0 |
| 47 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 16 |
| 0 | 0 | 0 | 0 | 18 | 4 |
| 33 | 37 | 0 | 0 | 0 | 0 |
| 25 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 16 |
| 0 | 0 | 0 | 0 | 41 | 4 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 30 | 1 |
G:=sub<GL(6,GF(61))| [31,47,0,0,0,0,14,37,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,57,18,0,0,0,0,16,4],[33,25,0,0,0,0,37,28,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,57,41,0,0,0,0,16,4],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,60,30,0,0,0,0,0,1] >;
D60⋊11C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes_{11}C_4 % in TeX
G:=Group("D60:11C4"); // GroupNames label
G:=SmallGroup(480,858);
// by ID
G=gap.SmallGroup(480,858);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,219,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^31,c*b*c^-1=a^30*b>;
// generators/relations