metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.1D20, C40.47D6, D20.20D6, C24.31D10, C60.97C23, C120.30C22, Dic3.12D20, Dic10.20D6, D60.28C22, Dic30.29C22, (S3×C8)⋊5D5, (S3×C40)⋊5C2, C40⋊C2⋊7S3, C15⋊4(C4○D8), C3⋊C8.28D10, C6.7(C2×D20), C8.14(S3×D5), C10.7(S3×D4), C3⋊D40⋊12C2, C30.22(C2×D4), C2.12(S3×D20), C24⋊D5⋊12C2, D60⋊C2⋊9C2, D20⋊5S3⋊9C2, C3⋊1(D40⋊7C2), (S3×C10).20D4, (C4×S3).39D10, C3⋊Dic20⋊11C2, C5⋊1(Q8.7D6), (C5×Dic3).23D4, C12.75(C22×D5), (S3×C20).45C22, C20.147(C22×S3), (C3×D20).22C22, (C3×Dic10).24C22, C4.96(C2×S3×D5), (C3×C40⋊C2)⋊4C2, (C5×C3⋊C8).32C22, SmallGroup(480,348)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.1D20
G = < a,b,c,d | a6=b2=1, c20=d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c19 >
Subgroups: 780 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, C3×D5, D15, C30, C4○D8, C40, C40, Dic10, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D4⋊2S3, Q8⋊3S3, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C40⋊C2, C40⋊C2, D40, Dic20, C2×C40, C4○D20, Q8.7D6, C5×C3⋊C8, C120, D5×Dic3, D30.C2, C15⋊D4, C5⋊D12, C3×Dic10, C3×D20, S3×C20, Dic30, D60, D40⋊7C2, C3⋊D40, C3⋊Dic20, C3×C40⋊C2, S3×C40, C24⋊D5, D20⋊5S3, D60⋊C2, D6.1D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C4○D8, D20, C22×D5, S3×D4, S3×D5, C2×D20, Q8.7D6, C2×S3×D5, D40⋊7C2, S3×D20, D6.1D20
►On 240 points
Generators in S240
(1 172 204 21 192 224)(2 173 205 22 193 225)(3 174 206 23 194 226)(4 175 207 24 195 227)(5 176 208 25 196 228)(6 177 209 26 197 229)(7 178 210 27 198 230)(8 179 211 28 199 231)(9 180 212 29 200 232)(10 181 213 30 161 233)(11 182 214 31 162 234)(12 183 215 32 163 235)(13 184 216 33 164 236)(14 185 217 34 165 237)(15 186 218 35 166 238)(16 187 219 36 167 239)(17 188 220 37 168 240)(18 189 221 38 169 201)(19 190 222 39 170 202)(20 191 223 40 171 203)(41 125 106 61 145 86)(42 126 107 62 146 87)(43 127 108 63 147 88)(44 128 109 64 148 89)(45 129 110 65 149 90)(46 130 111 66 150 91)(47 131 112 67 151 92)(48 132 113 68 152 93)(49 133 114 69 153 94)(50 134 115 70 154 95)(51 135 116 71 155 96)(52 136 117 72 156 97)(53 137 118 73 157 98)(54 138 119 74 158 99)(55 139 120 75 159 100)(56 140 81 76 160 101)(57 141 82 77 121 102)(58 142 83 78 122 103)(59 143 84 79 123 104)(60 144 85 80 124 105)
(1 224)(2 225)(3 226)(4 227)(5 228)(6 229)(7 230)(8 231)(9 232)(10 233)(11 234)(12 235)(13 236)(14 237)(15 238)(16 239)(17 240)(18 201)(19 202)(20 203)(21 204)(22 205)(23 206)(24 207)(25 208)(26 209)(27 210)(28 211)(29 212)(30 213)(31 214)(32 215)(33 216)(34 217)(35 218)(36 219)(37 220)(38 221)(39 222)(40 223)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 121)(58 122)(59 123)(60 124)(61 125)(62 126)(63 127)(64 128)(65 129)(66 130)(67 131)(68 132)(69 133)(70 134)(71 135)(72 136)(73 137)(74 138)(75 139)(76 140)(77 141)(78 142)(79 143)(80 144)(161 181)(162 182)(163 183)(164 184)(165 185)(166 186)(167 187)(168 188)(169 189)(170 190)(171 191)(172 192)(173 193)(174 194)(175 195)(176 196)(177 197)(178 198)(179 199)(180 200)
(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)
(1 133 21 153)(2 152 22 132)(3 131 23 151)(4 150 24 130)(5 129 25 149)(6 148 26 128)(7 127 27 147)(8 146 28 126)(9 125 29 145)(10 144 30 124)(11 123 31 143)(12 142 32 122)(13 121 33 141)(14 140 34 160)(15 159 35 139)(16 138 36 158)(17 157 37 137)(18 136 38 156)(19 155 39 135)(20 134 40 154)(41 212 61 232)(42 231 62 211)(43 210 63 230)(44 229 64 209)(45 208 65 228)(46 227 66 207)(47 206 67 226)(48 225 68 205)(49 204 69 224)(50 223 70 203)(51 202 71 222)(52 221 72 201)(53 240 73 220)(54 219 74 239)(55 238 75 218)(56 217 76 237)(57 236 77 216)(58 215 78 235)(59 234 79 214)(60 213 80 233)(81 165 101 185)(82 184 102 164)(83 163 103 183)(84 182 104 162)(85 161 105 181)(86 180 106 200)(87 199 107 179)(88 178 108 198)(89 197 109 177)(90 176 110 196)(91 195 111 175)(92 174 112 194)(93 193 113 173)(94 172 114 192)(95 191 115 171)(96 170 116 190)(97 189 117 169)(98 168 118 188)(99 187 119 167)(100 166 120 186)
G:=sub<Sym(240)| (1,172,204,21,192,224)(2,173,205,22,193,225)(3,174,206,23,194,226)(4,175,207,24,195,227)(5,176,208,25,196,228)(6,177,209,26,197,229)(7,178,210,27,198,230)(8,179,211,28,199,231)(9,180,212,29,200,232)(10,181,213,30,161,233)(11,182,214,31,162,234)(12,183,215,32,163,235)(13,184,216,33,164,236)(14,185,217,34,165,237)(15,186,218,35,166,238)(16,187,219,36,167,239)(17,188,220,37,168,240)(18,189,221,38,169,201)(19,190,222,39,170,202)(20,191,223,40,171,203)(41,125,106,61,145,86)(42,126,107,62,146,87)(43,127,108,63,147,88)(44,128,109,64,148,89)(45,129,110,65,149,90)(46,130,111,66,150,91)(47,131,112,67,151,92)(48,132,113,68,152,93)(49,133,114,69,153,94)(50,134,115,70,154,95)(51,135,116,71,155,96)(52,136,117,72,156,97)(53,137,118,73,157,98)(54,138,119,74,158,99)(55,139,120,75,159,100)(56,140,81,76,160,101)(57,141,82,77,121,102)(58,142,83,78,122,103)(59,143,84,79,123,104)(60,144,85,80,124,105), (1,224)(2,225)(3,226)(4,227)(5,228)(6,229)(7,230)(8,231)(9,232)(10,233)(11,234)(12,235)(13,236)(14,237)(15,238)(16,239)(17,240)(18,201)(19,202)(20,203)(21,204)(22,205)(23,206)(24,207)(25,208)(26,209)(27,210)(28,211)(29,212)(30,213)(31,214)(32,215)(33,216)(34,217)(35,218)(36,219)(37,220)(38,221)(39,222)(40,223)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,121)(58,122)(59,123)(60,124)(61,125)(62,126)(63,127)(64,128)(65,129)(66,130)(67,131)(68,132)(69,133)(70,134)(71,135)(72,136)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144)(161,181)(162,182)(163,183)(164,184)(165,185)(166,186)(167,187)(168,188)(169,189)(170,190)(171,191)(172,192)(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)(180,200), (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), (1,133,21,153)(2,152,22,132)(3,131,23,151)(4,150,24,130)(5,129,25,149)(6,148,26,128)(7,127,27,147)(8,146,28,126)(9,125,29,145)(10,144,30,124)(11,123,31,143)(12,142,32,122)(13,121,33,141)(14,140,34,160)(15,159,35,139)(16,138,36,158)(17,157,37,137)(18,136,38,156)(19,155,39,135)(20,134,40,154)(41,212,61,232)(42,231,62,211)(43,210,63,230)(44,229,64,209)(45,208,65,228)(46,227,66,207)(47,206,67,226)(48,225,68,205)(49,204,69,224)(50,223,70,203)(51,202,71,222)(52,221,72,201)(53,240,73,220)(54,219,74,239)(55,238,75,218)(56,217,76,237)(57,236,77,216)(58,215,78,235)(59,234,79,214)(60,213,80,233)(81,165,101,185)(82,184,102,164)(83,163,103,183)(84,182,104,162)(85,161,105,181)(86,180,106,200)(87,199,107,179)(88,178,108,198)(89,197,109,177)(90,176,110,196)(91,195,111,175)(92,174,112,194)(93,193,113,173)(94,172,114,192)(95,191,115,171)(96,170,116,190)(97,189,117,169)(98,168,118,188)(99,187,119,167)(100,166,120,186)>;
G:=Group( (1,172,204,21,192,224)(2,173,205,22,193,225)(3,174,206,23,194,226)(4,175,207,24,195,227)(5,176,208,25,196,228)(6,177,209,26,197,229)(7,178,210,27,198,230)(8,179,211,28,199,231)(9,180,212,29,200,232)(10,181,213,30,161,233)(11,182,214,31,162,234)(12,183,215,32,163,235)(13,184,216,33,164,236)(14,185,217,34,165,237)(15,186,218,35,166,238)(16,187,219,36,167,239)(17,188,220,37,168,240)(18,189,221,38,169,201)(19,190,222,39,170,202)(20,191,223,40,171,203)(41,125,106,61,145,86)(42,126,107,62,146,87)(43,127,108,63,147,88)(44,128,109,64,148,89)(45,129,110,65,149,90)(46,130,111,66,150,91)(47,131,112,67,151,92)(48,132,113,68,152,93)(49,133,114,69,153,94)(50,134,115,70,154,95)(51,135,116,71,155,96)(52,136,117,72,156,97)(53,137,118,73,157,98)(54,138,119,74,158,99)(55,139,120,75,159,100)(56,140,81,76,160,101)(57,141,82,77,121,102)(58,142,83,78,122,103)(59,143,84,79,123,104)(60,144,85,80,124,105), (1,224)(2,225)(3,226)(4,227)(5,228)(6,229)(7,230)(8,231)(9,232)(10,233)(11,234)(12,235)(13,236)(14,237)(15,238)(16,239)(17,240)(18,201)(19,202)(20,203)(21,204)(22,205)(23,206)(24,207)(25,208)(26,209)(27,210)(28,211)(29,212)(30,213)(31,214)(32,215)(33,216)(34,217)(35,218)(36,219)(37,220)(38,221)(39,222)(40,223)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,121)(58,122)(59,123)(60,124)(61,125)(62,126)(63,127)(64,128)(65,129)(66,130)(67,131)(68,132)(69,133)(70,134)(71,135)(72,136)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144)(161,181)(162,182)(163,183)(164,184)(165,185)(166,186)(167,187)(168,188)(169,189)(170,190)(171,191)(172,192)(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)(180,200), (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), (1,133,21,153)(2,152,22,132)(3,131,23,151)(4,150,24,130)(5,129,25,149)(6,148,26,128)(7,127,27,147)(8,146,28,126)(9,125,29,145)(10,144,30,124)(11,123,31,143)(12,142,32,122)(13,121,33,141)(14,140,34,160)(15,159,35,139)(16,138,36,158)(17,157,37,137)(18,136,38,156)(19,155,39,135)(20,134,40,154)(41,212,61,232)(42,231,62,211)(43,210,63,230)(44,229,64,209)(45,208,65,228)(46,227,66,207)(47,206,67,226)(48,225,68,205)(49,204,69,224)(50,223,70,203)(51,202,71,222)(52,221,72,201)(53,240,73,220)(54,219,74,239)(55,238,75,218)(56,217,76,237)(57,236,77,216)(58,215,78,235)(59,234,79,214)(60,213,80,233)(81,165,101,185)(82,184,102,164)(83,163,103,183)(84,182,104,162)(85,161,105,181)(86,180,106,200)(87,199,107,179)(88,178,108,198)(89,197,109,177)(90,176,110,196)(91,195,111,175)(92,174,112,194)(93,193,113,173)(94,172,114,192)(95,191,115,171)(96,170,116,190)(97,189,117,169)(98,168,118,188)(99,187,119,167)(100,166,120,186) );
G=PermutationGroup([[(1,172,204,21,192,224),(2,173,205,22,193,225),(3,174,206,23,194,226),(4,175,207,24,195,227),(5,176,208,25,196,228),(6,177,209,26,197,229),(7,178,210,27,198,230),(8,179,211,28,199,231),(9,180,212,29,200,232),(10,181,213,30,161,233),(11,182,214,31,162,234),(12,183,215,32,163,235),(13,184,216,33,164,236),(14,185,217,34,165,237),(15,186,218,35,166,238),(16,187,219,36,167,239),(17,188,220,37,168,240),(18,189,221,38,169,201),(19,190,222,39,170,202),(20,191,223,40,171,203),(41,125,106,61,145,86),(42,126,107,62,146,87),(43,127,108,63,147,88),(44,128,109,64,148,89),(45,129,110,65,149,90),(46,130,111,66,150,91),(47,131,112,67,151,92),(48,132,113,68,152,93),(49,133,114,69,153,94),(50,134,115,70,154,95),(51,135,116,71,155,96),(52,136,117,72,156,97),(53,137,118,73,157,98),(54,138,119,74,158,99),(55,139,120,75,159,100),(56,140,81,76,160,101),(57,141,82,77,121,102),(58,142,83,78,122,103),(59,143,84,79,123,104),(60,144,85,80,124,105)], [(1,224),(2,225),(3,226),(4,227),(5,228),(6,229),(7,230),(8,231),(9,232),(10,233),(11,234),(12,235),(13,236),(14,237),(15,238),(16,239),(17,240),(18,201),(19,202),(20,203),(21,204),(22,205),(23,206),(24,207),(25,208),(26,209),(27,210),(28,211),(29,212),(30,213),(31,214),(32,215),(33,216),(34,217),(35,218),(36,219),(37,220),(38,221),(39,222),(40,223),(41,145),(42,146),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(50,154),(51,155),(52,156),(53,157),(54,158),(55,159),(56,160),(57,121),(58,122),(59,123),(60,124),(61,125),(62,126),(63,127),(64,128),(65,129),(66,130),(67,131),(68,132),(69,133),(70,134),(71,135),(72,136),(73,137),(74,138),(75,139),(76,140),(77,141),(78,142),(79,143),(80,144),(161,181),(162,182),(163,183),(164,184),(165,185),(166,186),(167,187),(168,188),(169,189),(170,190),(171,191),(172,192),(173,193),(174,194),(175,195),(176,196),(177,197),(178,198),(179,199),(180,200)], [(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)], [(1,133,21,153),(2,152,22,132),(3,131,23,151),(4,150,24,130),(5,129,25,149),(6,148,26,128),(7,127,27,147),(8,146,28,126),(9,125,29,145),(10,144,30,124),(11,123,31,143),(12,142,32,122),(13,121,33,141),(14,140,34,160),(15,159,35,139),(16,138,36,158),(17,157,37,137),(18,136,38,156),(19,155,39,135),(20,134,40,154),(41,212,61,232),(42,231,62,211),(43,210,63,230),(44,229,64,209),(45,208,65,228),(46,227,66,207),(47,206,67,226),(48,225,68,205),(49,204,69,224),(50,223,70,203),(51,202,71,222),(52,221,72,201),(53,240,73,220),(54,219,74,239),(55,238,75,218),(56,217,76,237),(57,236,77,216),(58,215,78,235),(59,234,79,214),(60,213,80,233),(81,165,101,185),(82,184,102,164),(83,163,103,183),(84,182,104,162),(85,161,105,181),(86,180,106,200),(87,199,107,179),(88,178,108,198),(89,197,109,177),(90,176,110,196),(91,195,111,175),(92,174,112,194),(93,193,113,173),(94,172,114,192),(95,191,115,171),(96,170,116,190),(97,189,117,169),(98,168,118,188),(99,187,119,167),(100,166,120,186)]])
69 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 30A | 30B | 40A | ··· | 40H | 40I | ··· | 40P | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 6 | 20 | 60 | 2 | 2 | 3 | 3 | 20 | 60 | 2 | 2 | 2 | 40 | 2 | 2 | 6 | 6 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 40 | 4 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
69 irreducible representations
| dim | 1 | 1 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C4○D8 | D20 | D20 | D40⋊7C2 | S3×D4 | S3×D5 | Q8.7D6 | C2×S3×D5 | S3×D20 | D6.1D20 |
| kernel | D6.1D20 | C3⋊D40 | C3⋊Dic20 | C3×C40⋊C2 | S3×C40 | C24⋊D5 | D20⋊5S3 | D60⋊C2 | C40⋊C2 | C5×Dic3 | S3×C10 | S3×C8 | C40 | Dic10 | D20 | C3⋊C8 | C24 | C4×S3 | C15 | Dic3 | D6 | C3 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 16 | 1 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D6.1D20 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 169 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 211 | 0 | 0 | 0 | 0 | 0 |
| 172 | 233 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 51 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 201 | 135 | 0 | 0 | 0 | 0 |
| 222 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 190 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0],[240,169,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,240,1],[211,172,0,0,0,0,0,233,0,0,0,0,0,0,0,240,0,0,0,0,1,51,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[201,222,0,0,0,0,135,40,0,0,0,0,0,0,240,190,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D6.1D20 in GAP, Magma, Sage, TeX
D_6._1D_{20} % in TeX
G:=Group("D6.1D20"); // GroupNames label
G:=SmallGroup(480,348);
// by ID
G=gap.SmallGroup(480,348);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,58,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=1,c^20=d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^19>;
// generators/relations