metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.8D20, D6⋊C4⋊17D5, C4⋊Dic5⋊5S3, (S3×C10).6D4, C10.17(S3×D4), (C2×C20).21D6, C6.17(C2×D20), C2.19(S3×D20), C30.48(C2×D4), C5⋊3(D6.D4), D30⋊3C4⋊15C2, D30⋊4C4⋊15C2, C30.68(C4○D4), (C2×C12).226D10, C6.Dic10⋊21C2, C10.70(C4○D12), C6.45(D4⋊2D5), (C2×C30).117C23, (C2×C60).257C22, (C2×Dic5).111D6, (C2×Dic3).36D10, (C22×S3).41D10, C3⋊1(C22.D20), C2.18(D12⋊D5), C10.35(Q8⋊3S3), C15⋊10(C22.D4), (C6×Dic5).70C22, C2.16(Dic3.D10), (C10×Dic3).72C22, (C2×Dic15).95C22, (C22×D15).39C22, (C2×S3×Dic5)⋊6C2, (C5×D6⋊C4)⋊17C2, (C2×C4).51(S3×D5), (C3×C4⋊Dic5)⋊16C2, (C2×C5⋊D12).7C2, C22.181(C2×S3×D5), (S3×C2×C10).22C22, (C2×C6).129(C22×D5), (C2×C10).129(C22×S3), SmallGroup(480,503)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.D20
G = < a,b,c,d | a6=b2=c20=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >
Subgroups: 908 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C22.D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, S3×C10, D30, C2×C30, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, D6.D4, S3×Dic5, C5⋊D12, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, C22.D20, D30⋊4C4, C6.Dic10, C3×C4⋊Dic5, C5×D6⋊C4, D30⋊3C4, C2×S3×Dic5, C2×C5⋊D12, D6.D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C22.D4, D20, C22×D5, C4○D12, S3×D4, Q8⋊3S3, S3×D5, C2×D20, D4⋊2D5, D6.D4, C2×S3×D5, C22.D20, D12⋊D5, S3×D20, Dic3.D10, D6.D20
►On 240 points
Generators in S240
(1 60 198 114 159 63)(2 41 199 115 160 64)(3 42 200 116 141 65)(4 43 181 117 142 66)(5 44 182 118 143 67)(6 45 183 119 144 68)(7 46 184 120 145 69)(8 47 185 101 146 70)(9 48 186 102 147 71)(10 49 187 103 148 72)(11 50 188 104 149 73)(12 51 189 105 150 74)(13 52 190 106 151 75)(14 53 191 107 152 76)(15 54 192 108 153 77)(16 55 193 109 154 78)(17 56 194 110 155 79)(18 57 195 111 156 80)(19 58 196 112 157 61)(20 59 197 113 158 62)(21 224 164 128 203 87)(22 225 165 129 204 88)(23 226 166 130 205 89)(24 227 167 131 206 90)(25 228 168 132 207 91)(26 229 169 133 208 92)(27 230 170 134 209 93)(28 231 171 135 210 94)(29 232 172 136 211 95)(30 233 173 137 212 96)(31 234 174 138 213 97)(32 235 175 139 214 98)(33 236 176 140 215 99)(34 237 177 121 216 100)(35 238 178 122 217 81)(36 239 179 123 218 82)(37 240 180 124 219 83)(38 221 161 125 220 84)(39 222 162 126 201 85)(40 223 163 127 202 86)
(1 63)(2 199)(3 65)(4 181)(5 67)(6 183)(7 69)(8 185)(9 71)(10 187)(11 73)(12 189)(13 75)(14 191)(15 77)(16 193)(17 79)(18 195)(19 61)(20 197)(21 128)(23 130)(25 132)(27 134)(29 136)(31 138)(33 140)(35 122)(37 124)(39 126)(42 141)(44 143)(46 145)(48 147)(50 149)(52 151)(54 153)(56 155)(58 157)(60 159)(62 113)(64 115)(66 117)(68 119)(70 101)(72 103)(74 105)(76 107)(78 109)(80 111)(81 217)(82 239)(83 219)(84 221)(85 201)(86 223)(87 203)(88 225)(89 205)(90 227)(91 207)(92 229)(93 209)(94 231)(95 211)(96 233)(97 213)(98 235)(99 215)(100 237)(102 186)(104 188)(106 190)(108 192)(110 194)(112 196)(114 198)(116 200)(118 182)(120 184)(161 220)(162 222)(163 202)(164 224)(165 204)(166 226)(167 206)(168 228)(169 208)(170 230)(171 210)(172 232)(173 212)(174 234)(175 214)(176 236)(177 216)(178 238)(179 218)(180 240)
(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)
(1 169 114 92)(2 168 115 91)(3 167 116 90)(4 166 117 89)(5 165 118 88)(6 164 119 87)(7 163 120 86)(8 162 101 85)(9 161 102 84)(10 180 103 83)(11 179 104 82)(12 178 105 81)(13 177 106 100)(14 176 107 99)(15 175 108 98)(16 174 109 97)(17 173 110 96)(18 172 111 95)(19 171 112 94)(20 170 113 93)(21 45 128 144)(22 44 129 143)(23 43 130 142)(24 42 131 141)(25 41 132 160)(26 60 133 159)(27 59 134 158)(28 58 135 157)(29 57 136 156)(30 56 137 155)(31 55 138 154)(32 54 139 153)(33 53 140 152)(34 52 121 151)(35 51 122 150)(36 50 123 149)(37 49 124 148)(38 48 125 147)(39 47 126 146)(40 46 127 145)(61 231 196 210)(62 230 197 209)(63 229 198 208)(64 228 199 207)(65 227 200 206)(66 226 181 205)(67 225 182 204)(68 224 183 203)(69 223 184 202)(70 222 185 201)(71 221 186 220)(72 240 187 219)(73 239 188 218)(74 238 189 217)(75 237 190 216)(76 236 191 215)(77 235 192 214)(78 234 193 213)(79 233 194 212)(80 232 195 211)
G:=sub<Sym(240)| (1,60,198,114,159,63)(2,41,199,115,160,64)(3,42,200,116,141,65)(4,43,181,117,142,66)(5,44,182,118,143,67)(6,45,183,119,144,68)(7,46,184,120,145,69)(8,47,185,101,146,70)(9,48,186,102,147,71)(10,49,187,103,148,72)(11,50,188,104,149,73)(12,51,189,105,150,74)(13,52,190,106,151,75)(14,53,191,107,152,76)(15,54,192,108,153,77)(16,55,193,109,154,78)(17,56,194,110,155,79)(18,57,195,111,156,80)(19,58,196,112,157,61)(20,59,197,113,158,62)(21,224,164,128,203,87)(22,225,165,129,204,88)(23,226,166,130,205,89)(24,227,167,131,206,90)(25,228,168,132,207,91)(26,229,169,133,208,92)(27,230,170,134,209,93)(28,231,171,135,210,94)(29,232,172,136,211,95)(30,233,173,137,212,96)(31,234,174,138,213,97)(32,235,175,139,214,98)(33,236,176,140,215,99)(34,237,177,121,216,100)(35,238,178,122,217,81)(36,239,179,123,218,82)(37,240,180,124,219,83)(38,221,161,125,220,84)(39,222,162,126,201,85)(40,223,163,127,202,86), (1,63)(2,199)(3,65)(4,181)(5,67)(6,183)(7,69)(8,185)(9,71)(10,187)(11,73)(12,189)(13,75)(14,191)(15,77)(16,193)(17,79)(18,195)(19,61)(20,197)(21,128)(23,130)(25,132)(27,134)(29,136)(31,138)(33,140)(35,122)(37,124)(39,126)(42,141)(44,143)(46,145)(48,147)(50,149)(52,151)(54,153)(56,155)(58,157)(60,159)(62,113)(64,115)(66,117)(68,119)(70,101)(72,103)(74,105)(76,107)(78,109)(80,111)(81,217)(82,239)(83,219)(84,221)(85,201)(86,223)(87,203)(88,225)(89,205)(90,227)(91,207)(92,229)(93,209)(94,231)(95,211)(96,233)(97,213)(98,235)(99,215)(100,237)(102,186)(104,188)(106,190)(108,192)(110,194)(112,196)(114,198)(116,200)(118,182)(120,184)(161,220)(162,222)(163,202)(164,224)(165,204)(166,226)(167,206)(168,228)(169,208)(170,230)(171,210)(172,232)(173,212)(174,234)(175,214)(176,236)(177,216)(178,238)(179,218)(180,240), (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), (1,169,114,92)(2,168,115,91)(3,167,116,90)(4,166,117,89)(5,165,118,88)(6,164,119,87)(7,163,120,86)(8,162,101,85)(9,161,102,84)(10,180,103,83)(11,179,104,82)(12,178,105,81)(13,177,106,100)(14,176,107,99)(15,175,108,98)(16,174,109,97)(17,173,110,96)(18,172,111,95)(19,171,112,94)(20,170,113,93)(21,45,128,144)(22,44,129,143)(23,43,130,142)(24,42,131,141)(25,41,132,160)(26,60,133,159)(27,59,134,158)(28,58,135,157)(29,57,136,156)(30,56,137,155)(31,55,138,154)(32,54,139,153)(33,53,140,152)(34,52,121,151)(35,51,122,150)(36,50,123,149)(37,49,124,148)(38,48,125,147)(39,47,126,146)(40,46,127,145)(61,231,196,210)(62,230,197,209)(63,229,198,208)(64,228,199,207)(65,227,200,206)(66,226,181,205)(67,225,182,204)(68,224,183,203)(69,223,184,202)(70,222,185,201)(71,221,186,220)(72,240,187,219)(73,239,188,218)(74,238,189,217)(75,237,190,216)(76,236,191,215)(77,235,192,214)(78,234,193,213)(79,233,194,212)(80,232,195,211)>;
G:=Group( (1,60,198,114,159,63)(2,41,199,115,160,64)(3,42,200,116,141,65)(4,43,181,117,142,66)(5,44,182,118,143,67)(6,45,183,119,144,68)(7,46,184,120,145,69)(8,47,185,101,146,70)(9,48,186,102,147,71)(10,49,187,103,148,72)(11,50,188,104,149,73)(12,51,189,105,150,74)(13,52,190,106,151,75)(14,53,191,107,152,76)(15,54,192,108,153,77)(16,55,193,109,154,78)(17,56,194,110,155,79)(18,57,195,111,156,80)(19,58,196,112,157,61)(20,59,197,113,158,62)(21,224,164,128,203,87)(22,225,165,129,204,88)(23,226,166,130,205,89)(24,227,167,131,206,90)(25,228,168,132,207,91)(26,229,169,133,208,92)(27,230,170,134,209,93)(28,231,171,135,210,94)(29,232,172,136,211,95)(30,233,173,137,212,96)(31,234,174,138,213,97)(32,235,175,139,214,98)(33,236,176,140,215,99)(34,237,177,121,216,100)(35,238,178,122,217,81)(36,239,179,123,218,82)(37,240,180,124,219,83)(38,221,161,125,220,84)(39,222,162,126,201,85)(40,223,163,127,202,86), (1,63)(2,199)(3,65)(4,181)(5,67)(6,183)(7,69)(8,185)(9,71)(10,187)(11,73)(12,189)(13,75)(14,191)(15,77)(16,193)(17,79)(18,195)(19,61)(20,197)(21,128)(23,130)(25,132)(27,134)(29,136)(31,138)(33,140)(35,122)(37,124)(39,126)(42,141)(44,143)(46,145)(48,147)(50,149)(52,151)(54,153)(56,155)(58,157)(60,159)(62,113)(64,115)(66,117)(68,119)(70,101)(72,103)(74,105)(76,107)(78,109)(80,111)(81,217)(82,239)(83,219)(84,221)(85,201)(86,223)(87,203)(88,225)(89,205)(90,227)(91,207)(92,229)(93,209)(94,231)(95,211)(96,233)(97,213)(98,235)(99,215)(100,237)(102,186)(104,188)(106,190)(108,192)(110,194)(112,196)(114,198)(116,200)(118,182)(120,184)(161,220)(162,222)(163,202)(164,224)(165,204)(166,226)(167,206)(168,228)(169,208)(170,230)(171,210)(172,232)(173,212)(174,234)(175,214)(176,236)(177,216)(178,238)(179,218)(180,240), (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), (1,169,114,92)(2,168,115,91)(3,167,116,90)(4,166,117,89)(5,165,118,88)(6,164,119,87)(7,163,120,86)(8,162,101,85)(9,161,102,84)(10,180,103,83)(11,179,104,82)(12,178,105,81)(13,177,106,100)(14,176,107,99)(15,175,108,98)(16,174,109,97)(17,173,110,96)(18,172,111,95)(19,171,112,94)(20,170,113,93)(21,45,128,144)(22,44,129,143)(23,43,130,142)(24,42,131,141)(25,41,132,160)(26,60,133,159)(27,59,134,158)(28,58,135,157)(29,57,136,156)(30,56,137,155)(31,55,138,154)(32,54,139,153)(33,53,140,152)(34,52,121,151)(35,51,122,150)(36,50,123,149)(37,49,124,148)(38,48,125,147)(39,47,126,146)(40,46,127,145)(61,231,196,210)(62,230,197,209)(63,229,198,208)(64,228,199,207)(65,227,200,206)(66,226,181,205)(67,225,182,204)(68,224,183,203)(69,223,184,202)(70,222,185,201)(71,221,186,220)(72,240,187,219)(73,239,188,218)(74,238,189,217)(75,237,190,216)(76,236,191,215)(77,235,192,214)(78,234,193,213)(79,233,194,212)(80,232,195,211) );
G=PermutationGroup([[(1,60,198,114,159,63),(2,41,199,115,160,64),(3,42,200,116,141,65),(4,43,181,117,142,66),(5,44,182,118,143,67),(6,45,183,119,144,68),(7,46,184,120,145,69),(8,47,185,101,146,70),(9,48,186,102,147,71),(10,49,187,103,148,72),(11,50,188,104,149,73),(12,51,189,105,150,74),(13,52,190,106,151,75),(14,53,191,107,152,76),(15,54,192,108,153,77),(16,55,193,109,154,78),(17,56,194,110,155,79),(18,57,195,111,156,80),(19,58,196,112,157,61),(20,59,197,113,158,62),(21,224,164,128,203,87),(22,225,165,129,204,88),(23,226,166,130,205,89),(24,227,167,131,206,90),(25,228,168,132,207,91),(26,229,169,133,208,92),(27,230,170,134,209,93),(28,231,171,135,210,94),(29,232,172,136,211,95),(30,233,173,137,212,96),(31,234,174,138,213,97),(32,235,175,139,214,98),(33,236,176,140,215,99),(34,237,177,121,216,100),(35,238,178,122,217,81),(36,239,179,123,218,82),(37,240,180,124,219,83),(38,221,161,125,220,84),(39,222,162,126,201,85),(40,223,163,127,202,86)], [(1,63),(2,199),(3,65),(4,181),(5,67),(6,183),(7,69),(8,185),(9,71),(10,187),(11,73),(12,189),(13,75),(14,191),(15,77),(16,193),(17,79),(18,195),(19,61),(20,197),(21,128),(23,130),(25,132),(27,134),(29,136),(31,138),(33,140),(35,122),(37,124),(39,126),(42,141),(44,143),(46,145),(48,147),(50,149),(52,151),(54,153),(56,155),(58,157),(60,159),(62,113),(64,115),(66,117),(68,119),(70,101),(72,103),(74,105),(76,107),(78,109),(80,111),(81,217),(82,239),(83,219),(84,221),(85,201),(86,223),(87,203),(88,225),(89,205),(90,227),(91,207),(92,229),(93,209),(94,231),(95,211),(96,233),(97,213),(98,235),(99,215),(100,237),(102,186),(104,188),(106,190),(108,192),(110,194),(112,196),(114,198),(116,200),(118,182),(120,184),(161,220),(162,222),(163,202),(164,224),(165,204),(166,226),(167,206),(168,228),(169,208),(170,230),(171,210),(172,232),(173,212),(174,234),(175,214),(176,236),(177,216),(178,238),(179,218),(180,240)], [(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)], [(1,169,114,92),(2,168,115,91),(3,167,116,90),(4,166,117,89),(5,165,118,88),(6,164,119,87),(7,163,120,86),(8,162,101,85),(9,161,102,84),(10,180,103,83),(11,179,104,82),(12,178,105,81),(13,177,106,100),(14,176,107,99),(15,175,108,98),(16,174,109,97),(17,173,110,96),(18,172,111,95),(19,171,112,94),(20,170,113,93),(21,45,128,144),(22,44,129,143),(23,43,130,142),(24,42,131,141),(25,41,132,160),(26,60,133,159),(27,59,134,158),(28,58,135,157),(29,57,136,156),(30,56,137,155),(31,55,138,154),(32,54,139,153),(33,53,140,152),(34,52,121,151),(35,51,122,150),(36,50,123,149),(37,49,124,148),(38,48,125,147),(39,47,126,146),(40,46,127,145),(61,231,196,210),(62,230,197,209),(63,229,198,208),(64,228,199,207),(65,227,200,206),(66,226,181,205),(67,225,182,204),(68,224,183,203),(69,223,184,202),(70,222,185,201),(71,221,186,220),(72,240,187,219),(73,239,188,218),(74,238,189,217),(75,237,190,216),(76,236,191,215),(77,235,192,214),(78,234,193,213),(79,233,194,212),(80,232,195,211)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 60 | 2 | 4 | 10 | 10 | 12 | 20 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D20 | C4○D12 | S3×D4 | Q8⋊3S3 | S3×D5 | D4⋊2D5 | C2×S3×D5 | D12⋊D5 | S3×D20 | Dic3.D10 |
| kernel | D6.D20 | D30⋊4C4 | C6.Dic10 | C3×C4⋊Dic5 | C5×D6⋊C4 | D30⋊3C4 | C2×S3×Dic5 | C2×C5⋊D12 | C4⋊Dic5 | S3×C10 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C10 | C10 | C10 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D6.D20 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 60 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 60 |
| 7 | 32 | 0 | 0 |
| 29 | 2 | 0 | 0 |
| 0 | 0 | 52 | 43 |
| 0 | 0 | 18 | 9 |
| 53 | 20 | 0 | 0 |
| 6 | 8 | 0 | 0 |
| 0 | 0 | 38 | 15 |
| 0 | 0 | 46 | 23 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,1,60,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,1,60],[7,29,0,0,32,2,0,0,0,0,52,18,0,0,43,9],[53,6,0,0,20,8,0,0,0,0,38,46,0,0,15,23] >;
D6.D20 in GAP, Magma, Sage, TeX
D_6.D_{20} % in TeX
G:=Group("D6.D20"); // GroupNames label
G:=SmallGroup(480,503);
// by ID
G=gap.SmallGroup(480,503);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^20=1,d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations