metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊1D20, (S3×C10)⋊9D4, C5⋊4(Dic3⋊D4), C3⋊2(C20⋊7D4), C15⋊9(C4⋊D4), C2.21(S3×D20), C6.20(C2×D20), D10⋊C4⋊3S3, (C5×Dic3)⋊13D4, (C2×C20).269D6, C30.149(C2×D4), C10.136(S3×D4), D30⋊3C4⋊33C2, C30.88(C4○D4), C6.40(C4○D20), (C2×C12).200D10, Dic3⋊4(C5⋊D4), C6.Dic10⋊25C2, (C2×Dic5).45D6, (C22×D5).15D6, C10.43(C4○D12), (C2×C60).413C22, (C2×C30).144C23, (C22×S3).74D10, (C2×Dic3).155D10, (C6×Dic5).86C22, C2.29(D6.D10), (C22×D15).49C22, (C10×Dic3).189C22, (C2×Dic15).111C22, (S3×C2×C4)⋊13D5, (S3×C2×C20)⋊21C2, (C2×C5⋊D12)⋊5C2, (C2×C15⋊D4)⋊5C2, (C2×C3⋊D20)⋊4C2, (C2×C4).81(S3×D5), C6.38(C2×C5⋊D4), C2.17(S3×C5⋊D4), (D5×C2×C6).29C22, C22.196(C2×S3×D5), (S3×C2×C10).89C22, (C3×D10⋊C4)⋊34C2, (C2×C6).156(C22×D5), (C2×C10).156(C22×S3), SmallGroup(480,530)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊D20
G = < a,b,c,d | a6=b2=c20=d2=1, bab=cac-1=dad=a-1, cbc-1=a4b, dbd=ab, dcd=c-1 >
Subgroups: 1132 in 188 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, 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, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C4⋊D4, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, S3×C10, D30, C2×C30, C4⋊Dic5, D10⋊C4, D10⋊C4, C2×D20, C2×C5⋊D4, C22×C20, Dic3⋊D4, C15⋊D4, C3⋊D20, C5⋊D12, C6×Dic5, S3×C20, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C22×D15, C20⋊7D4, C6.Dic10, C3×D10⋊C4, D30⋊3C4, C2×C15⋊D4, C2×C3⋊D20, C2×C5⋊D12, S3×C2×C20, D6⋊D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C4⋊D4, D20, C5⋊D4, C22×D5, C4○D12, S3×D4, S3×D5, C2×D20, C4○D20, C2×C5⋊D4, Dic3⋊D4, C2×S3×D5, C20⋊7D4, D6.D10, S3×D20, S3×C5⋊D4, D6⋊D20
►On 240 points
Generators in S240
(1 191 62 136 34 155)(2 156 35 137 63 192)(3 193 64 138 36 157)(4 158 37 139 65 194)(5 195 66 140 38 159)(6 160 39 121 67 196)(7 197 68 122 40 141)(8 142 21 123 69 198)(9 199 70 124 22 143)(10 144 23 125 71 200)(11 181 72 126 24 145)(12 146 25 127 73 182)(13 183 74 128 26 147)(14 148 27 129 75 184)(15 185 76 130 28 149)(16 150 29 131 77 186)(17 187 78 132 30 151)(18 152 31 133 79 188)(19 189 80 134 32 153)(20 154 33 135 61 190)(41 227 202 106 83 178)(42 179 84 107 203 228)(43 229 204 108 85 180)(44 161 86 109 205 230)(45 231 206 110 87 162)(46 163 88 111 207 232)(47 233 208 112 89 164)(48 165 90 113 209 234)(49 235 210 114 91 166)(50 167 92 115 211 236)(51 237 212 116 93 168)(52 169 94 117 213 238)(53 239 214 118 95 170)(54 171 96 119 215 240)(55 221 216 120 97 172)(56 173 98 101 217 222)(57 223 218 102 99 174)(58 175 100 103 219 224)(59 225 220 104 81 176)(60 177 82 105 201 226)
(1 72)(2 25)(3 74)(4 27)(5 76)(6 29)(7 78)(8 31)(9 80)(10 33)(11 62)(12 35)(13 64)(14 37)(15 66)(16 39)(17 68)(18 21)(19 70)(20 23)(22 32)(24 34)(26 36)(28 38)(30 40)(41 116)(42 169)(43 118)(44 171)(45 120)(46 173)(47 102)(48 175)(49 104)(50 177)(51 106)(52 179)(53 108)(54 161)(55 110)(56 163)(57 112)(58 165)(59 114)(60 167)(61 71)(63 73)(65 75)(67 77)(69 79)(81 166)(82 236)(83 168)(84 238)(85 170)(86 240)(87 172)(88 222)(89 174)(90 224)(91 176)(92 226)(93 178)(94 228)(95 180)(96 230)(97 162)(98 232)(99 164)(100 234)(101 207)(103 209)(105 211)(107 213)(109 215)(111 217)(113 219)(115 201)(117 203)(119 205)(121 186)(122 151)(123 188)(124 153)(125 190)(126 155)(127 192)(128 157)(129 194)(130 159)(131 196)(132 141)(133 198)(134 143)(135 200)(136 145)(137 182)(138 147)(139 184)(140 149)(142 152)(144 154)(146 156)(148 158)(150 160)(181 191)(183 193)(185 195)(187 197)(189 199)(202 237)(204 239)(206 221)(208 223)(210 225)(212 227)(214 229)(216 231)(218 233)(220 235)
(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)
(1 211)(2 210)(3 209)(4 208)(5 207)(6 206)(7 205)(8 204)(9 203)(10 202)(11 201)(12 220)(13 219)(14 218)(15 217)(16 216)(17 215)(18 214)(19 213)(20 212)(21 43)(22 42)(23 41)(24 60)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(71 83)(72 82)(73 81)(74 100)(75 99)(76 98)(77 97)(78 96)(79 95)(80 94)(101 185)(102 184)(103 183)(104 182)(105 181)(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)(121 162)(122 161)(123 180)(124 179)(125 178)(126 177)(127 176)(128 175)(129 174)(130 173)(131 172)(132 171)(133 170)(134 169)(135 168)(136 167)(137 166)(138 165)(139 164)(140 163)(141 230)(142 229)(143 228)(144 227)(145 226)(146 225)(147 224)(148 223)(149 222)(150 221)(151 240)(152 239)(153 238)(154 237)(155 236)(156 235)(157 234)(158 233)(159 232)(160 231)
G:=sub<Sym(240)| (1,191,62,136,34,155)(2,156,35,137,63,192)(3,193,64,138,36,157)(4,158,37,139,65,194)(5,195,66,140,38,159)(6,160,39,121,67,196)(7,197,68,122,40,141)(8,142,21,123,69,198)(9,199,70,124,22,143)(10,144,23,125,71,200)(11,181,72,126,24,145)(12,146,25,127,73,182)(13,183,74,128,26,147)(14,148,27,129,75,184)(15,185,76,130,28,149)(16,150,29,131,77,186)(17,187,78,132,30,151)(18,152,31,133,79,188)(19,189,80,134,32,153)(20,154,33,135,61,190)(41,227,202,106,83,178)(42,179,84,107,203,228)(43,229,204,108,85,180)(44,161,86,109,205,230)(45,231,206,110,87,162)(46,163,88,111,207,232)(47,233,208,112,89,164)(48,165,90,113,209,234)(49,235,210,114,91,166)(50,167,92,115,211,236)(51,237,212,116,93,168)(52,169,94,117,213,238)(53,239,214,118,95,170)(54,171,96,119,215,240)(55,221,216,120,97,172)(56,173,98,101,217,222)(57,223,218,102,99,174)(58,175,100,103,219,224)(59,225,220,104,81,176)(60,177,82,105,201,226), (1,72)(2,25)(3,74)(4,27)(5,76)(6,29)(7,78)(8,31)(9,80)(10,33)(11,62)(12,35)(13,64)(14,37)(15,66)(16,39)(17,68)(18,21)(19,70)(20,23)(22,32)(24,34)(26,36)(28,38)(30,40)(41,116)(42,169)(43,118)(44,171)(45,120)(46,173)(47,102)(48,175)(49,104)(50,177)(51,106)(52,179)(53,108)(54,161)(55,110)(56,163)(57,112)(58,165)(59,114)(60,167)(61,71)(63,73)(65,75)(67,77)(69,79)(81,166)(82,236)(83,168)(84,238)(85,170)(86,240)(87,172)(88,222)(89,174)(90,224)(91,176)(92,226)(93,178)(94,228)(95,180)(96,230)(97,162)(98,232)(99,164)(100,234)(101,207)(103,209)(105,211)(107,213)(109,215)(111,217)(113,219)(115,201)(117,203)(119,205)(121,186)(122,151)(123,188)(124,153)(125,190)(126,155)(127,192)(128,157)(129,194)(130,159)(131,196)(132,141)(133,198)(134,143)(135,200)(136,145)(137,182)(138,147)(139,184)(140,149)(142,152)(144,154)(146,156)(148,158)(150,160)(181,191)(183,193)(185,195)(187,197)(189,199)(202,237)(204,239)(206,221)(208,223)(210,225)(212,227)(214,229)(216,231)(218,233)(220,235), (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), (1,211)(2,210)(3,209)(4,208)(5,207)(6,206)(7,205)(8,204)(9,203)(10,202)(11,201)(12,220)(13,219)(14,218)(15,217)(16,216)(17,215)(18,214)(19,213)(20,212)(21,43)(22,42)(23,41)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)(73,81)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(101,185)(102,184)(103,183)(104,182)(105,181)(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)(121,162)(122,161)(123,180)(124,179)(125,178)(126,177)(127,176)(128,175)(129,174)(130,173)(131,172)(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,230)(142,229)(143,228)(144,227)(145,226)(146,225)(147,224)(148,223)(149,222)(150,221)(151,240)(152,239)(153,238)(154,237)(155,236)(156,235)(157,234)(158,233)(159,232)(160,231)>;
G:=Group( (1,191,62,136,34,155)(2,156,35,137,63,192)(3,193,64,138,36,157)(4,158,37,139,65,194)(5,195,66,140,38,159)(6,160,39,121,67,196)(7,197,68,122,40,141)(8,142,21,123,69,198)(9,199,70,124,22,143)(10,144,23,125,71,200)(11,181,72,126,24,145)(12,146,25,127,73,182)(13,183,74,128,26,147)(14,148,27,129,75,184)(15,185,76,130,28,149)(16,150,29,131,77,186)(17,187,78,132,30,151)(18,152,31,133,79,188)(19,189,80,134,32,153)(20,154,33,135,61,190)(41,227,202,106,83,178)(42,179,84,107,203,228)(43,229,204,108,85,180)(44,161,86,109,205,230)(45,231,206,110,87,162)(46,163,88,111,207,232)(47,233,208,112,89,164)(48,165,90,113,209,234)(49,235,210,114,91,166)(50,167,92,115,211,236)(51,237,212,116,93,168)(52,169,94,117,213,238)(53,239,214,118,95,170)(54,171,96,119,215,240)(55,221,216,120,97,172)(56,173,98,101,217,222)(57,223,218,102,99,174)(58,175,100,103,219,224)(59,225,220,104,81,176)(60,177,82,105,201,226), (1,72)(2,25)(3,74)(4,27)(5,76)(6,29)(7,78)(8,31)(9,80)(10,33)(11,62)(12,35)(13,64)(14,37)(15,66)(16,39)(17,68)(18,21)(19,70)(20,23)(22,32)(24,34)(26,36)(28,38)(30,40)(41,116)(42,169)(43,118)(44,171)(45,120)(46,173)(47,102)(48,175)(49,104)(50,177)(51,106)(52,179)(53,108)(54,161)(55,110)(56,163)(57,112)(58,165)(59,114)(60,167)(61,71)(63,73)(65,75)(67,77)(69,79)(81,166)(82,236)(83,168)(84,238)(85,170)(86,240)(87,172)(88,222)(89,174)(90,224)(91,176)(92,226)(93,178)(94,228)(95,180)(96,230)(97,162)(98,232)(99,164)(100,234)(101,207)(103,209)(105,211)(107,213)(109,215)(111,217)(113,219)(115,201)(117,203)(119,205)(121,186)(122,151)(123,188)(124,153)(125,190)(126,155)(127,192)(128,157)(129,194)(130,159)(131,196)(132,141)(133,198)(134,143)(135,200)(136,145)(137,182)(138,147)(139,184)(140,149)(142,152)(144,154)(146,156)(148,158)(150,160)(181,191)(183,193)(185,195)(187,197)(189,199)(202,237)(204,239)(206,221)(208,223)(210,225)(212,227)(214,229)(216,231)(218,233)(220,235), (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), (1,211)(2,210)(3,209)(4,208)(5,207)(6,206)(7,205)(8,204)(9,203)(10,202)(11,201)(12,220)(13,219)(14,218)(15,217)(16,216)(17,215)(18,214)(19,213)(20,212)(21,43)(22,42)(23,41)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)(73,81)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(101,185)(102,184)(103,183)(104,182)(105,181)(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)(121,162)(122,161)(123,180)(124,179)(125,178)(126,177)(127,176)(128,175)(129,174)(130,173)(131,172)(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,230)(142,229)(143,228)(144,227)(145,226)(146,225)(147,224)(148,223)(149,222)(150,221)(151,240)(152,239)(153,238)(154,237)(155,236)(156,235)(157,234)(158,233)(159,232)(160,231) );
G=PermutationGroup([[(1,191,62,136,34,155),(2,156,35,137,63,192),(3,193,64,138,36,157),(4,158,37,139,65,194),(5,195,66,140,38,159),(6,160,39,121,67,196),(7,197,68,122,40,141),(8,142,21,123,69,198),(9,199,70,124,22,143),(10,144,23,125,71,200),(11,181,72,126,24,145),(12,146,25,127,73,182),(13,183,74,128,26,147),(14,148,27,129,75,184),(15,185,76,130,28,149),(16,150,29,131,77,186),(17,187,78,132,30,151),(18,152,31,133,79,188),(19,189,80,134,32,153),(20,154,33,135,61,190),(41,227,202,106,83,178),(42,179,84,107,203,228),(43,229,204,108,85,180),(44,161,86,109,205,230),(45,231,206,110,87,162),(46,163,88,111,207,232),(47,233,208,112,89,164),(48,165,90,113,209,234),(49,235,210,114,91,166),(50,167,92,115,211,236),(51,237,212,116,93,168),(52,169,94,117,213,238),(53,239,214,118,95,170),(54,171,96,119,215,240),(55,221,216,120,97,172),(56,173,98,101,217,222),(57,223,218,102,99,174),(58,175,100,103,219,224),(59,225,220,104,81,176),(60,177,82,105,201,226)], [(1,72),(2,25),(3,74),(4,27),(5,76),(6,29),(7,78),(8,31),(9,80),(10,33),(11,62),(12,35),(13,64),(14,37),(15,66),(16,39),(17,68),(18,21),(19,70),(20,23),(22,32),(24,34),(26,36),(28,38),(30,40),(41,116),(42,169),(43,118),(44,171),(45,120),(46,173),(47,102),(48,175),(49,104),(50,177),(51,106),(52,179),(53,108),(54,161),(55,110),(56,163),(57,112),(58,165),(59,114),(60,167),(61,71),(63,73),(65,75),(67,77),(69,79),(81,166),(82,236),(83,168),(84,238),(85,170),(86,240),(87,172),(88,222),(89,174),(90,224),(91,176),(92,226),(93,178),(94,228),(95,180),(96,230),(97,162),(98,232),(99,164),(100,234),(101,207),(103,209),(105,211),(107,213),(109,215),(111,217),(113,219),(115,201),(117,203),(119,205),(121,186),(122,151),(123,188),(124,153),(125,190),(126,155),(127,192),(128,157),(129,194),(130,159),(131,196),(132,141),(133,198),(134,143),(135,200),(136,145),(137,182),(138,147),(139,184),(140,149),(142,152),(144,154),(146,156),(148,158),(150,160),(181,191),(183,193),(185,195),(187,197),(189,199),(202,237),(204,239),(206,221),(208,223),(210,225),(212,227),(214,229),(216,231),(218,233),(220,235)], [(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)], [(1,211),(2,210),(3,209),(4,208),(5,207),(6,206),(7,205),(8,204),(9,203),(10,202),(11,201),(12,220),(13,219),(14,218),(15,217),(16,216),(17,215),(18,214),(19,213),(20,212),(21,43),(22,42),(23,41),(24,60),(25,59),(26,58),(27,57),(28,56),(29,55),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(61,93),(62,92),(63,91),(64,90),(65,89),(66,88),(67,87),(68,86),(69,85),(70,84),(71,83),(72,82),(73,81),(74,100),(75,99),(76,98),(77,97),(78,96),(79,95),(80,94),(101,185),(102,184),(103,183),(104,182),(105,181),(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),(121,162),(122,161),(123,180),(124,179),(125,178),(126,177),(127,176),(128,175),(129,174),(130,173),(131,172),(132,171),(133,170),(134,169),(135,168),(136,167),(137,166),(138,165),(139,164),(140,163),(141,230),(142,229),(143,228),(144,227),(145,226),(146,225),(147,224),(148,223),(149,222),(150,221),(151,240),(152,239),(153,238),(154,237),(155,236),(156,235),(157,234),(158,233),(159,232),(160,231)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 20 | 60 | 2 | 2 | 2 | 6 | 6 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 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 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D20 | C4○D12 | C4○D20 | S3×D4 | S3×D5 | C2×S3×D5 | D6.D10 | S3×D20 | S3×C5⋊D4 |
| kernel | D6⋊D20 | C6.Dic10 | C3×D10⋊C4 | D30⋊3C4 | C2×C15⋊D4 | C2×C3⋊D20 | C2×C5⋊D12 | S3×C2×C20 | D10⋊C4 | C5×Dic3 | S3×C10 | S3×C2×C4 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C22×S3 | Dic3 | D6 | C10 | C6 | C10 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 4 | 8 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D6⋊D20 ►in GL6(𝔽61)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 12 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 15 | 0 | 0 | 0 | 0 |
| 12 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 39 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 0 |
| 0 | 0 | 0 | 0 | 58 | 37 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 53 |
| 0 | 0 | 0 | 0 | 12 | 55 |
G:=sub<GL(6,GF(61))| [1,12,0,0,0,0,15,59,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,12,0,0,0,0,15,59,0,0,0,0,0,0,1,39,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,15,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,58,0,0,0,0,0,37],[1,0,0,0,0,0,15,60,0,0,0,0,0,0,60,0,0,0,0,0,11,1,0,0,0,0,0,0,6,12,0,0,0,0,53,55] >;
D6⋊D20 in GAP, Magma, Sage, TeX
D_6\rtimes D_{20} % in TeX
G:=Group("D6:D20"); // GroupNames label
G:=SmallGroup(480,530);
// by ID
G=gap.SmallGroup(480,530);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^20=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^4*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations