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## G = D6⋊D20order 480 = 25·3·5

### 1st semidirect product of D6 and D20 acting via D20/C20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D6⋊D20
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C15⋊D4 — D6⋊D20
 Lower central C15 — C2×C30 — D6⋊D20
 Upper central C1 — C22 — C2×C4

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 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], C5, S3 [×3], C6 [×3], C6, C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], D5 [×2], C10 [×3], C10 [×2], Dic3 [×2], Dic3, C12 [×2], D6 [×2], D6 [×5], C2×C6, C2×C6 [×3], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], C20 [×3], D10 [×6], C2×C10, C2×C10 [×4], C4×S3 [×2], D12 [×2], C2×Dic3, C2×Dic3, C3⋊D4 [×4], C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3 [×2], C3×D5, D15, C30 [×3], C4⋊D4, D20 [×2], C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20 [×3], C22×D5, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4 [×2], C5×Dic3 [×2], C3×Dic5, Dic15, C60, C6×D5 [×3], S3×C10 [×2], S3×C10 [×2], D30 [×3], C2×C30, C4⋊Dic5, D10⋊C4, D10⋊C4, C2×D20, C2×C5⋊D4 [×2], C22×C20, Dic3⋊D4, C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C6×Dic5, S3×C20 [×2], C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C22×D15, C207D4, C6.Dic10, C3×D10⋊C4, D303C4, C2×C15⋊D4, C2×C3⋊D20, C2×C5⋊D12, S3×C2×C20, D6⋊D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C22×S3, C4⋊D4, D20 [×2], C5⋊D4 [×2], C22×D5, C4○D12, S3×D4 [×2], S3×D5, C2×D20, C4○D20, C2×C5⋊D4, Dic3⋊D4, C2×S3×D5, C207D4, D6.D10, S3×D20, S3×C5⋊D4, D6⋊D20

Smallest permutation representation of D6⋊D20
On 240 points
Generators in S240
```(1 226 217 56 70 180)(2 161 71 57 218 227)(3 228 219 58 72 162)(4 163 73 59 220 229)(5 230 201 60 74 164)(6 165 75 41 202 231)(7 232 203 42 76 166)(8 167 77 43 204 233)(9 234 205 44 78 168)(10 169 79 45 206 235)(11 236 207 46 80 170)(12 171 61 47 208 237)(13 238 209 48 62 172)(14 173 63 49 210 239)(15 240 211 50 64 174)(16 175 65 51 212 221)(17 222 213 52 66 176)(18 177 67 53 214 223)(19 224 215 54 68 178)(20 179 69 55 216 225)(21 194 98 158 133 107)(22 108 134 159 99 195)(23 196 100 160 135 109)(24 110 136 141 81 197)(25 198 82 142 137 111)(26 112 138 143 83 199)(27 200 84 144 139 113)(28 114 140 145 85 181)(29 182 86 146 121 115)(30 116 122 147 87 183)(31 184 88 148 123 117)(32 118 124 149 89 185)(33 186 90 150 125 119)(34 120 126 151 91 187)(35 188 92 152 127 101)(36 102 128 153 93 189)(37 190 94 154 129 103)(38 104 130 155 95 191)(39 192 96 156 131 105)(40 106 132 157 97 193)
(1 207)(2 61)(3 209)(4 63)(5 211)(6 65)(7 213)(8 67)(9 215)(10 69)(11 217)(12 71)(13 219)(14 73)(15 201)(16 75)(17 203)(18 77)(19 205)(20 79)(21 148)(22 118)(23 150)(24 120)(25 152)(26 102)(27 154)(28 104)(29 156)(30 106)(31 158)(32 108)(33 160)(34 110)(35 142)(36 112)(37 144)(38 114)(39 146)(40 116)(41 221)(42 176)(43 223)(44 178)(45 225)(46 180)(47 227)(48 162)(49 229)(50 164)(51 231)(52 166)(53 233)(54 168)(55 235)(56 170)(57 237)(58 172)(59 239)(60 174)(62 72)(64 74)(66 76)(68 78)(70 80)(81 151)(82 188)(83 153)(84 190)(85 155)(86 192)(87 157)(88 194)(89 159)(90 196)(91 141)(92 198)(93 143)(94 200)(95 145)(96 182)(97 147)(98 184)(99 149)(100 186)(101 137)(103 139)(105 121)(107 123)(109 125)(111 127)(113 129)(115 131)(117 133)(119 135)(122 193)(124 195)(126 197)(128 199)(130 181)(132 183)(134 185)(136 187)(138 189)(140 191)(161 171)(163 173)(165 175)(167 177)(169 179)(202 212)(204 214)(206 216)(208 218)(210 220)(222 232)(224 234)(226 236)(228 238)(230 240)
(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 95)(2 94)(3 93)(4 92)(5 91)(6 90)(7 89)(8 88)(9 87)(10 86)(11 85)(12 84)(13 83)(14 82)(15 81)(16 100)(17 99)(18 98)(19 97)(20 96)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 70)(39 69)(40 68)(41 119)(42 118)(43 117)(44 116)(45 115)(46 114)(47 113)(48 112)(49 111)(50 110)(51 109)(52 108)(53 107)(54 106)(55 105)(56 104)(57 103)(58 102)(59 101)(60 120)(121 206)(122 205)(123 204)(124 203)(125 202)(126 201)(127 220)(128 219)(129 218)(130 217)(131 216)(132 215)(133 214)(134 213)(135 212)(136 211)(137 210)(138 209)(139 208)(140 207)(141 240)(142 239)(143 238)(144 237)(145 236)(146 235)(147 234)(148 233)(149 232)(150 231)(151 230)(152 229)(153 228)(154 227)(155 226)(156 225)(157 224)(158 223)(159 222)(160 221)(161 190)(162 189)(163 188)(164 187)(165 186)(166 185)(167 184)(168 183)(169 182)(170 181)(171 200)(172 199)(173 198)(174 197)(175 196)(176 195)(177 194)(178 193)(179 192)(180 191)```

`G:=sub<Sym(240)| (1,226,217,56,70,180)(2,161,71,57,218,227)(3,228,219,58,72,162)(4,163,73,59,220,229)(5,230,201,60,74,164)(6,165,75,41,202,231)(7,232,203,42,76,166)(8,167,77,43,204,233)(9,234,205,44,78,168)(10,169,79,45,206,235)(11,236,207,46,80,170)(12,171,61,47,208,237)(13,238,209,48,62,172)(14,173,63,49,210,239)(15,240,211,50,64,174)(16,175,65,51,212,221)(17,222,213,52,66,176)(18,177,67,53,214,223)(19,224,215,54,68,178)(20,179,69,55,216,225)(21,194,98,158,133,107)(22,108,134,159,99,195)(23,196,100,160,135,109)(24,110,136,141,81,197)(25,198,82,142,137,111)(26,112,138,143,83,199)(27,200,84,144,139,113)(28,114,140,145,85,181)(29,182,86,146,121,115)(30,116,122,147,87,183)(31,184,88,148,123,117)(32,118,124,149,89,185)(33,186,90,150,125,119)(34,120,126,151,91,187)(35,188,92,152,127,101)(36,102,128,153,93,189)(37,190,94,154,129,103)(38,104,130,155,95,191)(39,192,96,156,131,105)(40,106,132,157,97,193), (1,207)(2,61)(3,209)(4,63)(5,211)(6,65)(7,213)(8,67)(9,215)(10,69)(11,217)(12,71)(13,219)(14,73)(15,201)(16,75)(17,203)(18,77)(19,205)(20,79)(21,148)(22,118)(23,150)(24,120)(25,152)(26,102)(27,154)(28,104)(29,156)(30,106)(31,158)(32,108)(33,160)(34,110)(35,142)(36,112)(37,144)(38,114)(39,146)(40,116)(41,221)(42,176)(43,223)(44,178)(45,225)(46,180)(47,227)(48,162)(49,229)(50,164)(51,231)(52,166)(53,233)(54,168)(55,235)(56,170)(57,237)(58,172)(59,239)(60,174)(62,72)(64,74)(66,76)(68,78)(70,80)(81,151)(82,188)(83,153)(84,190)(85,155)(86,192)(87,157)(88,194)(89,159)(90,196)(91,141)(92,198)(93,143)(94,200)(95,145)(96,182)(97,147)(98,184)(99,149)(100,186)(101,137)(103,139)(105,121)(107,123)(109,125)(111,127)(113,129)(115,131)(117,133)(119,135)(122,193)(124,195)(126,197)(128,199)(130,181)(132,183)(134,185)(136,187)(138,189)(140,191)(161,171)(163,173)(165,175)(167,177)(169,179)(202,212)(204,214)(206,216)(208,218)(210,220)(222,232)(224,234)(226,236)(228,238)(230,240), (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), (1,95)(2,94)(3,93)(4,92)(5,91)(6,90)(7,89)(8,88)(9,87)(10,86)(11,85)(12,84)(13,83)(14,82)(15,81)(16,100)(17,99)(18,98)(19,97)(20,96)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,119)(42,118)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,111)(50,110)(51,109)(52,108)(53,107)(54,106)(55,105)(56,104)(57,103)(58,102)(59,101)(60,120)(121,206)(122,205)(123,204)(124,203)(125,202)(126,201)(127,220)(128,219)(129,218)(130,217)(131,216)(132,215)(133,214)(134,213)(135,212)(136,211)(137,210)(138,209)(139,208)(140,207)(141,240)(142,239)(143,238)(144,237)(145,236)(146,235)(147,234)(148,233)(149,232)(150,231)(151,230)(152,229)(153,228)(154,227)(155,226)(156,225)(157,224)(158,223)(159,222)(160,221)(161,190)(162,189)(163,188)(164,187)(165,186)(166,185)(167,184)(168,183)(169,182)(170,181)(171,200)(172,199)(173,198)(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191)>;`

`G:=Group( (1,226,217,56,70,180)(2,161,71,57,218,227)(3,228,219,58,72,162)(4,163,73,59,220,229)(5,230,201,60,74,164)(6,165,75,41,202,231)(7,232,203,42,76,166)(8,167,77,43,204,233)(9,234,205,44,78,168)(10,169,79,45,206,235)(11,236,207,46,80,170)(12,171,61,47,208,237)(13,238,209,48,62,172)(14,173,63,49,210,239)(15,240,211,50,64,174)(16,175,65,51,212,221)(17,222,213,52,66,176)(18,177,67,53,214,223)(19,224,215,54,68,178)(20,179,69,55,216,225)(21,194,98,158,133,107)(22,108,134,159,99,195)(23,196,100,160,135,109)(24,110,136,141,81,197)(25,198,82,142,137,111)(26,112,138,143,83,199)(27,200,84,144,139,113)(28,114,140,145,85,181)(29,182,86,146,121,115)(30,116,122,147,87,183)(31,184,88,148,123,117)(32,118,124,149,89,185)(33,186,90,150,125,119)(34,120,126,151,91,187)(35,188,92,152,127,101)(36,102,128,153,93,189)(37,190,94,154,129,103)(38,104,130,155,95,191)(39,192,96,156,131,105)(40,106,132,157,97,193), (1,207)(2,61)(3,209)(4,63)(5,211)(6,65)(7,213)(8,67)(9,215)(10,69)(11,217)(12,71)(13,219)(14,73)(15,201)(16,75)(17,203)(18,77)(19,205)(20,79)(21,148)(22,118)(23,150)(24,120)(25,152)(26,102)(27,154)(28,104)(29,156)(30,106)(31,158)(32,108)(33,160)(34,110)(35,142)(36,112)(37,144)(38,114)(39,146)(40,116)(41,221)(42,176)(43,223)(44,178)(45,225)(46,180)(47,227)(48,162)(49,229)(50,164)(51,231)(52,166)(53,233)(54,168)(55,235)(56,170)(57,237)(58,172)(59,239)(60,174)(62,72)(64,74)(66,76)(68,78)(70,80)(81,151)(82,188)(83,153)(84,190)(85,155)(86,192)(87,157)(88,194)(89,159)(90,196)(91,141)(92,198)(93,143)(94,200)(95,145)(96,182)(97,147)(98,184)(99,149)(100,186)(101,137)(103,139)(105,121)(107,123)(109,125)(111,127)(113,129)(115,131)(117,133)(119,135)(122,193)(124,195)(126,197)(128,199)(130,181)(132,183)(134,185)(136,187)(138,189)(140,191)(161,171)(163,173)(165,175)(167,177)(169,179)(202,212)(204,214)(206,216)(208,218)(210,220)(222,232)(224,234)(226,236)(228,238)(230,240), (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), (1,95)(2,94)(3,93)(4,92)(5,91)(6,90)(7,89)(8,88)(9,87)(10,86)(11,85)(12,84)(13,83)(14,82)(15,81)(16,100)(17,99)(18,98)(19,97)(20,96)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,119)(42,118)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,111)(50,110)(51,109)(52,108)(53,107)(54,106)(55,105)(56,104)(57,103)(58,102)(59,101)(60,120)(121,206)(122,205)(123,204)(124,203)(125,202)(126,201)(127,220)(128,219)(129,218)(130,217)(131,216)(132,215)(133,214)(134,213)(135,212)(136,211)(137,210)(138,209)(139,208)(140,207)(141,240)(142,239)(143,238)(144,237)(145,236)(146,235)(147,234)(148,233)(149,232)(150,231)(151,230)(152,229)(153,228)(154,227)(155,226)(156,225)(157,224)(158,223)(159,222)(160,221)(161,190)(162,189)(163,188)(164,187)(165,186)(166,185)(167,184)(168,183)(169,182)(170,181)(171,200)(172,199)(173,198)(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191) );`

`G=PermutationGroup([(1,226,217,56,70,180),(2,161,71,57,218,227),(3,228,219,58,72,162),(4,163,73,59,220,229),(5,230,201,60,74,164),(6,165,75,41,202,231),(7,232,203,42,76,166),(8,167,77,43,204,233),(9,234,205,44,78,168),(10,169,79,45,206,235),(11,236,207,46,80,170),(12,171,61,47,208,237),(13,238,209,48,62,172),(14,173,63,49,210,239),(15,240,211,50,64,174),(16,175,65,51,212,221),(17,222,213,52,66,176),(18,177,67,53,214,223),(19,224,215,54,68,178),(20,179,69,55,216,225),(21,194,98,158,133,107),(22,108,134,159,99,195),(23,196,100,160,135,109),(24,110,136,141,81,197),(25,198,82,142,137,111),(26,112,138,143,83,199),(27,200,84,144,139,113),(28,114,140,145,85,181),(29,182,86,146,121,115),(30,116,122,147,87,183),(31,184,88,148,123,117),(32,118,124,149,89,185),(33,186,90,150,125,119),(34,120,126,151,91,187),(35,188,92,152,127,101),(36,102,128,153,93,189),(37,190,94,154,129,103),(38,104,130,155,95,191),(39,192,96,156,131,105),(40,106,132,157,97,193)], [(1,207),(2,61),(3,209),(4,63),(5,211),(6,65),(7,213),(8,67),(9,215),(10,69),(11,217),(12,71),(13,219),(14,73),(15,201),(16,75),(17,203),(18,77),(19,205),(20,79),(21,148),(22,118),(23,150),(24,120),(25,152),(26,102),(27,154),(28,104),(29,156),(30,106),(31,158),(32,108),(33,160),(34,110),(35,142),(36,112),(37,144),(38,114),(39,146),(40,116),(41,221),(42,176),(43,223),(44,178),(45,225),(46,180),(47,227),(48,162),(49,229),(50,164),(51,231),(52,166),(53,233),(54,168),(55,235),(56,170),(57,237),(58,172),(59,239),(60,174),(62,72),(64,74),(66,76),(68,78),(70,80),(81,151),(82,188),(83,153),(84,190),(85,155),(86,192),(87,157),(88,194),(89,159),(90,196),(91,141),(92,198),(93,143),(94,200),(95,145),(96,182),(97,147),(98,184),(99,149),(100,186),(101,137),(103,139),(105,121),(107,123),(109,125),(111,127),(113,129),(115,131),(117,133),(119,135),(122,193),(124,195),(126,197),(128,199),(130,181),(132,183),(134,185),(136,187),(138,189),(140,191),(161,171),(163,173),(165,175),(167,177),(169,179),(202,212),(204,214),(206,216),(208,218),(210,220),(222,232),(224,234),(226,236),(228,238),(230,240)], [(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)], [(1,95),(2,94),(3,93),(4,92),(5,91),(6,90),(7,89),(8,88),(9,87),(10,86),(11,85),(12,84),(13,83),(14,82),(15,81),(16,100),(17,99),(18,98),(19,97),(20,96),(21,67),(22,66),(23,65),(24,64),(25,63),(26,62),(27,61),(28,80),(29,79),(30,78),(31,77),(32,76),(33,75),(34,74),(35,73),(36,72),(37,71),(38,70),(39,69),(40,68),(41,119),(42,118),(43,117),(44,116),(45,115),(46,114),(47,113),(48,112),(49,111),(50,110),(51,109),(52,108),(53,107),(54,106),(55,105),(56,104),(57,103),(58,102),(59,101),(60,120),(121,206),(122,205),(123,204),(124,203),(125,202),(126,201),(127,220),(128,219),(129,218),(130,217),(131,216),(132,215),(133,214),(134,213),(135,212),(136,211),(137,210),(138,209),(139,208),(140,207),(141,240),(142,239),(143,238),(144,237),(145,236),(146,235),(147,234),(148,233),(149,232),(150,231),(151,230),(152,229),(153,228),(154,227),(155,226),(156,225),(157,224),(158,223),(159,222),(160,221),(161,190),(162,189),(163,188),(164,187),(165,186),(166,185),(167,184),(168,183),(169,182),(170,181),(171,200),(172,199),(173,198),(174,197),(175,196),(176,195),(177,194),(178,193),(179,192),(180,191)])`

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`

׿
×
𝔽