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## G = D20⋊8Dic3order 480 = 25·3·5

### 5th semidirect product of D20 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D20⋊8Dic3
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — D20⋊8Dic3
 Lower central C15 — C30 — D20⋊8Dic3
 Upper central C1 — C22 — C2×C4

Generators and relations for D208Dic3
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, ac=ca, dad=a19, bc=cb, bd=db, dcd=c-1 >

Subgroups: 892 in 188 conjugacy classes, 70 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C3×D5, C30, C4×D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C6×D4, C5×Dic3, Dic15, Dic15, C60, C6×D5, C6×D5, C2×C30, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, D4×Dic3, D5×Dic3, C3×D20, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, D208C4, D10⋊Dic3, C5×C4⋊Dic3, C4×Dic15, C2×D5×Dic3, C6×D20, D208Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, Dic3, D6, C22×C4, C2×D4, C4○D4, D10, C2×Dic3, C22×S3, C4×D4, C4×D5, C22×D5, S3×D4, D42S3, C22×Dic3, S3×D5, C2×C4×D5, D4×D5, Q82D5, D4×Dic3, D5×Dic3, C2×S3×D5, D208C4, D20⋊S3, C20⋊D6, C2×D5×Dic3, D208Dic3

Smallest permutation representation of D208Dic3
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)
(1 237 16 222)(2 236 17 221)(3 235 18 220)(4 234 19 219)(5 233 20 218)(6 232 21 217)(7 231 22 216)(8 230 23 215)(9 229 24 214)(10 228 25 213)(11 227 26 212)(12 226 27 211)(13 225 28 240)(14 224 29 239)(15 223 30 238)(31 165 46 180)(32 164 47 179)(33 163 48 178)(34 162 49 177)(35 161 50 176)(36 160 51 175)(37 159 52 174)(38 158 53 173)(39 157 54 172)(40 156 55 171)(41 155 56 170)(42 154 57 169)(43 153 58 168)(44 152 59 167)(45 151 60 166)(61 100 76 115)(62 99 77 114)(63 98 78 113)(64 97 79 112)(65 96 80 111)(66 95 81 110)(67 94 82 109)(68 93 83 108)(69 92 84 107)(70 91 85 106)(71 120 86 105)(72 119 87 104)(73 118 88 103)(74 117 89 102)(75 116 90 101)(121 199 136 184)(122 198 137 183)(123 197 138 182)(124 196 139 181)(125 195 140 210)(126 194 141 209)(127 193 142 208)(128 192 143 207)(129 191 144 206)(130 190 145 205)(131 189 146 204)(132 188 147 203)(133 187 148 202)(134 186 149 201)(135 185 150 200)
(1 163 135 86)(2 164 136 87)(3 165 137 88)(4 166 138 89)(5 167 139 90)(6 168 140 61)(7 169 141 62)(8 170 142 63)(9 171 143 64)(10 172 144 65)(11 173 145 66)(12 174 146 67)(13 175 147 68)(14 176 148 69)(15 177 149 70)(16 178 150 71)(17 179 121 72)(18 180 122 73)(19 151 123 74)(20 152 124 75)(21 153 125 76)(22 154 126 77)(23 155 127 78)(24 156 128 79)(25 157 129 80)(26 158 130 81)(27 159 131 82)(28 160 132 83)(29 161 133 84)(30 162 134 85)(31 198 118 220)(32 199 119 221)(33 200 120 222)(34 201 91 223)(35 202 92 224)(36 203 93 225)(37 204 94 226)(38 205 95 227)(39 206 96 228)(40 207 97 229)(41 208 98 230)(42 209 99 231)(43 210 100 232)(44 181 101 233)(45 182 102 234)(46 183 103 235)(47 184 104 236)(48 185 105 237)(49 186 106 238)(50 187 107 239)(51 188 108 240)(52 189 109 211)(53 190 110 212)(54 191 111 213)(55 192 112 214)(56 193 113 215)(57 194 114 216)(58 195 115 217)(59 196 116 218)(60 197 117 219)
(1 163)(2 152)(3 171)(4 160)(5 179)(6 168)(7 157)(8 176)(9 165)(10 154)(11 173)(12 162)(13 151)(14 170)(15 159)(16 178)(17 167)(18 156)(19 175)(20 164)(21 153)(22 172)(23 161)(24 180)(25 169)(26 158)(27 177)(28 166)(29 155)(30 174)(31 214)(32 233)(33 222)(34 211)(35 230)(36 219)(37 238)(38 227)(39 216)(40 235)(41 224)(42 213)(43 232)(44 221)(45 240)(46 229)(47 218)(48 237)(49 226)(50 215)(51 234)(52 223)(53 212)(54 231)(55 220)(56 239)(57 228)(58 217)(59 236)(60 225)(61 140)(62 129)(63 148)(64 137)(65 126)(66 145)(67 134)(68 123)(69 142)(70 131)(71 150)(72 139)(73 128)(74 147)(75 136)(76 125)(77 144)(78 133)(79 122)(80 141)(81 130)(82 149)(83 138)(84 127)(85 146)(86 135)(87 124)(88 143)(89 132)(90 121)(91 189)(92 208)(93 197)(94 186)(95 205)(96 194)(97 183)(98 202)(99 191)(100 210)(101 199)(102 188)(103 207)(104 196)(105 185)(106 204)(107 193)(108 182)(109 201)(110 190)(111 209)(112 198)(113 187)(114 206)(115 195)(116 184)(117 203)(118 192)(119 181)(120 200)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 12A 12B 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 10 10 2 2 2 6 6 6 6 15 15 15 15 30 30 2 2 2 2 2 20 20 20 20 2 ··· 2 4 4 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + + - + + + + + - + + + - + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D5 Dic3 D6 D6 C4○D4 D10 D10 C4×D5 S3×D4 D4⋊2S3 S3×D5 D4×D5 Q8⋊2D5 D5×Dic3 C2×S3×D5 D20⋊S3 C20⋊D6 kernel D20⋊8Dic3 D10⋊Dic3 C5×C4⋊Dic3 C4×Dic15 C2×D5×Dic3 C6×D20 C3×D20 C2×D20 Dic15 C4⋊Dic3 D20 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 C12 C10 C10 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 1 2 1 8 1 2 2 4 1 2 2 4 2 8 1 1 2 2 2 4 2 4 4

Matrix representation of D208Dic3 in GL6(𝔽61)

 0 60 0 0 0 0 1 44 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 1 0 0 0 0 60 0
,
 11 0 0 0 0 0 4 50 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 34 37 0 0 0 0 10 27
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 46 0 0 0 0 53 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 60 0 0 0 0 0 44 1 0 0 0 0 0 0 60 15 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[11,4,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,34,10,0,0,0,0,37,27],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,53,0,0,0,0,46,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,44,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,15,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

D208Dic3 in GAP, Magma, Sage, TeX

D_{20}\rtimes_8{\rm Dic}_3
% in TeX

G:=Group("D20:8Dic3");
// GroupNames label

G:=SmallGroup(480,510);
// by ID

G=gap.SmallGroup(480,510);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,422,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=d^2=1,b^2=a^15,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^19,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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