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## G = Dic20⋊S3order 480 = 25·3·5

### 2nd semidirect product of Dic20 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic20⋊S3
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — S3×Dic10 — Dic20⋊S3
 Lower central C15 — C30 — C60 — Dic20⋊S3
 Upper central C1 — C2 — C4 — C8

Generators and relations for Dic20⋊S3
G = < a,b,c,d | a40=c3=d2=1, b2=a20, bab-1=a-1, ac=ca, dad=a21, bc=cb, bd=db, dcd=c-1 >

Subgroups: 732 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3 [×2], C6, C8, C8, C2×C4 [×3], D4 [×2], Q8 [×4], D5, C10, C10, Dic3, Dic3, C12, C12 [×2], D6, D6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×3], C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6 [×2], C4×S3, C4×S3 [×2], D12 [×2], C3×Q8 [×2], C5×S3, D15, C30, C8.C22, C40, C40, Dic10 [×2], Dic10 [×2], C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C8⋊S3, C24⋊C2, Q82S3, C3⋊Q16, C3×Q16, S3×Q8, Q83S3, C5×Dic3, C3×Dic5 [×2], Dic15, C60, S3×C10, D30, C40⋊C2 [×2], Dic20, Dic20, C5×M4(2), C2×Dic10, C4○D20, Q16⋊S3, C5×C3⋊C8, C120, S3×Dic5, D30.C2, C5⋊D12, C15⋊Q8, C3×Dic10 [×2], S3×C20, Dic30, D60, C8.D10, C15⋊SD16, C3⋊Dic20, C3×Dic20, C5×C8⋊S3, C24⋊D5, S3×Dic10, D60⋊C2, Dic20⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8.C22, D20 [×2], C22×D5, S3×D4, S3×D5, C2×D20, Q16⋊S3, C2×S3×D5, C8.D10, S3×D20, Dic20⋊S3

Smallest permutation representation of Dic20⋊S3
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)
(1 166 21 186)(2 165 22 185)(3 164 23 184)(4 163 24 183)(5 162 25 182)(6 161 26 181)(7 200 27 180)(8 199 28 179)(9 198 29 178)(10 197 30 177)(11 196 31 176)(12 195 32 175)(13 194 33 174)(14 193 34 173)(15 192 35 172)(16 191 36 171)(17 190 37 170)(18 189 38 169)(19 188 39 168)(20 187 40 167)(41 100 61 120)(42 99 62 119)(43 98 63 118)(44 97 64 117)(45 96 65 116)(46 95 66 115)(47 94 67 114)(48 93 68 113)(49 92 69 112)(50 91 70 111)(51 90 71 110)(52 89 72 109)(53 88 73 108)(54 87 74 107)(55 86 75 106)(56 85 76 105)(57 84 77 104)(58 83 78 103)(59 82 79 102)(60 81 80 101)(121 220 141 240)(122 219 142 239)(123 218 143 238)(124 217 144 237)(125 216 145 236)(126 215 146 235)(127 214 147 234)(128 213 148 233)(129 212 149 232)(130 211 150 231)(131 210 151 230)(132 209 152 229)(133 208 153 228)(134 207 154 227)(135 206 155 226)(136 205 156 225)(137 204 157 224)(138 203 158 223)(139 202 159 222)(140 201 160 221)
(1 122 80)(2 123 41)(3 124 42)(4 125 43)(5 126 44)(6 127 45)(7 128 46)(8 129 47)(9 130 48)(10 131 49)(11 132 50)(12 133 51)(13 134 52)(14 135 53)(15 136 54)(16 137 55)(17 138 56)(18 139 57)(19 140 58)(20 141 59)(21 142 60)(22 143 61)(23 144 62)(24 145 63)(25 146 64)(26 147 65)(27 148 66)(28 149 67)(29 150 68)(30 151 69)(31 152 70)(32 153 71)(33 154 72)(34 155 73)(35 156 74)(36 157 75)(37 158 76)(38 159 77)(39 160 78)(40 121 79)(81 186 239)(82 187 240)(83 188 201)(84 189 202)(85 190 203)(86 191 204)(87 192 205)(88 193 206)(89 194 207)(90 195 208)(91 196 209)(92 197 210)(93 198 211)(94 199 212)(95 200 213)(96 161 214)(97 162 215)(98 163 216)(99 164 217)(100 165 218)(101 166 219)(102 167 220)(103 168 221)(104 169 222)(105 170 223)(106 171 224)(107 172 225)(108 173 226)(109 174 227)(110 175 228)(111 176 229)(112 177 230)(113 178 231)(114 179 232)(115 180 233)(116 181 234)(117 182 235)(118 183 236)(119 184 237)(120 185 238)
(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 34)(16 36)(18 38)(20 40)(41 143)(42 124)(43 145)(44 126)(45 147)(46 128)(47 149)(48 130)(49 151)(50 132)(51 153)(52 134)(53 155)(54 136)(55 157)(56 138)(57 159)(58 140)(59 121)(60 142)(61 123)(62 144)(63 125)(64 146)(65 127)(66 148)(67 129)(68 150)(69 131)(70 152)(71 133)(72 154)(73 135)(74 156)(75 137)(76 158)(77 139)(78 160)(79 141)(80 122)(81 239)(82 220)(83 201)(84 222)(85 203)(86 224)(87 205)(88 226)(89 207)(90 228)(91 209)(92 230)(93 211)(94 232)(95 213)(96 234)(97 215)(98 236)(99 217)(100 238)(101 219)(102 240)(103 221)(104 202)(105 223)(106 204)(107 225)(108 206)(109 227)(110 208)(111 229)(112 210)(113 231)(114 212)(115 233)(116 214)(117 235)(118 216)(119 237)(120 218)(161 181)(163 183)(165 185)(167 187)(169 189)(171 191)(173 193)(175 195)(177 197)(179 199)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 5A 5B 6 8A 8B 10A 10B 10C 10D 12A 12B 12C 15A 15B 20A 20B 20C 20D 20E 20F 24A 24B 30A 30B 40A 40B 40C 40D 40E 40F 40G 40H 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 3 4 4 4 4 4 5 5 6 8 8 10 10 10 10 12 12 12 15 15 20 20 20 20 20 20 24 24 30 30 40 40 40 40 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 6 60 2 2 6 20 20 60 2 2 2 4 12 2 2 12 12 4 40 40 4 4 2 2 2 2 12 12 4 4 4 4 4 4 4 4 12 12 12 12 4 4 4 4 4 ··· 4

54 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 D4 D5 D6 D6 D10 D10 D10 D20 D20 C8.C22 S3×D4 S3×D5 Q16⋊S3 C2×S3×D5 C8.D10 S3×D20 Dic20⋊S3 kernel Dic20⋊S3 C15⋊SD16 C3⋊Dic20 C3×Dic20 C5×C8⋊S3 C24⋊D5 S3×Dic10 D60⋊C2 Dic20 C5×Dic3 S3×C10 C8⋊S3 C40 Dic10 C3⋊C8 C24 C4×S3 Dic3 D6 C15 C10 C8 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 4 4 1 1 2 2 2 4 4 8

Matrix representation of Dic20⋊S3 in GL4(𝔽241) generated by

 102 48 204 96 193 16 145 32 37 145 139 193 96 209 48 225
,
 183 148 0 0 31 58 0 0 0 0 183 148 0 0 31 58
,
 240 0 240 0 0 240 0 240 1 0 0 0 0 1 0 0
,
 1 0 0 0 0 1 0 0 240 0 240 0 0 240 0 240
G:=sub<GL(4,GF(241))| [102,193,37,96,48,16,145,209,204,145,139,48,96,32,193,225],[183,31,0,0,148,58,0,0,0,0,183,31,0,0,148,58],[240,0,1,0,0,240,0,1,240,0,0,0,0,240,0,0],[1,0,240,0,0,1,0,240,0,0,240,0,0,0,0,240] >;

Dic20⋊S3 in GAP, Magma, Sage, TeX

{\rm Dic}_{20}\rtimes S_3
% in TeX

G:=Group("Dic20:S3");
// GroupNames label

G:=SmallGroup(480,339);
// by ID

G=gap.SmallGroup(480,339);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,219,142,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^40=c^3=d^2=1,b^2=a^20,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^21,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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