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## G = S3×Dic20order 480 = 25·3·5

### Direct product of S3 and Dic20

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×Dic20
G = < a,b,c,d | a3=b2=c40=1, d2=c20, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 636 in 120 conjugacy classes, 44 normal (30 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×5], C22, C5, S3 [×2], C6, C8, C8, C2×C4 [×3], Q8 [×6], C10, C10 [×2], Dic3, Dic3 [×2], C12, C12 [×2], D6, C15, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5 [×4], C20, C20, C2×C10, C3⋊C8, C24, Dic6 [×4], C4×S3, C4×S3 [×2], C3×Q8 [×2], C5×S3 [×2], C30, C2×Q16, C40, C40, Dic10 [×2], Dic10 [×4], C2×Dic5 [×2], C2×C20, S3×C8, Dic12, C3⋊Q16 [×2], C3×Q16, S3×Q8 [×2], C5×Dic3, C3×Dic5 [×2], Dic15 [×2], C60, S3×C10, Dic20, Dic20 [×3], C2×C40, C2×Dic10 [×2], S3×Q16, C5×C3⋊C8, C120, S3×Dic5 [×2], C15⋊Q8 [×2], C3×Dic10 [×2], S3×C20, Dic30 [×2], C2×Dic20, C3⋊Dic20 [×2], C3×Dic20, S3×C40, Dic60, S3×Dic10 [×2], S3×Dic20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], Q16 [×2], C2×D4, D10 [×3], C22×S3, C2×Q16, D20 [×2], C22×D5, S3×D4, S3×D5, Dic20 [×2], C2×D20, S3×Q16, C2×S3×D5, C2×Dic20, S3×D20, S3×Dic20

Smallest permutation representation of S3×Dic20
On 240 points
Generators in S240
(1 99 163)(2 100 164)(3 101 165)(4 102 166)(5 103 167)(6 104 168)(7 105 169)(8 106 170)(9 107 171)(10 108 172)(11 109 173)(12 110 174)(13 111 175)(14 112 176)(15 113 177)(16 114 178)(17 115 179)(18 116 180)(19 117 181)(20 118 182)(21 119 183)(22 120 184)(23 81 185)(24 82 186)(25 83 187)(26 84 188)(27 85 189)(28 86 190)(29 87 191)(30 88 192)(31 89 193)(32 90 194)(33 91 195)(34 92 196)(35 93 197)(36 94 198)(37 95 199)(38 96 200)(39 97 161)(40 98 162)(41 139 221)(42 140 222)(43 141 223)(44 142 224)(45 143 225)(46 144 226)(47 145 227)(48 146 228)(49 147 229)(50 148 230)(51 149 231)(52 150 232)(53 151 233)(54 152 234)(55 153 235)(56 154 236)(57 155 237)(58 156 238)(59 157 239)(60 158 240)(61 159 201)(62 160 202)(63 121 203)(64 122 204)(65 123 205)(66 124 206)(67 125 207)(68 126 208)(69 127 209)(70 128 210)(71 129 211)(72 130 212)(73 131 213)(74 132 214)(75 133 215)(76 134 216)(77 135 217)(78 136 218)(79 137 219)(80 138 220)
(81 185)(82 186)(83 187)(84 188)(85 189)(86 190)(87 191)(88 192)(89 193)(90 194)(91 195)(92 196)(93 197)(94 198)(95 199)(96 200)(97 161)(98 162)(99 163)(100 164)(101 165)(102 166)(103 167)(104 168)(105 169)(106 170)(107 171)(108 172)(109 173)(110 174)(111 175)(112 176)(113 177)(114 178)(115 179)(116 180)(117 181)(118 182)(119 183)(120 184)(121 203)(122 204)(123 205)(124 206)(125 207)(126 208)(127 209)(128 210)(129 211)(130 212)(131 213)(132 214)(133 215)(134 216)(135 217)(136 218)(137 219)(138 220)(139 221)(140 222)(141 223)(142 224)(143 225)(144 226)(145 227)(146 228)(147 229)(148 230)(149 231)(150 232)(151 233)(152 234)(153 235)(154 236)(155 237)(156 238)(157 239)(158 240)(159 201)(160 202)
(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 51 21 71)(2 50 22 70)(3 49 23 69)(4 48 24 68)(5 47 25 67)(6 46 26 66)(7 45 27 65)(8 44 28 64)(9 43 29 63)(10 42 30 62)(11 41 31 61)(12 80 32 60)(13 79 33 59)(14 78 34 58)(15 77 35 57)(16 76 36 56)(17 75 37 55)(18 74 38 54)(19 73 39 53)(20 72 40 52)(81 127 101 147)(82 126 102 146)(83 125 103 145)(84 124 104 144)(85 123 105 143)(86 122 106 142)(87 121 107 141)(88 160 108 140)(89 159 109 139)(90 158 110 138)(91 157 111 137)(92 156 112 136)(93 155 113 135)(94 154 114 134)(95 153 115 133)(96 152 116 132)(97 151 117 131)(98 150 118 130)(99 149 119 129)(100 148 120 128)(161 233 181 213)(162 232 182 212)(163 231 183 211)(164 230 184 210)(165 229 185 209)(166 228 186 208)(167 227 187 207)(168 226 188 206)(169 225 189 205)(170 224 190 204)(171 223 191 203)(172 222 192 202)(173 221 193 201)(174 220 194 240)(175 219 195 239)(176 218 196 238)(177 217 197 237)(178 216 198 236)(179 215 199 235)(180 214 200 234)

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

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

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

69 conjugacy classes

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

69 irreducible representations

 dim 1 1 1 1 1 1 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 S3 D4 D4 D5 D6 D6 Q16 D10 D10 D10 D20 D20 Dic20 S3×D4 S3×D5 S3×Q16 C2×S3×D5 S3×D20 S3×Dic20 kernel S3×Dic20 C3⋊Dic20 C3×Dic20 S3×C40 Dic60 S3×Dic10 Dic20 C5×Dic3 S3×C10 S3×C8 C40 Dic10 C5×S3 C3⋊C8 C24 C4×S3 Dic3 D6 S3 C10 C8 C5 C4 C2 C1 # reps 1 2 1 1 1 2 1 1 1 2 1 2 4 2 2 2 4 4 16 1 2 2 2 4 8

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

 240 240 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 240 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 29 20 0 0 221 194
,
 240 0 0 0 0 240 0 0 0 0 78 22 0 0 63 163
G:=sub<GL(4,GF(241))| [240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,29,221,0,0,20,194],[240,0,0,0,0,240,0,0,0,0,78,63,0,0,22,163] >;

S3×Dic20 in GAP, Magma, Sage, TeX

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

G:=Group("S3xDic20");
// GroupNames label

G:=SmallGroup(480,338);
// by ID

G=gap.SmallGroup(480,338);
# by ID

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

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

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