metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.1D6, D6.6D20, Dic20⋊2S3, C24.28D10, C60.96C23, Dic3.8D20, C120.26C22, Dic10.19D6, D60.27C22, Dic30.28C22, C3⋊C8.1D10, C8.3(S3×D5), C8⋊S3⋊4D5, C6.6(C2×D20), C10.6(S3×D4), C24⋊D5⋊4C2, (S3×C10).3D4, (C4×S3).3D10, C2.11(S3×D20), C30.17(C2×D4), C5⋊1(Q16⋊S3), (S3×Dic10)⋊9C2, (C3×Dic20)⋊4C2, C3⋊Dic20⋊10C2, C3⋊2(C8.D10), C15⋊3(C8.C22), (C5×Dic3).3D4, D60⋊C2.2C2, C15⋊SD16⋊11C2, C12.73(C22×D5), (S3×C20).26C22, C20.146(C22×S3), (C3×Dic10).23C22, C4.95(C2×S3×D5), (C5×C8⋊S3)⋊4C2, (C5×C3⋊C8).19C22, SmallGroup(480,339)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C4, C22, C5, S3, C6, C8, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, C3×Q8, C5×S3, D15, C30, C8.C22, C40, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C8⋊S3, C24⋊C2, Q8⋊2S3, C3⋊Q16, C3×Q16, S3×Q8, Q8⋊3S3, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, D30, C40⋊C2, 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, 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, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8.C22, D20, C22×D5, S3×D4, S3×D5, C2×D20, Q16⋊S3, C2×S3×D5, C8.D10, S3×D20, 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)
(1 145 21 125)(2 144 22 124)(3 143 23 123)(4 142 24 122)(5 141 25 121)(6 140 26 160)(7 139 27 159)(8 138 28 158)(9 137 29 157)(10 136 30 156)(11 135 31 155)(12 134 32 154)(13 133 33 153)(14 132 34 152)(15 131 35 151)(16 130 36 150)(17 129 37 149)(18 128 38 148)(19 127 39 147)(20 126 40 146)(41 236 61 216)(42 235 62 215)(43 234 63 214)(44 233 64 213)(45 232 65 212)(46 231 66 211)(47 230 67 210)(48 229 68 209)(49 228 69 208)(50 227 70 207)(51 226 71 206)(52 225 72 205)(53 224 73 204)(54 223 74 203)(55 222 75 202)(56 221 76 201)(57 220 77 240)(58 219 78 239)(59 218 79 238)(60 217 80 237)(81 164 101 184)(82 163 102 183)(83 162 103 182)(84 161 104 181)(85 200 105 180)(86 199 106 179)(87 198 107 178)(88 197 108 177)(89 196 109 176)(90 195 110 175)(91 194 111 174)(92 193 112 173)(93 192 113 172)(94 191 114 171)(95 190 115 170)(96 189 116 169)(97 188 117 168)(98 187 118 167)(99 186 119 166)(100 185 120 165)
(1 191 47)(2 192 48)(3 193 49)(4 194 50)(5 195 51)(6 196 52)(7 197 53)(8 198 54)(9 199 55)(10 200 56)(11 161 57)(12 162 58)(13 163 59)(14 164 60)(15 165 61)(16 166 62)(17 167 63)(18 168 64)(19 169 65)(20 170 66)(21 171 67)(22 172 68)(23 173 69)(24 174 70)(25 175 71)(26 176 72)(27 177 73)(28 178 74)(29 179 75)(30 180 76)(31 181 77)(32 182 78)(33 183 79)(34 184 80)(35 185 41)(36 186 42)(37 187 43)(38 188 44)(39 189 45)(40 190 46)(81 237 152)(82 238 153)(83 239 154)(84 240 155)(85 201 156)(86 202 157)(87 203 158)(88 204 159)(89 205 160)(90 206 121)(91 207 122)(92 208 123)(93 209 124)(94 210 125)(95 211 126)(96 212 127)(97 213 128)(98 214 129)(99 215 130)(100 216 131)(101 217 132)(102 218 133)(103 219 134)(104 220 135)(105 221 136)(106 222 137)(107 223 138)(108 224 139)(109 225 140)(110 226 141)(111 227 142)(112 228 143)(113 229 144)(114 230 145)(115 231 146)(116 232 147)(117 233 148)(118 234 149)(119 235 150)(120 236 151)
(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 34)(16 36)(18 38)(20 40)(41 185)(42 166)(43 187)(44 168)(45 189)(46 170)(47 191)(48 172)(49 193)(50 174)(51 195)(52 176)(53 197)(54 178)(55 199)(56 180)(57 161)(58 182)(59 163)(60 184)(61 165)(62 186)(63 167)(64 188)(65 169)(66 190)(67 171)(68 192)(69 173)(70 194)(71 175)(72 196)(73 177)(74 198)(75 179)(76 200)(77 181)(78 162)(79 183)(80 164)(81 217)(82 238)(83 219)(84 240)(85 221)(86 202)(87 223)(88 204)(89 225)(90 206)(91 227)(92 208)(93 229)(94 210)(95 231)(96 212)(97 233)(98 214)(99 235)(100 216)(101 237)(102 218)(103 239)(104 220)(105 201)(106 222)(107 203)(108 224)(109 205)(110 226)(111 207)(112 228)(113 209)(114 230)(115 211)(116 232)(117 213)(118 234)(119 215)(120 236)(122 142)(124 144)(126 146)(128 148)(130 150)(132 152)(134 154)(136 156)(138 158)(140 160)
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), (1,145,21,125)(2,144,22,124)(3,143,23,123)(4,142,24,122)(5,141,25,121)(6,140,26,160)(7,139,27,159)(8,138,28,158)(9,137,29,157)(10,136,30,156)(11,135,31,155)(12,134,32,154)(13,133,33,153)(14,132,34,152)(15,131,35,151)(16,130,36,150)(17,129,37,149)(18,128,38,148)(19,127,39,147)(20,126,40,146)(41,236,61,216)(42,235,62,215)(43,234,63,214)(44,233,64,213)(45,232,65,212)(46,231,66,211)(47,230,67,210)(48,229,68,209)(49,228,69,208)(50,227,70,207)(51,226,71,206)(52,225,72,205)(53,224,73,204)(54,223,74,203)(55,222,75,202)(56,221,76,201)(57,220,77,240)(58,219,78,239)(59,218,79,238)(60,217,80,237)(81,164,101,184)(82,163,102,183)(83,162,103,182)(84,161,104,181)(85,200,105,180)(86,199,106,179)(87,198,107,178)(88,197,108,177)(89,196,109,176)(90,195,110,175)(91,194,111,174)(92,193,112,173)(93,192,113,172)(94,191,114,171)(95,190,115,170)(96,189,116,169)(97,188,117,168)(98,187,118,167)(99,186,119,166)(100,185,120,165), (1,191,47)(2,192,48)(3,193,49)(4,194,50)(5,195,51)(6,196,52)(7,197,53)(8,198,54)(9,199,55)(10,200,56)(11,161,57)(12,162,58)(13,163,59)(14,164,60)(15,165,61)(16,166,62)(17,167,63)(18,168,64)(19,169,65)(20,170,66)(21,171,67)(22,172,68)(23,173,69)(24,174,70)(25,175,71)(26,176,72)(27,177,73)(28,178,74)(29,179,75)(30,180,76)(31,181,77)(32,182,78)(33,183,79)(34,184,80)(35,185,41)(36,186,42)(37,187,43)(38,188,44)(39,189,45)(40,190,46)(81,237,152)(82,238,153)(83,239,154)(84,240,155)(85,201,156)(86,202,157)(87,203,158)(88,204,159)(89,205,160)(90,206,121)(91,207,122)(92,208,123)(93,209,124)(94,210,125)(95,211,126)(96,212,127)(97,213,128)(98,214,129)(99,215,130)(100,216,131)(101,217,132)(102,218,133)(103,219,134)(104,220,135)(105,221,136)(106,222,137)(107,223,138)(108,224,139)(109,225,140)(110,226,141)(111,227,142)(112,228,143)(113,229,144)(114,230,145)(115,231,146)(116,232,147)(117,233,148)(118,234,149)(119,235,150)(120,236,151), (2,22)(4,24)(6,26)(8,28)(10,30)(12,32)(14,34)(16,36)(18,38)(20,40)(41,185)(42,166)(43,187)(44,168)(45,189)(46,170)(47,191)(48,172)(49,193)(50,174)(51,195)(52,176)(53,197)(54,178)(55,199)(56,180)(57,161)(58,182)(59,163)(60,184)(61,165)(62,186)(63,167)(64,188)(65,169)(66,190)(67,171)(68,192)(69,173)(70,194)(71,175)(72,196)(73,177)(74,198)(75,179)(76,200)(77,181)(78,162)(79,183)(80,164)(81,217)(82,238)(83,219)(84,240)(85,221)(86,202)(87,223)(88,204)(89,225)(90,206)(91,227)(92,208)(93,229)(94,210)(95,231)(96,212)(97,233)(98,214)(99,235)(100,216)(101,237)(102,218)(103,239)(104,220)(105,201)(106,222)(107,203)(108,224)(109,205)(110,226)(111,207)(112,228)(113,209)(114,230)(115,211)(116,232)(117,213)(118,234)(119,215)(120,236)(122,142)(124,144)(126,146)(128,148)(130,150)(132,152)(134,154)(136,156)(138,158)(140,160)>;
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), (1,145,21,125)(2,144,22,124)(3,143,23,123)(4,142,24,122)(5,141,25,121)(6,140,26,160)(7,139,27,159)(8,138,28,158)(9,137,29,157)(10,136,30,156)(11,135,31,155)(12,134,32,154)(13,133,33,153)(14,132,34,152)(15,131,35,151)(16,130,36,150)(17,129,37,149)(18,128,38,148)(19,127,39,147)(20,126,40,146)(41,236,61,216)(42,235,62,215)(43,234,63,214)(44,233,64,213)(45,232,65,212)(46,231,66,211)(47,230,67,210)(48,229,68,209)(49,228,69,208)(50,227,70,207)(51,226,71,206)(52,225,72,205)(53,224,73,204)(54,223,74,203)(55,222,75,202)(56,221,76,201)(57,220,77,240)(58,219,78,239)(59,218,79,238)(60,217,80,237)(81,164,101,184)(82,163,102,183)(83,162,103,182)(84,161,104,181)(85,200,105,180)(86,199,106,179)(87,198,107,178)(88,197,108,177)(89,196,109,176)(90,195,110,175)(91,194,111,174)(92,193,112,173)(93,192,113,172)(94,191,114,171)(95,190,115,170)(96,189,116,169)(97,188,117,168)(98,187,118,167)(99,186,119,166)(100,185,120,165), (1,191,47)(2,192,48)(3,193,49)(4,194,50)(5,195,51)(6,196,52)(7,197,53)(8,198,54)(9,199,55)(10,200,56)(11,161,57)(12,162,58)(13,163,59)(14,164,60)(15,165,61)(16,166,62)(17,167,63)(18,168,64)(19,169,65)(20,170,66)(21,171,67)(22,172,68)(23,173,69)(24,174,70)(25,175,71)(26,176,72)(27,177,73)(28,178,74)(29,179,75)(30,180,76)(31,181,77)(32,182,78)(33,183,79)(34,184,80)(35,185,41)(36,186,42)(37,187,43)(38,188,44)(39,189,45)(40,190,46)(81,237,152)(82,238,153)(83,239,154)(84,240,155)(85,201,156)(86,202,157)(87,203,158)(88,204,159)(89,205,160)(90,206,121)(91,207,122)(92,208,123)(93,209,124)(94,210,125)(95,211,126)(96,212,127)(97,213,128)(98,214,129)(99,215,130)(100,216,131)(101,217,132)(102,218,133)(103,219,134)(104,220,135)(105,221,136)(106,222,137)(107,223,138)(108,224,139)(109,225,140)(110,226,141)(111,227,142)(112,228,143)(113,229,144)(114,230,145)(115,231,146)(116,232,147)(117,233,148)(118,234,149)(119,235,150)(120,236,151), (2,22)(4,24)(6,26)(8,28)(10,30)(12,32)(14,34)(16,36)(18,38)(20,40)(41,185)(42,166)(43,187)(44,168)(45,189)(46,170)(47,191)(48,172)(49,193)(50,174)(51,195)(52,176)(53,197)(54,178)(55,199)(56,180)(57,161)(58,182)(59,163)(60,184)(61,165)(62,186)(63,167)(64,188)(65,169)(66,190)(67,171)(68,192)(69,173)(70,194)(71,175)(72,196)(73,177)(74,198)(75,179)(76,200)(77,181)(78,162)(79,183)(80,164)(81,217)(82,238)(83,219)(84,240)(85,221)(86,202)(87,223)(88,204)(89,225)(90,206)(91,227)(92,208)(93,229)(94,210)(95,231)(96,212)(97,233)(98,214)(99,235)(100,216)(101,237)(102,218)(103,239)(104,220)(105,201)(106,222)(107,203)(108,224)(109,205)(110,226)(111,207)(112,228)(113,209)(114,230)(115,211)(116,232)(117,213)(118,234)(119,215)(120,236)(122,142)(124,144)(126,146)(128,148)(130,150)(132,152)(134,154)(136,156)(138,158)(140,160) );
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)], [(1,145,21,125),(2,144,22,124),(3,143,23,123),(4,142,24,122),(5,141,25,121),(6,140,26,160),(7,139,27,159),(8,138,28,158),(9,137,29,157),(10,136,30,156),(11,135,31,155),(12,134,32,154),(13,133,33,153),(14,132,34,152),(15,131,35,151),(16,130,36,150),(17,129,37,149),(18,128,38,148),(19,127,39,147),(20,126,40,146),(41,236,61,216),(42,235,62,215),(43,234,63,214),(44,233,64,213),(45,232,65,212),(46,231,66,211),(47,230,67,210),(48,229,68,209),(49,228,69,208),(50,227,70,207),(51,226,71,206),(52,225,72,205),(53,224,73,204),(54,223,74,203),(55,222,75,202),(56,221,76,201),(57,220,77,240),(58,219,78,239),(59,218,79,238),(60,217,80,237),(81,164,101,184),(82,163,102,183),(83,162,103,182),(84,161,104,181),(85,200,105,180),(86,199,106,179),(87,198,107,178),(88,197,108,177),(89,196,109,176),(90,195,110,175),(91,194,111,174),(92,193,112,173),(93,192,113,172),(94,191,114,171),(95,190,115,170),(96,189,116,169),(97,188,117,168),(98,187,118,167),(99,186,119,166),(100,185,120,165)], [(1,191,47),(2,192,48),(3,193,49),(4,194,50),(5,195,51),(6,196,52),(7,197,53),(8,198,54),(9,199,55),(10,200,56),(11,161,57),(12,162,58),(13,163,59),(14,164,60),(15,165,61),(16,166,62),(17,167,63),(18,168,64),(19,169,65),(20,170,66),(21,171,67),(22,172,68),(23,173,69),(24,174,70),(25,175,71),(26,176,72),(27,177,73),(28,178,74),(29,179,75),(30,180,76),(31,181,77),(32,182,78),(33,183,79),(34,184,80),(35,185,41),(36,186,42),(37,187,43),(38,188,44),(39,189,45),(40,190,46),(81,237,152),(82,238,153),(83,239,154),(84,240,155),(85,201,156),(86,202,157),(87,203,158),(88,204,159),(89,205,160),(90,206,121),(91,207,122),(92,208,123),(93,209,124),(94,210,125),(95,211,126),(96,212,127),(97,213,128),(98,214,129),(99,215,130),(100,216,131),(101,217,132),(102,218,133),(103,219,134),(104,220,135),(105,221,136),(106,222,137),(107,223,138),(108,224,139),(109,225,140),(110,226,141),(111,227,142),(112,228,143),(113,229,144),(114,230,145),(115,231,146),(116,232,147),(117,233,148),(118,234,149),(119,235,150),(120,236,151)], [(2,22),(4,24),(6,26),(8,28),(10,30),(12,32),(14,34),(16,36),(18,38),(20,40),(41,185),(42,166),(43,187),(44,168),(45,189),(46,170),(47,191),(48,172),(49,193),(50,174),(51,195),(52,176),(53,197),(54,178),(55,199),(56,180),(57,161),(58,182),(59,163),(60,184),(61,165),(62,186),(63,167),(64,188),(65,169),(66,190),(67,171),(68,192),(69,173),(70,194),(71,175),(72,196),(73,177),(74,198),(75,179),(76,200),(77,181),(78,162),(79,183),(80,164),(81,217),(82,238),(83,219),(84,240),(85,221),(86,202),(87,223),(88,204),(89,225),(90,206),(91,227),(92,208),(93,229),(94,210),(95,231),(96,212),(97,233),(98,214),(99,235),(100,216),(101,237),(102,218),(103,239),(104,220),(105,201),(106,222),(107,203),(108,224),(109,205),(110,226),(111,207),(112,228),(113,209),(114,230),(115,211),(116,232),(117,213),(118,234),(119,215),(120,236),(122,142),(124,144),(126,146),(128,148),(130,150),(132,152),(134,154),(136,156),(138,158),(140,160)]])
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