D40:5S3 - GroupNames

G = D₄₀⋊₅S₃ order 480 = 2⁵·3·5

5^th semidirect product of D₄₀ and S₃ acting via S₃/C₃=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄₀⋊₅S₃, D₂₀.₄D₆, C₄₀.₂₀D₆, Dic₁₂⋊₅D₅, D₃₀.₂₄D₄, C₂₄.₂₀D₁₀, Dic₆.₃D₁₀, C₆₀.₁₄₅C₂³, C₁₂₀.₃₇C₂², Dic₁₅.₂₉D₄, (C₃×D₄₀)⋊₈C₂, (C₈×D₁₅)⋊₇C₂, C₁₅⋊₈(C₄○D₈), C₆.₃₄(D₄×D₅), C₈.₃₀(S₃×D₅), C₅⋊₂(D₈⋊₃S₃), C₃₀.₂₇(C₂×D₄), C₁₀.₃₄(S₃×D₄), C₃⋊₂(Q₈.D₁₀), (C₅×Dic₁₂)⋊₇C₂, D₂₀⋊S₃⋊₁₀C₂, C₃₀.D₄⋊₁₁C₂, C₂₀.₈₀(C₂²×S₃), C₁₂.₈₀(C₂²×D₅), C₂.₁₂(C₂₀⋊D₆), C₁₅⋊₃C₈.₄₂C₂², (C₃×D₂₀).₂₈C₂², (C₄×D₁₅).₅₇C₂², (C₅×Dic₆).₂₉C₂², C₄.₁₁₈(C₂×S₃×D₅), SmallGroup(480,353)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆₀ — D₄₀⋊₅S₃

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₆₀ — C₃×D₂₀ — D₂₀⋊S₃ — D₄₀⋊₅S₃

Lower central

C₁₅ — C₃₀ — C₆₀ — D₄₀⋊₅S₃

Upper central

C₁ — C₂ — C₄ — C₈

Generators and relations for D₄₀⋊₅S₃
G = < a,b,c,d | a⁴⁰=b²=c³=d²=1, bab=a^-1, ac=ca, dad=a⁹, bc=cb, dbd=a²⁸b, dcd=c^-1 >

Subgroups: 732 in 124 conjugacy classes, 38 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₅, S₃, C₆, C₆, C₈, C₈, C₂×C₄, D₄, Q₈, D₅, C₁₀, Dic₃, C₁₂, D₆, C₂×C₆, C₁₅, C₂×C₈, D₈, SD₁₆, Q₁₆, C₄○D₄, Dic₅, C₂₀, C₂₀, D₁₀, C₃⋊C₈, C₂₄, Dic₆, C₄×S₃, C₂×Dic₃, C₃⋊D₄, C₃×D₄, C₃×D₅, D₁₅, C₃₀, C₄○D₈, C₅⋊₂C₈, C₄₀, C₄×D₅, D₂₀, D₂₀, C₅×Q₈, S₃×C₈, Dic₁₂, D₄.S₃, C₃×D₈, D₄⋊₂S₃, C₅×Dic₃, Dic₁₅, C₆₀, C₆×D₅, D₃₀, C₈×D₅, D₄₀, Q₈⋊D₅, C₅×Q₁₆, Q₈⋊₂D₅, D₈⋊₃S₃, C₁₅⋊₃C₈, C₁₂₀, D₅×Dic₃, C₃⋊D₂₀, C₃×D₂₀, C₅×Dic₆, C₄×D₁₅, Q₈.D₁₀, C₃₀.D₄, C₃×D₄₀, C₅×Dic₁₂, C₈×D₁₅, D₂₀⋊S₃, D₄₀⋊₅S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, C₂×D₄, D₁₀, C₂²×S₃, C₄○D₈, C₂²×D₅, S₃×D₄, S₃×D₅, D₄×D₅, D₈⋊₃S₃, C₂×S₃×D₅, Q₈.D₁₀, C₂₀⋊D₆, D₄₀⋊₅S₃

Smallest permutation representation of D₄₀⋊₅S₃
►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)

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

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

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

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

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

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

51 conjugacy classes

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

51 irreducible representations

dim	1	1	1	1	1	1	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	S₃	D₄	D₄	D₅	D₆	D₆	D₁₀	D₁₀	C₄○D₈	S₃×D₄	S₃×D₅	D₄×D₅	D₈⋊₃S₃	C₂×S₃×D₅	Q₈.D₁₀	C₂₀⋊D₆	D₄₀⋊₅S₃
kernel	D₄₀⋊₅S₃	C₃₀.D₄	C₃×D₄₀	C₅×Dic₁₂	C₈×D₁₅	D₂₀⋊S₃	D₄₀	Dic₁₅	D₃₀	Dic₁₂	C₄₀	D₂₀	C₂₄	Dic₆	C₁₅	C₁₀	C₈	C₆	C₅	C₄	C₃	C₂	C₁
# reps	1	2	1	1	1	2	1	1	1	2	1	2	2	4	4	1	2	2	2	2	4	4	8

Matrix representation of D₄₀⋊₅S₃ ►in GL₆(𝔽₂₄₁)

189	190	0	0	0	0
52	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	30	0
0	0	0	0	0	233

52	1	0	0	0	0
189	189	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	24
0	0	0	0	231	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	240	1	0	0
0	0	240	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

189	190	0	0	0	0
53	52	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	240

G:=sub<GL(6,GF(241))| [189,52,0,0,0,0,190,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,233],[52,189,0,0,0,0,1,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,231,0,0,0,0,24,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[189,53,0,0,0,0,190,52,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240] >;

D₄₀⋊₅S₃ in GAP, Magma, Sage, TeX

D_{40}\rtimes_5S_3

% in TeX

G:=Group("D40:5S3");

// GroupNames label

G:=SmallGroup(480,353);

// by ID

G=gap.SmallGroup(480,353);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,135,142,675,346,80,1356,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^40=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^28*b,d*c*d=c^-1>;

// generators/relations

G = D40⋊5S3 order 480 = 25·3·5

5th semidirect product of D40 and S3 acting via S3/C3=C2

G = D₄₀⋊₅S₃ order 480 = 2⁵·3·5

5^th semidirect product of D₄₀ and S₃ acting via S₃/C₃=C₂