metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊7S3, D6.2D20, C40.45D6, Dic60⋊5C2, D20.21D6, C24.18D10, C60.98C23, C120.7C22, Dic3.13D20, Dic30.30C22, (S3×C8)⋊2D5, (S3×C40)⋊2C2, (C3×D40)⋊3C2, C15⋊5(C4○D8), C3⋊C8.29D10, C6.8(C2×D20), C10.8(S3×D4), C8.12(S3×D5), C5⋊1(D8⋊3S3), C30.23(C2×D4), C2.13(S3×D20), C3⋊2(D40⋊7C2), (S3×C10).21D4, (C4×S3).40D10, D20⋊5S3⋊10C2, C6.D20⋊11C2, (C5×Dic3).24D4, C12.76(C22×D5), (S3×C20).46C22, C20.148(C22×S3), (C3×D20).23C22, C4.97(C2×S3×D5), (C5×C3⋊C8).33C22, SmallGroup(480,349)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40⋊7S3
G = < a,b,c,d | a40=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a20b, dcd=c-1 >
Subgroups: 732 in 124 conjugacy classes, 40 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C5×S3, C3×D5, C30, C4○D8, C40, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, S3×C8, Dic12, D4.S3, C3×D8, D4⋊2S3, C5×Dic3, Dic15, C60, C6×D5, S3×C10, C40⋊C2, D40, Dic20, C2×C40, C4○D20, D8⋊3S3, C5×C3⋊C8, C120, D5×Dic3, C15⋊D4, C3×D20, S3×C20, Dic30, D40⋊7C2, C6.D20, C3×D40, S3×C40, Dic60, D20⋊5S3, D40⋊7S3
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C4○D8, D20, C22×D5, S3×D4, S3×D5, C2×D20, D8⋊3S3, C2×S3×D5, D40⋊7C2, S3×D20, D40⋊7S3
►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)
(1 105)(2 104)(3 103)(4 102)(5 101)(6 100)(7 99)(8 98)(9 97)(10 96)(11 95)(12 94)(13 93)(14 92)(15 91)(16 90)(17 89)(18 88)(19 87)(20 86)(21 85)(22 84)(23 83)(24 82)(25 81)(26 120)(27 119)(28 118)(29 117)(30 116)(31 115)(32 114)(33 113)(34 112)(35 111)(36 110)(37 109)(38 108)(39 107)(40 106)(41 198)(42 197)(43 196)(44 195)(45 194)(46 193)(47 192)(48 191)(49 190)(50 189)(51 188)(52 187)(53 186)(54 185)(55 184)(56 183)(57 182)(58 181)(59 180)(60 179)(61 178)(62 177)(63 176)(64 175)(65 174)(66 173)(67 172)(68 171)(69 170)(70 169)(71 168)(72 167)(73 166)(74 165)(75 164)(76 163)(77 162)(78 161)(79 200)(80 199)(121 217)(122 216)(123 215)(124 214)(125 213)(126 212)(127 211)(128 210)(129 209)(130 208)(131 207)(132 206)(133 205)(134 204)(135 203)(136 202)(137 201)(138 240)(139 239)(140 238)(141 237)(142 236)(143 235)(144 234)(145 233)(146 232)(147 231)(148 230)(149 229)(150 228)(151 227)(152 226)(153 225)(154 224)(155 223)(156 222)(157 221)(158 220)(159 219)(160 218)
(1 238 59)(2 239 60)(3 240 61)(4 201 62)(5 202 63)(6 203 64)(7 204 65)(8 205 66)(9 206 67)(10 207 68)(11 208 69)(12 209 70)(13 210 71)(14 211 72)(15 212 73)(16 213 74)(17 214 75)(18 215 76)(19 216 77)(20 217 78)(21 218 79)(22 219 80)(23 220 41)(24 221 42)(25 222 43)(26 223 44)(27 224 45)(28 225 46)(29 226 47)(30 227 48)(31 228 49)(32 229 50)(33 230 51)(34 231 52)(35 232 53)(36 233 54)(37 234 55)(38 235 56)(39 236 57)(40 237 58)(81 156 196)(82 157 197)(83 158 198)(84 159 199)(85 160 200)(86 121 161)(87 122 162)(88 123 163)(89 124 164)(90 125 165)(91 126 166)(92 127 167)(93 128 168)(94 129 169)(95 130 170)(96 131 171)(97 132 172)(98 133 173)(99 134 174)(100 135 175)(101 136 176)(102 137 177)(103 138 178)(104 139 179)(105 140 180)(106 141 181)(107 142 182)(108 143 183)(109 144 184)(110 145 185)(111 146 186)(112 147 187)(113 148 188)(114 149 189)(115 150 190)(116 151 191)(117 152 192)(118 153 193)(119 154 194)(120 155 195)
(41 220)(42 221)(43 222)(44 223)(45 224)(46 225)(47 226)(48 227)(49 228)(50 229)(51 230)(52 231)(53 232)(54 233)(55 234)(56 235)(57 236)(58 237)(59 238)(60 239)(61 240)(62 201)(63 202)(64 203)(65 204)(66 205)(67 206)(68 207)(69 208)(70 209)(71 210)(72 211)(73 212)(74 213)(75 214)(76 215)(77 216)(78 217)(79 218)(80 219)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 181)(122 182)(123 183)(124 184)(125 185)(126 186)(127 187)(128 188)(129 189)(130 190)(131 191)(132 192)(133 193)(134 194)(135 195)(136 196)(137 197)(138 198)(139 199)(140 200)(141 161)(142 162)(143 163)(144 164)(145 165)(146 166)(147 167)(148 168)(149 169)(150 170)(151 171)(152 172)(153 173)(154 174)(155 175)(156 176)(157 177)(158 178)(159 179)(160 180)
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), (1,105)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,84)(23,83)(24,82)(25,81)(26,120)(27,119)(28,118)(29,117)(30,116)(31,115)(32,114)(33,113)(34,112)(35,111)(36,110)(37,109)(38,108)(39,107)(40,106)(41,198)(42,197)(43,196)(44,195)(45,194)(46,193)(47,192)(48,191)(49,190)(50,189)(51,188)(52,187)(53,186)(54,185)(55,184)(56,183)(57,182)(58,181)(59,180)(60,179)(61,178)(62,177)(63,176)(64,175)(65,174)(66,173)(67,172)(68,171)(69,170)(70,169)(71,168)(72,167)(73,166)(74,165)(75,164)(76,163)(77,162)(78,161)(79,200)(80,199)(121,217)(122,216)(123,215)(124,214)(125,213)(126,212)(127,211)(128,210)(129,209)(130,208)(131,207)(132,206)(133,205)(134,204)(135,203)(136,202)(137,201)(138,240)(139,239)(140,238)(141,237)(142,236)(143,235)(144,234)(145,233)(146,232)(147,231)(148,230)(149,229)(150,228)(151,227)(152,226)(153,225)(154,224)(155,223)(156,222)(157,221)(158,220)(159,219)(160,218), (1,238,59)(2,239,60)(3,240,61)(4,201,62)(5,202,63)(6,203,64)(7,204,65)(8,205,66)(9,206,67)(10,207,68)(11,208,69)(12,209,70)(13,210,71)(14,211,72)(15,212,73)(16,213,74)(17,214,75)(18,215,76)(19,216,77)(20,217,78)(21,218,79)(22,219,80)(23,220,41)(24,221,42)(25,222,43)(26,223,44)(27,224,45)(28,225,46)(29,226,47)(30,227,48)(31,228,49)(32,229,50)(33,230,51)(34,231,52)(35,232,53)(36,233,54)(37,234,55)(38,235,56)(39,236,57)(40,237,58)(81,156,196)(82,157,197)(83,158,198)(84,159,199)(85,160,200)(86,121,161)(87,122,162)(88,123,163)(89,124,164)(90,125,165)(91,126,166)(92,127,167)(93,128,168)(94,129,169)(95,130,170)(96,131,171)(97,132,172)(98,133,173)(99,134,174)(100,135,175)(101,136,176)(102,137,177)(103,138,178)(104,139,179)(105,140,180)(106,141,181)(107,142,182)(108,143,183)(109,144,184)(110,145,185)(111,146,186)(112,147,187)(113,148,188)(114,149,189)(115,150,190)(116,151,191)(117,152,192)(118,153,193)(119,154,194)(120,155,195), (41,220)(42,221)(43,222)(44,223)(45,224)(46,225)(47,226)(48,227)(49,228)(50,229)(51,230)(52,231)(53,232)(54,233)(55,234)(56,235)(57,236)(58,237)(59,238)(60,239)(61,240)(62,201)(63,202)(64,203)(65,204)(66,205)(67,206)(68,207)(69,208)(70,209)(71,210)(72,211)(73,212)(74,213)(75,214)(76,215)(77,216)(78,217)(79,218)(80,219)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)(136,196)(137,197)(138,198)(139,199)(140,200)(141,161)(142,162)(143,163)(144,164)(145,165)(146,166)(147,167)(148,168)(149,169)(150,170)(151,171)(152,172)(153,173)(154,174)(155,175)(156,176)(157,177)(158,178)(159,179)(160,180)>;
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), (1,105)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,84)(23,83)(24,82)(25,81)(26,120)(27,119)(28,118)(29,117)(30,116)(31,115)(32,114)(33,113)(34,112)(35,111)(36,110)(37,109)(38,108)(39,107)(40,106)(41,198)(42,197)(43,196)(44,195)(45,194)(46,193)(47,192)(48,191)(49,190)(50,189)(51,188)(52,187)(53,186)(54,185)(55,184)(56,183)(57,182)(58,181)(59,180)(60,179)(61,178)(62,177)(63,176)(64,175)(65,174)(66,173)(67,172)(68,171)(69,170)(70,169)(71,168)(72,167)(73,166)(74,165)(75,164)(76,163)(77,162)(78,161)(79,200)(80,199)(121,217)(122,216)(123,215)(124,214)(125,213)(126,212)(127,211)(128,210)(129,209)(130,208)(131,207)(132,206)(133,205)(134,204)(135,203)(136,202)(137,201)(138,240)(139,239)(140,238)(141,237)(142,236)(143,235)(144,234)(145,233)(146,232)(147,231)(148,230)(149,229)(150,228)(151,227)(152,226)(153,225)(154,224)(155,223)(156,222)(157,221)(158,220)(159,219)(160,218), (1,238,59)(2,239,60)(3,240,61)(4,201,62)(5,202,63)(6,203,64)(7,204,65)(8,205,66)(9,206,67)(10,207,68)(11,208,69)(12,209,70)(13,210,71)(14,211,72)(15,212,73)(16,213,74)(17,214,75)(18,215,76)(19,216,77)(20,217,78)(21,218,79)(22,219,80)(23,220,41)(24,221,42)(25,222,43)(26,223,44)(27,224,45)(28,225,46)(29,226,47)(30,227,48)(31,228,49)(32,229,50)(33,230,51)(34,231,52)(35,232,53)(36,233,54)(37,234,55)(38,235,56)(39,236,57)(40,237,58)(81,156,196)(82,157,197)(83,158,198)(84,159,199)(85,160,200)(86,121,161)(87,122,162)(88,123,163)(89,124,164)(90,125,165)(91,126,166)(92,127,167)(93,128,168)(94,129,169)(95,130,170)(96,131,171)(97,132,172)(98,133,173)(99,134,174)(100,135,175)(101,136,176)(102,137,177)(103,138,178)(104,139,179)(105,140,180)(106,141,181)(107,142,182)(108,143,183)(109,144,184)(110,145,185)(111,146,186)(112,147,187)(113,148,188)(114,149,189)(115,150,190)(116,151,191)(117,152,192)(118,153,193)(119,154,194)(120,155,195), (41,220)(42,221)(43,222)(44,223)(45,224)(46,225)(47,226)(48,227)(49,228)(50,229)(51,230)(52,231)(53,232)(54,233)(55,234)(56,235)(57,236)(58,237)(59,238)(60,239)(61,240)(62,201)(63,202)(64,203)(65,204)(66,205)(67,206)(68,207)(69,208)(70,209)(71,210)(72,211)(73,212)(74,213)(75,214)(76,215)(77,216)(78,217)(79,218)(80,219)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)(136,196)(137,197)(138,198)(139,199)(140,200)(141,161)(142,162)(143,163)(144,164)(145,165)(146,166)(147,167)(148,168)(149,169)(150,170)(151,171)(152,172)(153,173)(154,174)(155,175)(156,176)(157,177)(158,178)(159,179)(160,180) );
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)], [(1,105),(2,104),(3,103),(4,102),(5,101),(6,100),(7,99),(8,98),(9,97),(10,96),(11,95),(12,94),(13,93),(14,92),(15,91),(16,90),(17,89),(18,88),(19,87),(20,86),(21,85),(22,84),(23,83),(24,82),(25,81),(26,120),(27,119),(28,118),(29,117),(30,116),(31,115),(32,114),(33,113),(34,112),(35,111),(36,110),(37,109),(38,108),(39,107),(40,106),(41,198),(42,197),(43,196),(44,195),(45,194),(46,193),(47,192),(48,191),(49,190),(50,189),(51,188),(52,187),(53,186),(54,185),(55,184),(56,183),(57,182),(58,181),(59,180),(60,179),(61,178),(62,177),(63,176),(64,175),(65,174),(66,173),(67,172),(68,171),(69,170),(70,169),(71,168),(72,167),(73,166),(74,165),(75,164),(76,163),(77,162),(78,161),(79,200),(80,199),(121,217),(122,216),(123,215),(124,214),(125,213),(126,212),(127,211),(128,210),(129,209),(130,208),(131,207),(132,206),(133,205),(134,204),(135,203),(136,202),(137,201),(138,240),(139,239),(140,238),(141,237),(142,236),(143,235),(144,234),(145,233),(146,232),(147,231),(148,230),(149,229),(150,228),(151,227),(152,226),(153,225),(154,224),(155,223),(156,222),(157,221),(158,220),(159,219),(160,218)], [(1,238,59),(2,239,60),(3,240,61),(4,201,62),(5,202,63),(6,203,64),(7,204,65),(8,205,66),(9,206,67),(10,207,68),(11,208,69),(12,209,70),(13,210,71),(14,211,72),(15,212,73),(16,213,74),(17,214,75),(18,215,76),(19,216,77),(20,217,78),(21,218,79),(22,219,80),(23,220,41),(24,221,42),(25,222,43),(26,223,44),(27,224,45),(28,225,46),(29,226,47),(30,227,48),(31,228,49),(32,229,50),(33,230,51),(34,231,52),(35,232,53),(36,233,54),(37,234,55),(38,235,56),(39,236,57),(40,237,58),(81,156,196),(82,157,197),(83,158,198),(84,159,199),(85,160,200),(86,121,161),(87,122,162),(88,123,163),(89,124,164),(90,125,165),(91,126,166),(92,127,167),(93,128,168),(94,129,169),(95,130,170),(96,131,171),(97,132,172),(98,133,173),(99,134,174),(100,135,175),(101,136,176),(102,137,177),(103,138,178),(104,139,179),(105,140,180),(106,141,181),(107,142,182),(108,143,183),(109,144,184),(110,145,185),(111,146,186),(112,147,187),(113,148,188),(114,149,189),(115,150,190),(116,151,191),(117,152,192),(118,153,193),(119,154,194),(120,155,195)], [(41,220),(42,221),(43,222),(44,223),(45,224),(46,225),(47,226),(48,227),(49,228),(50,229),(51,230),(52,231),(53,232),(54,233),(55,234),(56,235),(57,236),(58,237),(59,238),(60,239),(61,240),(62,201),(63,202),(64,203),(65,204),(66,205),(67,206),(68,207),(69,208),(70,209),(71,210),(72,211),(73,212),(74,213),(75,214),(76,215),(77,216),(78,217),(79,218),(80,219),(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(121,181),(122,182),(123,183),(124,184),(125,185),(126,186),(127,187),(128,188),(129,189),(130,190),(131,191),(132,192),(133,193),(134,194),(135,195),(136,196),(137,197),(138,198),(139,199),(140,200),(141,161),(142,162),(143,163),(144,164),(145,165),(146,166),(147,167),(148,168),(149,169),(150,170),(151,171),(152,172),(153,173),(154,174),(155,175),(156,176),(157,177),(158,178),(159,179),(160,180)]])
69 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 | 10C | 10D | 10E | 10F | 12 | 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 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 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 | 6 | 20 | 20 | 2 | 2 | 3 | 3 | 60 | 60 | 2 | 2 | 2 | 40 | 40 | 2 | 2 | 6 | 6 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 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 | D10 | D10 | D10 | C4○D8 | D20 | D20 | D40⋊7C2 | S3×D4 | S3×D5 | D8⋊3S3 | C2×S3×D5 | S3×D20 | D40⋊7S3 |
| kernel | D40⋊7S3 | C6.D20 | C3×D40 | S3×C40 | Dic60 | D20⋊5S3 | D40 | C5×Dic3 | S3×C10 | S3×C8 | C40 | D20 | C3⋊C8 | C24 | C4×S3 | C15 | Dic3 | D6 | C3 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 16 | 1 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D40⋊7S3 ►in GL4(𝔽241) generated by
| 116 | 218 | 0 | 0 |
| 0 | 214 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 67 | 132 | 0 | 0 |
| 207 | 174 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 240 |
| 0 | 0 | 1 | 0 |
| 1 | 89 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 240 | 240 |
G:=sub<GL(4,GF(241))| [116,0,0,0,218,214,0,0,0,0,1,0,0,0,0,1],[67,207,0,0,132,174,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,240,1,0,0,240,0],[1,0,0,0,89,240,0,0,0,0,1,240,0,0,0,240] >;
D40⋊7S3 in GAP, Magma, Sage, TeX
D_{40}\rtimes_7S_3 % in TeX
G:=Group("D40:7S3"); // GroupNames label
G:=SmallGroup(480,349);
// by ID
G=gap.SmallGroup(480,349);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,142,675,80,1356,18822]);
// 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,a*d=d*a,b*c=c*b,d*b*d=a^20*b,d*c*d=c^-1>;
// generators/relations