direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C3⋊D40, C30⋊2D8, C6⋊2D40, D20⋊18D6, C60.37D4, C12.13D20, D60⋊29C22, C60.101C23, C15⋊5(C2×D8), C3⋊3(C2×D40), C3⋊C8⋊25D10, (C2×D20)⋊1S3, (C6×D20)⋊4C2, C10⋊1(D4⋊S3), (C2×D60)⋊20C2, (C2×C6).38D20, C6.46(C2×D20), (C2×C30).43D4, C30.75(C2×D4), (C2×C20).285D6, (C2×C12).88D10, C4.5(C3⋊D20), (C3×D20)⋊20C22, C20.53(C3⋊D4), C12.88(C22×D5), (C2×C60).104C22, C20.151(C22×S3), C22.19(C3⋊D20), C5⋊1(C2×D4⋊S3), (C2×C3⋊C8)⋊5D5, (C10×C3⋊C8)⋊6C2, C4.100(C2×S3×D5), (C5×C3⋊C8)⋊29C22, (C2×C4).94(S3×D5), C10.1(C2×C3⋊D4), C2.5(C2×C3⋊D20), (C2×C10).30(C3⋊D4), SmallGroup(480,376)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C3⋊D40
G = < a,b,c,d | a2=b3=c40=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1148 in 152 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C12, D6, C2×C6, C2×C6, C15, C2×C8, D8, C2×D4, C20, D10, C2×C10, C3⋊C8, D12, C2×C12, C3×D4, C22×S3, C22×C6, C3×D5, D15, C30, C30, C2×D8, C40, D20, D20, C2×C20, C22×D5, C2×C3⋊C8, D4⋊S3, C2×D12, C6×D4, C60, C6×D5, D30, C2×C30, D40, C2×C40, C2×D20, C2×D20, C2×D4⋊S3, C5×C3⋊C8, C3×D20, C3×D20, D60, D60, C2×C60, D5×C2×C6, C22×D15, C2×D40, C3⋊D40, C10×C3⋊C8, C6×D20, C2×D60, C2×C3⋊D40
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, C3⋊D4, C22×S3, C2×D8, D20, C22×D5, D4⋊S3, C2×C3⋊D4, S3×D5, D40, C2×D20, C2×D4⋊S3, C3⋊D20, C2×S3×D5, C2×D40, C3⋊D40, C2×C3⋊D20, C2×C3⋊D40
►On 240 points
Generators in S240
(1 145)(2 146)(3 147)(4 148)(5 149)(6 150)(7 151)(8 152)(9 153)(10 154)(11 155)(12 156)(13 157)(14 158)(15 159)(16 160)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 141)(38 142)(39 143)(40 144)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 113)(73 114)(74 115)(75 116)(76 117)(77 118)(78 119)(79 120)(80 81)(161 213)(162 214)(163 215)(164 216)(165 217)(166 218)(167 219)(168 220)(169 221)(170 222)(171 223)(172 224)(173 225)(174 226)(175 227)(176 228)(177 229)(178 230)(179 231)(180 232)(181 233)(182 234)(183 235)(184 236)(185 237)(186 238)(187 239)(188 240)(189 201)(190 202)(191 203)(192 204)(193 205)(194 206)(195 207)(196 208)(197 209)(198 210)(199 211)(200 212)
(1 109 222)(2 223 110)(3 111 224)(4 225 112)(5 113 226)(6 227 114)(7 115 228)(8 229 116)(9 117 230)(10 231 118)(11 119 232)(12 233 120)(13 81 234)(14 235 82)(15 83 236)(16 237 84)(17 85 238)(18 239 86)(19 87 240)(20 201 88)(21 89 202)(22 203 90)(23 91 204)(24 205 92)(25 93 206)(26 207 94)(27 95 208)(28 209 96)(29 97 210)(30 211 98)(31 99 212)(32 213 100)(33 101 214)(34 215 102)(35 103 216)(36 217 104)(37 105 218)(38 219 106)(39 107 220)(40 221 108)(41 158 183)(42 184 159)(43 160 185)(44 186 121)(45 122 187)(46 188 123)(47 124 189)(48 190 125)(49 126 191)(50 192 127)(51 128 193)(52 194 129)(53 130 195)(54 196 131)(55 132 197)(56 198 133)(57 134 199)(58 200 135)(59 136 161)(60 162 137)(61 138 163)(62 164 139)(63 140 165)(64 166 141)(65 142 167)(66 168 143)(67 144 169)(68 170 145)(69 146 171)(70 172 147)(71 148 173)(72 174 149)(73 150 175)(74 176 151)(75 152 177)(76 178 153)(77 154 179)(78 180 155)(79 156 181)(80 182 157)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 171)(42 170)(43 169)(44 168)(45 167)(46 166)(47 165)(48 164)(49 163)(50 162)(51 161)(52 200)(53 199)(54 198)(55 197)(56 196)(57 195)(58 194)(59 193)(60 192)(61 191)(62 190)(63 189)(64 188)(65 187)(66 186)(67 185)(68 184)(69 183)(70 182)(71 181)(72 180)(73 179)(74 178)(75 177)(76 176)(77 175)(78 174)(79 173)(80 172)(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)(121 143)(122 142)(123 141)(124 140)(125 139)(126 138)(127 137)(128 136)(129 135)(130 134)(131 133)(144 160)(145 159)(146 158)(147 157)(148 156)(149 155)(150 154)(151 153)
G:=sub<Sym(240)| (1,145)(2,146)(3,147)(4,148)(5,149)(6,150)(7,151)(8,152)(9,153)(10,154)(11,155)(12,156)(13,157)(14,158)(15,159)(16,160)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,141)(38,142)(39,143)(40,144)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(73,114)(74,115)(75,116)(76,117)(77,118)(78,119)(79,120)(80,81)(161,213)(162,214)(163,215)(164,216)(165,217)(166,218)(167,219)(168,220)(169,221)(170,222)(171,223)(172,224)(173,225)(174,226)(175,227)(176,228)(177,229)(178,230)(179,231)(180,232)(181,233)(182,234)(183,235)(184,236)(185,237)(186,238)(187,239)(188,240)(189,201)(190,202)(191,203)(192,204)(193,205)(194,206)(195,207)(196,208)(197,209)(198,210)(199,211)(200,212), (1,109,222)(2,223,110)(3,111,224)(4,225,112)(5,113,226)(6,227,114)(7,115,228)(8,229,116)(9,117,230)(10,231,118)(11,119,232)(12,233,120)(13,81,234)(14,235,82)(15,83,236)(16,237,84)(17,85,238)(18,239,86)(19,87,240)(20,201,88)(21,89,202)(22,203,90)(23,91,204)(24,205,92)(25,93,206)(26,207,94)(27,95,208)(28,209,96)(29,97,210)(30,211,98)(31,99,212)(32,213,100)(33,101,214)(34,215,102)(35,103,216)(36,217,104)(37,105,218)(38,219,106)(39,107,220)(40,221,108)(41,158,183)(42,184,159)(43,160,185)(44,186,121)(45,122,187)(46,188,123)(47,124,189)(48,190,125)(49,126,191)(50,192,127)(51,128,193)(52,194,129)(53,130,195)(54,196,131)(55,132,197)(56,198,133)(57,134,199)(58,200,135)(59,136,161)(60,162,137)(61,138,163)(62,164,139)(63,140,165)(64,166,141)(65,142,167)(66,168,143)(67,144,169)(68,170,145)(69,146,171)(70,172,147)(71,148,173)(72,174,149)(73,150,175)(74,176,151)(75,152,177)(76,178,153)(77,154,179)(78,180,155)(79,156,181)(80,182,157), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,171)(42,170)(43,169)(44,168)(45,167)(46,166)(47,165)(48,164)(49,163)(50,162)(51,161)(52,200)(53,199)(54,198)(55,197)(56,196)(57,195)(58,194)(59,193)(60,192)(61,191)(62,190)(63,189)(64,188)(65,187)(66,186)(67,185)(68,184)(69,183)(70,182)(71,181)(72,180)(73,179)(74,178)(75,177)(76,176)(77,175)(78,174)(79,173)(80,172)(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)(121,143)(122,142)(123,141)(124,140)(125,139)(126,138)(127,137)(128,136)(129,135)(130,134)(131,133)(144,160)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154)(151,153)>;
G:=Group( (1,145)(2,146)(3,147)(4,148)(5,149)(6,150)(7,151)(8,152)(9,153)(10,154)(11,155)(12,156)(13,157)(14,158)(15,159)(16,160)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,141)(38,142)(39,143)(40,144)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(73,114)(74,115)(75,116)(76,117)(77,118)(78,119)(79,120)(80,81)(161,213)(162,214)(163,215)(164,216)(165,217)(166,218)(167,219)(168,220)(169,221)(170,222)(171,223)(172,224)(173,225)(174,226)(175,227)(176,228)(177,229)(178,230)(179,231)(180,232)(181,233)(182,234)(183,235)(184,236)(185,237)(186,238)(187,239)(188,240)(189,201)(190,202)(191,203)(192,204)(193,205)(194,206)(195,207)(196,208)(197,209)(198,210)(199,211)(200,212), (1,109,222)(2,223,110)(3,111,224)(4,225,112)(5,113,226)(6,227,114)(7,115,228)(8,229,116)(9,117,230)(10,231,118)(11,119,232)(12,233,120)(13,81,234)(14,235,82)(15,83,236)(16,237,84)(17,85,238)(18,239,86)(19,87,240)(20,201,88)(21,89,202)(22,203,90)(23,91,204)(24,205,92)(25,93,206)(26,207,94)(27,95,208)(28,209,96)(29,97,210)(30,211,98)(31,99,212)(32,213,100)(33,101,214)(34,215,102)(35,103,216)(36,217,104)(37,105,218)(38,219,106)(39,107,220)(40,221,108)(41,158,183)(42,184,159)(43,160,185)(44,186,121)(45,122,187)(46,188,123)(47,124,189)(48,190,125)(49,126,191)(50,192,127)(51,128,193)(52,194,129)(53,130,195)(54,196,131)(55,132,197)(56,198,133)(57,134,199)(58,200,135)(59,136,161)(60,162,137)(61,138,163)(62,164,139)(63,140,165)(64,166,141)(65,142,167)(66,168,143)(67,144,169)(68,170,145)(69,146,171)(70,172,147)(71,148,173)(72,174,149)(73,150,175)(74,176,151)(75,152,177)(76,178,153)(77,154,179)(78,180,155)(79,156,181)(80,182,157), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,171)(42,170)(43,169)(44,168)(45,167)(46,166)(47,165)(48,164)(49,163)(50,162)(51,161)(52,200)(53,199)(54,198)(55,197)(56,196)(57,195)(58,194)(59,193)(60,192)(61,191)(62,190)(63,189)(64,188)(65,187)(66,186)(67,185)(68,184)(69,183)(70,182)(71,181)(72,180)(73,179)(74,178)(75,177)(76,176)(77,175)(78,174)(79,173)(80,172)(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)(121,143)(122,142)(123,141)(124,140)(125,139)(126,138)(127,137)(128,136)(129,135)(130,134)(131,133)(144,160)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154)(151,153) );
G=PermutationGroup([[(1,145),(2,146),(3,147),(4,148),(5,149),(6,150),(7,151),(8,152),(9,153),(10,154),(11,155),(12,156),(13,157),(14,158),(15,159),(16,160),(17,121),(18,122),(19,123),(20,124),(21,125),(22,126),(23,127),(24,128),(25,129),(26,130),(27,131),(28,132),(29,133),(30,134),(31,135),(32,136),(33,137),(34,138),(35,139),(36,140),(37,141),(38,142),(39,143),(40,144),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(47,88),(48,89),(49,90),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,97),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,113),(73,114),(74,115),(75,116),(76,117),(77,118),(78,119),(79,120),(80,81),(161,213),(162,214),(163,215),(164,216),(165,217),(166,218),(167,219),(168,220),(169,221),(170,222),(171,223),(172,224),(173,225),(174,226),(175,227),(176,228),(177,229),(178,230),(179,231),(180,232),(181,233),(182,234),(183,235),(184,236),(185,237),(186,238),(187,239),(188,240),(189,201),(190,202),(191,203),(192,204),(193,205),(194,206),(195,207),(196,208),(197,209),(198,210),(199,211),(200,212)], [(1,109,222),(2,223,110),(3,111,224),(4,225,112),(5,113,226),(6,227,114),(7,115,228),(8,229,116),(9,117,230),(10,231,118),(11,119,232),(12,233,120),(13,81,234),(14,235,82),(15,83,236),(16,237,84),(17,85,238),(18,239,86),(19,87,240),(20,201,88),(21,89,202),(22,203,90),(23,91,204),(24,205,92),(25,93,206),(26,207,94),(27,95,208),(28,209,96),(29,97,210),(30,211,98),(31,99,212),(32,213,100),(33,101,214),(34,215,102),(35,103,216),(36,217,104),(37,105,218),(38,219,106),(39,107,220),(40,221,108),(41,158,183),(42,184,159),(43,160,185),(44,186,121),(45,122,187),(46,188,123),(47,124,189),(48,190,125),(49,126,191),(50,192,127),(51,128,193),(52,194,129),(53,130,195),(54,196,131),(55,132,197),(56,198,133),(57,134,199),(58,200,135),(59,136,161),(60,162,137),(61,138,163),(62,164,139),(63,140,165),(64,166,141),(65,142,167),(66,168,143),(67,144,169),(68,170,145),(69,146,171),(70,172,147),(71,148,173),(72,174,149),(73,150,175),(74,176,151),(75,152,177),(76,178,153),(77,154,179),(78,180,155),(79,156,181),(80,182,157)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(41,171),(42,170),(43,169),(44,168),(45,167),(46,166),(47,165),(48,164),(49,163),(50,162),(51,161),(52,200),(53,199),(54,198),(55,197),(56,196),(57,195),(58,194),(59,193),(60,192),(61,191),(62,190),(63,189),(64,188),(65,187),(66,186),(67,185),(68,184),(69,183),(70,182),(71,181),(72,180),(73,179),(74,178),(75,177),(76,176),(77,175),(78,174),(79,173),(80,172),(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),(121,143),(122,142),(123,141),(124,140),(125,139),(126,138),(127,137),(128,136),(129,135),(130,134),(131,133),(144,160),(145,159),(146,158),(147,157),(148,156),(149,155),(150,154),(151,153)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 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 | S3 | D4 | D4 | D5 | D6 | D6 | D8 | D10 | D10 | C3⋊D4 | C3⋊D4 | D20 | D20 | D40 | D4⋊S3 | S3×D5 | C3⋊D20 | C2×S3×D5 | C3⋊D20 | C3⋊D40 |
| kernel | C2×C3⋊D40 | C3⋊D40 | C10×C3⋊C8 | C6×D20 | C2×D60 | C2×D20 | C60 | C2×C30 | C2×C3⋊C8 | D20 | C2×C20 | C30 | C3⋊C8 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C6 | C10 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C2×C3⋊D40 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 54 | 0 | 0 |
| 174 | 239 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 |
| 67 | 1 | 0 | 0 |
| 0 | 0 | 228 | 214 |
| 0 | 0 | 27 | 56 |
| 1 | 0 | 0 | 0 |
| 174 | 240 | 0 | 0 |
| 0 | 0 | 197 | 78 |
| 0 | 0 | 3 | 44 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,174,0,0,54,239,0,0,0,0,1,0,0,0,0,1],[240,67,0,0,0,1,0,0,0,0,228,27,0,0,214,56],[1,174,0,0,0,240,0,0,0,0,197,3,0,0,78,44] >;
C2×C3⋊D40 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes D_{40} % in TeX
G:=Group("C2xC3:D40"); // GroupNames label
G:=SmallGroup(480,376);
// by ID
G=gap.SmallGroup(480,376);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,176,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^40=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations