direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C40⋊S3, C8⋊9D30, C40⋊32D6, C24⋊33D10, C120⋊40C22, C30⋊10M4(2), C60.253C23, (C2×C40)⋊9S3, (C2×C8)⋊6D15, (C2×C24)⋊11D5, (C2×C120)⋊15C2, C6⋊2(C8⋊D5), C20.94(C4×S3), (C4×D15).9C4, (C2×C4).99D30, C4.24(C4×D15), C12.62(C4×D5), C10⋊4(C8⋊S3), C60.197(C2×C4), D30.35(C2×C4), (C2×C20).414D6, C15⋊23(C2×M4(2)), C15⋊3C8⋊32C22, (C2×C12).416D10, (C22×D15).9C4, C22.14(C4×D15), C4.35(C22×D15), (C2×C60).499C22, C30.164(C22×C4), C20.223(C22×S3), (C2×Dic15).15C4, Dic15.43(C2×C4), (C4×D15).49C22, C12.225(C22×D5), C5⋊6(C2×C8⋊S3), C3⋊3(C2×C8⋊D5), C6.69(C2×C4×D5), C2.14(C2×C4×D15), C10.101(S3×C2×C4), (C2×C4×D15).12C2, (C2×C15⋊3C8)⋊11C2, (C2×C6).32(C4×D5), (C2×C10).57(C4×S3), (C2×C30).139(C2×C4), SmallGroup(480,865)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C40⋊S3
G = < a,b,c,d | a2=b40=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b29, dcd=c-1 >
Subgroups: 692 in 136 conjugacy classes, 63 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, D15, C30, C30, C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, Dic15, C60, D30, D30, C2×C30, C8⋊D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, C2×C8⋊S3, C15⋊3C8, C120, C4×D15, C2×Dic15, C2×C60, C22×D15, C2×C8⋊D5, C40⋊S3, C2×C15⋊3C8, C2×C120, C2×C4×D15, C2×C40⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, M4(2), C22×C4, D10, C4×S3, C22×S3, D15, C2×M4(2), C4×D5, C22×D5, C8⋊S3, S3×C2×C4, D30, C8⋊D5, C2×C4×D5, C2×C8⋊S3, C4×D15, C22×D15, C2×C8⋊D5, C40⋊S3, C2×C4×D15, C2×C40⋊S3
►On 240 points
Generators in S240
(1 161)(2 162)(3 163)(4 164)(5 165)(6 166)(7 167)(8 168)(9 169)(10 170)(11 171)(12 172)(13 173)(14 174)(15 175)(16 176)(17 177)(18 178)(19 179)(20 180)(21 181)(22 182)(23 183)(24 184)(25 185)(26 186)(27 187)(28 188)(29 189)(30 190)(31 191)(32 192)(33 193)(34 194)(35 195)(36 196)(37 197)(38 198)(39 199)(40 200)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(121 238)(122 239)(123 240)(124 201)(125 202)(126 203)(127 204)(128 205)(129 206)(130 207)(131 208)(132 209)(133 210)(134 211)(135 212)(136 213)(137 214)(138 215)(139 216)(140 217)(141 218)(142 219)(143 220)(144 221)(145 222)(146 223)(147 224)(148 225)(149 226)(150 227)(151 228)(152 229)(153 230)(154 231)(155 232)(156 233)(157 234)(158 235)(159 236)(160 237)
(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 79 145)(2 80 146)(3 41 147)(4 42 148)(5 43 149)(6 44 150)(7 45 151)(8 46 152)(9 47 153)(10 48 154)(11 49 155)(12 50 156)(13 51 157)(14 52 158)(15 53 159)(16 54 160)(17 55 121)(18 56 122)(19 57 123)(20 58 124)(21 59 125)(22 60 126)(23 61 127)(24 62 128)(25 63 129)(26 64 130)(27 65 131)(28 66 132)(29 67 133)(30 68 134)(31 69 135)(32 70 136)(33 71 137)(34 72 138)(35 73 139)(36 74 140)(37 75 141)(38 76 142)(39 77 143)(40 78 144)(81 240 179)(82 201 180)(83 202 181)(84 203 182)(85 204 183)(86 205 184)(87 206 185)(88 207 186)(89 208 187)(90 209 188)(91 210 189)(92 211 190)(93 212 191)(94 213 192)(95 214 193)(96 215 194)(97 216 195)(98 217 196)(99 218 197)(100 219 198)(101 220 199)(102 221 200)(103 222 161)(104 223 162)(105 224 163)(106 225 164)(107 226 165)(108 227 166)(109 228 167)(110 229 168)(111 230 169)(112 231 170)(113 232 171)(114 233 172)(115 234 173)(116 235 174)(117 236 175)(118 237 176)(119 238 177)(120 239 178)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 123)(42 152)(43 141)(44 130)(45 159)(46 148)(47 137)(48 126)(49 155)(50 144)(51 133)(52 122)(53 151)(54 140)(55 129)(56 158)(57 147)(58 136)(59 125)(60 154)(61 143)(62 132)(63 121)(64 150)(65 139)(66 128)(67 157)(68 146)(69 135)(70 124)(71 153)(72 142)(73 131)(74 160)(75 149)(76 138)(77 127)(78 156)(79 145)(80 134)(81 224)(82 213)(83 202)(84 231)(85 220)(86 209)(87 238)(88 227)(89 216)(90 205)(91 234)(92 223)(93 212)(94 201)(95 230)(96 219)(97 208)(98 237)(99 226)(100 215)(101 204)(102 233)(103 222)(104 211)(105 240)(106 229)(107 218)(108 207)(109 236)(110 225)(111 214)(112 203)(113 232)(114 221)(115 210)(116 239)(117 228)(118 217)(119 206)(120 235)(162 190)(163 179)(164 168)(165 197)(166 186)(167 175)(169 193)(170 182)(172 200)(173 189)(174 178)(176 196)(177 185)(180 192)(183 199)(184 188)(187 195)(194 198)
G:=sub<Sym(240)| (1,161)(2,162)(3,163)(4,164)(5,165)(6,166)(7,167)(8,168)(9,169)(10,170)(11,171)(12,172)(13,173)(14,174)(15,175)(16,176)(17,177)(18,178)(19,179)(20,180)(21,181)(22,182)(23,183)(24,184)(25,185)(26,186)(27,187)(28,188)(29,189)(30,190)(31,191)(32,192)(33,193)(34,194)(35,195)(36,196)(37,197)(38,198)(39,199)(40,200)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(121,238)(122,239)(123,240)(124,201)(125,202)(126,203)(127,204)(128,205)(129,206)(130,207)(131,208)(132,209)(133,210)(134,211)(135,212)(136,213)(137,214)(138,215)(139,216)(140,217)(141,218)(142,219)(143,220)(144,221)(145,222)(146,223)(147,224)(148,225)(149,226)(150,227)(151,228)(152,229)(153,230)(154,231)(155,232)(156,233)(157,234)(158,235)(159,236)(160,237), (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), (1,79,145)(2,80,146)(3,41,147)(4,42,148)(5,43,149)(6,44,150)(7,45,151)(8,46,152)(9,47,153)(10,48,154)(11,49,155)(12,50,156)(13,51,157)(14,52,158)(15,53,159)(16,54,160)(17,55,121)(18,56,122)(19,57,123)(20,58,124)(21,59,125)(22,60,126)(23,61,127)(24,62,128)(25,63,129)(26,64,130)(27,65,131)(28,66,132)(29,67,133)(30,68,134)(31,69,135)(32,70,136)(33,71,137)(34,72,138)(35,73,139)(36,74,140)(37,75,141)(38,76,142)(39,77,143)(40,78,144)(81,240,179)(82,201,180)(83,202,181)(84,203,182)(85,204,183)(86,205,184)(87,206,185)(88,207,186)(89,208,187)(90,209,188)(91,210,189)(92,211,190)(93,212,191)(94,213,192)(95,214,193)(96,215,194)(97,216,195)(98,217,196)(99,218,197)(100,219,198)(101,220,199)(102,221,200)(103,222,161)(104,223,162)(105,224,163)(106,225,164)(107,226,165)(108,227,166)(109,228,167)(110,229,168)(111,230,169)(112,231,170)(113,232,171)(114,233,172)(115,234,173)(116,235,174)(117,236,175)(118,237,176)(119,238,177)(120,239,178), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,123)(42,152)(43,141)(44,130)(45,159)(46,148)(47,137)(48,126)(49,155)(50,144)(51,133)(52,122)(53,151)(54,140)(55,129)(56,158)(57,147)(58,136)(59,125)(60,154)(61,143)(62,132)(63,121)(64,150)(65,139)(66,128)(67,157)(68,146)(69,135)(70,124)(71,153)(72,142)(73,131)(74,160)(75,149)(76,138)(77,127)(78,156)(79,145)(80,134)(81,224)(82,213)(83,202)(84,231)(85,220)(86,209)(87,238)(88,227)(89,216)(90,205)(91,234)(92,223)(93,212)(94,201)(95,230)(96,219)(97,208)(98,237)(99,226)(100,215)(101,204)(102,233)(103,222)(104,211)(105,240)(106,229)(107,218)(108,207)(109,236)(110,225)(111,214)(112,203)(113,232)(114,221)(115,210)(116,239)(117,228)(118,217)(119,206)(120,235)(162,190)(163,179)(164,168)(165,197)(166,186)(167,175)(169,193)(170,182)(172,200)(173,189)(174,178)(176,196)(177,185)(180,192)(183,199)(184,188)(187,195)(194,198)>;
G:=Group( (1,161)(2,162)(3,163)(4,164)(5,165)(6,166)(7,167)(8,168)(9,169)(10,170)(11,171)(12,172)(13,173)(14,174)(15,175)(16,176)(17,177)(18,178)(19,179)(20,180)(21,181)(22,182)(23,183)(24,184)(25,185)(26,186)(27,187)(28,188)(29,189)(30,190)(31,191)(32,192)(33,193)(34,194)(35,195)(36,196)(37,197)(38,198)(39,199)(40,200)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(121,238)(122,239)(123,240)(124,201)(125,202)(126,203)(127,204)(128,205)(129,206)(130,207)(131,208)(132,209)(133,210)(134,211)(135,212)(136,213)(137,214)(138,215)(139,216)(140,217)(141,218)(142,219)(143,220)(144,221)(145,222)(146,223)(147,224)(148,225)(149,226)(150,227)(151,228)(152,229)(153,230)(154,231)(155,232)(156,233)(157,234)(158,235)(159,236)(160,237), (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), (1,79,145)(2,80,146)(3,41,147)(4,42,148)(5,43,149)(6,44,150)(7,45,151)(8,46,152)(9,47,153)(10,48,154)(11,49,155)(12,50,156)(13,51,157)(14,52,158)(15,53,159)(16,54,160)(17,55,121)(18,56,122)(19,57,123)(20,58,124)(21,59,125)(22,60,126)(23,61,127)(24,62,128)(25,63,129)(26,64,130)(27,65,131)(28,66,132)(29,67,133)(30,68,134)(31,69,135)(32,70,136)(33,71,137)(34,72,138)(35,73,139)(36,74,140)(37,75,141)(38,76,142)(39,77,143)(40,78,144)(81,240,179)(82,201,180)(83,202,181)(84,203,182)(85,204,183)(86,205,184)(87,206,185)(88,207,186)(89,208,187)(90,209,188)(91,210,189)(92,211,190)(93,212,191)(94,213,192)(95,214,193)(96,215,194)(97,216,195)(98,217,196)(99,218,197)(100,219,198)(101,220,199)(102,221,200)(103,222,161)(104,223,162)(105,224,163)(106,225,164)(107,226,165)(108,227,166)(109,228,167)(110,229,168)(111,230,169)(112,231,170)(113,232,171)(114,233,172)(115,234,173)(116,235,174)(117,236,175)(118,237,176)(119,238,177)(120,239,178), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,123)(42,152)(43,141)(44,130)(45,159)(46,148)(47,137)(48,126)(49,155)(50,144)(51,133)(52,122)(53,151)(54,140)(55,129)(56,158)(57,147)(58,136)(59,125)(60,154)(61,143)(62,132)(63,121)(64,150)(65,139)(66,128)(67,157)(68,146)(69,135)(70,124)(71,153)(72,142)(73,131)(74,160)(75,149)(76,138)(77,127)(78,156)(79,145)(80,134)(81,224)(82,213)(83,202)(84,231)(85,220)(86,209)(87,238)(88,227)(89,216)(90,205)(91,234)(92,223)(93,212)(94,201)(95,230)(96,219)(97,208)(98,237)(99,226)(100,215)(101,204)(102,233)(103,222)(104,211)(105,240)(106,229)(107,218)(108,207)(109,236)(110,225)(111,214)(112,203)(113,232)(114,221)(115,210)(116,239)(117,228)(118,217)(119,206)(120,235)(162,190)(163,179)(164,168)(165,197)(166,186)(167,175)(169,193)(170,182)(172,200)(173,189)(174,178)(176,196)(177,185)(180,192)(183,199)(184,188)(187,195)(194,198) );
G=PermutationGroup([[(1,161),(2,162),(3,163),(4,164),(5,165),(6,166),(7,167),(8,168),(9,169),(10,170),(11,171),(12,172),(13,173),(14,174),(15,175),(16,176),(17,177),(18,178),(19,179),(20,180),(21,181),(22,182),(23,183),(24,184),(25,185),(26,186),(27,187),(28,188),(29,189),(30,190),(31,191),(32,192),(33,193),(34,194),(35,195),(36,196),(37,197),(38,198),(39,199),(40,200),(41,105),(42,106),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(49,113),(50,114),(51,115),(52,116),(53,117),(54,118),(55,119),(56,120),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96),(73,97),(74,98),(75,99),(76,100),(77,101),(78,102),(79,103),(80,104),(121,238),(122,239),(123,240),(124,201),(125,202),(126,203),(127,204),(128,205),(129,206),(130,207),(131,208),(132,209),(133,210),(134,211),(135,212),(136,213),(137,214),(138,215),(139,216),(140,217),(141,218),(142,219),(143,220),(144,221),(145,222),(146,223),(147,224),(148,225),(149,226),(150,227),(151,228),(152,229),(153,230),(154,231),(155,232),(156,233),(157,234),(158,235),(159,236),(160,237)], [(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)], [(1,79,145),(2,80,146),(3,41,147),(4,42,148),(5,43,149),(6,44,150),(7,45,151),(8,46,152),(9,47,153),(10,48,154),(11,49,155),(12,50,156),(13,51,157),(14,52,158),(15,53,159),(16,54,160),(17,55,121),(18,56,122),(19,57,123),(20,58,124),(21,59,125),(22,60,126),(23,61,127),(24,62,128),(25,63,129),(26,64,130),(27,65,131),(28,66,132),(29,67,133),(30,68,134),(31,69,135),(32,70,136),(33,71,137),(34,72,138),(35,73,139),(36,74,140),(37,75,141),(38,76,142),(39,77,143),(40,78,144),(81,240,179),(82,201,180),(83,202,181),(84,203,182),(85,204,183),(86,205,184),(87,206,185),(88,207,186),(89,208,187),(90,209,188),(91,210,189),(92,211,190),(93,212,191),(94,213,192),(95,214,193),(96,215,194),(97,216,195),(98,217,196),(99,218,197),(100,219,198),(101,220,199),(102,221,200),(103,222,161),(104,223,162),(105,224,163),(106,225,164),(107,226,165),(108,227,166),(109,228,167),(110,229,168),(111,230,169),(112,231,170),(113,232,171),(114,233,172),(115,234,173),(116,235,174),(117,236,175),(118,237,176),(119,238,177),(120,239,178)], [(2,30),(3,19),(4,8),(5,37),(6,26),(7,15),(9,33),(10,22),(12,40),(13,29),(14,18),(16,36),(17,25),(20,32),(23,39),(24,28),(27,35),(34,38),(41,123),(42,152),(43,141),(44,130),(45,159),(46,148),(47,137),(48,126),(49,155),(50,144),(51,133),(52,122),(53,151),(54,140),(55,129),(56,158),(57,147),(58,136),(59,125),(60,154),(61,143),(62,132),(63,121),(64,150),(65,139),(66,128),(67,157),(68,146),(69,135),(70,124),(71,153),(72,142),(73,131),(74,160),(75,149),(76,138),(77,127),(78,156),(79,145),(80,134),(81,224),(82,213),(83,202),(84,231),(85,220),(86,209),(87,238),(88,227),(89,216),(90,205),(91,234),(92,223),(93,212),(94,201),(95,230),(96,219),(97,208),(98,237),(99,226),(100,215),(101,204),(102,233),(103,222),(104,211),(105,240),(106,229),(107,218),(108,207),(109,236),(110,225),(111,214),(112,203),(113,232),(114,221),(115,210),(116,239),(117,228),(118,217),(119,206),(120,235),(162,190),(163,179),(164,168),(165,197),(166,186),(167,175),(169,193),(170,182),(172,200),(173,189),(174,178),(176,196),(177,185),(180,192),(183,199),(184,188),(187,195),(194,198)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | M4(2) | D10 | D10 | C4×S3 | C4×S3 | D15 | C4×D5 | C4×D5 | C8⋊S3 | D30 | D30 | C8⋊D5 | C4×D15 | C4×D15 | C40⋊S3 |
| kernel | C2×C40⋊S3 | C40⋊S3 | C2×C15⋊3C8 | C2×C120 | C2×C4×D15 | C4×D15 | C2×Dic15 | C22×D15 | C2×C40 | C2×C24 | C40 | C2×C20 | C30 | C24 | C2×C12 | C20 | C2×C10 | C2×C8 | C12 | C2×C6 | C10 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of C2×C40⋊S3 ►in GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 240 |
| 64 | 0 | 0 |
| 0 | 196 | 45 |
| 0 | 196 | 25 |
| 1 | 0 | 0 |
| 0 | 63 | 211 |
| 0 | 30 | 177 |
| 240 | 0 | 0 |
| 0 | 189 | 240 |
| 0 | 52 | 52 |
G:=sub<GL(3,GF(241))| [240,0,0,0,240,0,0,0,240],[64,0,0,0,196,196,0,45,25],[1,0,0,0,63,30,0,211,177],[240,0,0,0,189,52,0,240,52] >;
C2×C40⋊S3 in GAP, Magma, Sage, TeX
C_2\times C_{40}\rtimes S_3 % in TeX
G:=Group("C2xC40:S3"); // GroupNames label
G:=SmallGroup(480,865);
// by ID
G=gap.SmallGroup(480,865);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,58,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^40=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^29,d*c*d=c^-1>;
// generators/relations