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 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×2], C10, C10 [×2], Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8, C2×C8, M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, D15 [×2], C30, C30 [×2], C2×M4(2), C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C8⋊S3 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, Dic15 [×2], C60 [×2], D30 [×2], D30 [×2], C2×C30, C8⋊D5 [×4], C2×C5⋊2C8, C2×C40, C2×C4×D5, C2×C8⋊S3, C15⋊3C8 [×2], C120 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, C2×C8⋊D5, C40⋊S3 [×4], C2×C15⋊3C8, C2×C120, C2×C4×D15, C2×C40⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], M4(2) [×2], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, D15, C2×M4(2), C4×D5 [×2], C22×D5, C8⋊S3 [×2], S3×C2×C4, D30 [×3], C8⋊D5 [×2], C2×C4×D5, C2×C8⋊S3, C4×D15 [×2], C22×D15, C2×C8⋊D5, C40⋊S3 [×2], C2×C4×D15, C2×C40⋊S3
►On 240 points
Generators in S240
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 81)(41 224)(42 225)(43 226)(44 227)(45 228)(46 229)(47 230)(48 231)(49 232)(50 233)(51 234)(52 235)(53 236)(54 237)(55 238)(56 239)(57 240)(58 201)(59 202)(60 203)(61 204)(62 205)(63 206)(64 207)(65 208)(66 209)(67 210)(68 211)(69 212)(70 213)(71 214)(72 215)(73 216)(74 217)(75 218)(76 219)(77 220)(78 221)(79 222)(80 223)(121 189)(122 190)(123 191)(124 192)(125 193)(126 194)(127 195)(128 196)(129 197)(130 198)(131 199)(132 200)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(141 169)(142 170)(143 171)(144 172)(145 173)(146 174)(147 175)(148 176)(149 177)(150 178)(151 179)(152 180)(153 181)(154 182)(155 183)(156 184)(157 185)(158 186)(159 187)(160 188)
(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 57 143)(2 58 144)(3 59 145)(4 60 146)(5 61 147)(6 62 148)(7 63 149)(8 64 150)(9 65 151)(10 66 152)(11 67 153)(12 68 154)(13 69 155)(14 70 156)(15 71 157)(16 72 158)(17 73 159)(18 74 160)(19 75 121)(20 76 122)(21 77 123)(22 78 124)(23 79 125)(24 80 126)(25 41 127)(26 42 128)(27 43 129)(28 44 130)(29 45 131)(30 46 132)(31 47 133)(32 48 134)(33 49 135)(34 50 136)(35 51 137)(36 52 138)(37 53 139)(38 54 140)(39 55 141)(40 56 142)(81 239 170)(82 240 171)(83 201 172)(84 202 173)(85 203 174)(86 204 175)(87 205 176)(88 206 177)(89 207 178)(90 208 179)(91 209 180)(92 210 181)(93 211 182)(94 212 183)(95 213 184)(96 214 185)(97 215 186)(98 216 187)(99 217 188)(100 218 189)(101 219 190)(102 220 191)(103 221 192)(104 222 193)(105 223 194)(106 224 195)(107 225 196)(108 226 197)(109 227 198)(110 228 199)(111 229 200)(112 230 161)(113 231 162)(114 232 163)(115 233 164)(116 234 165)(117 235 166)(118 236 167)(119 237 168)(120 238 169)
(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)(41 159)(42 148)(43 137)(44 126)(45 155)(46 144)(47 133)(48 122)(49 151)(50 140)(51 129)(52 158)(53 147)(54 136)(55 125)(56 154)(57 143)(58 132)(59 121)(60 150)(61 139)(62 128)(63 157)(64 146)(65 135)(66 124)(67 153)(68 142)(69 131)(70 160)(71 149)(72 138)(73 127)(74 156)(75 145)(76 134)(77 123)(78 152)(79 141)(80 130)(81 93)(83 111)(84 100)(85 89)(86 118)(87 107)(88 96)(90 114)(91 103)(94 110)(95 99)(97 117)(98 106)(101 113)(104 120)(105 109)(108 116)(115 119)(161 230)(162 219)(163 208)(164 237)(165 226)(166 215)(167 204)(168 233)(169 222)(170 211)(171 240)(172 229)(173 218)(174 207)(175 236)(176 225)(177 214)(178 203)(179 232)(180 221)(181 210)(182 239)(183 228)(184 217)(185 206)(186 235)(187 224)(188 213)(189 202)(190 231)(191 220)(192 209)(193 238)(194 227)(195 216)(196 205)(197 234)(198 223)(199 212)(200 201)
G:=sub<Sym(240)| (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,81)(41,224)(42,225)(43,226)(44,227)(45,228)(46,229)(47,230)(48,231)(49,232)(50,233)(51,234)(52,235)(53,236)(54,237)(55,238)(56,239)(57,240)(58,201)(59,202)(60,203)(61,204)(62,205)(63,206)(64,207)(65,208)(66,209)(67,210)(68,211)(69,212)(70,213)(71,214)(72,215)(73,216)(74,217)(75,218)(76,219)(77,220)(78,221)(79,222)(80,223)(121,189)(122,190)(123,191)(124,192)(125,193)(126,194)(127,195)(128,196)(129,197)(130,198)(131,199)(132,200)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(141,169)(142,170)(143,171)(144,172)(145,173)(146,174)(147,175)(148,176)(149,177)(150,178)(151,179)(152,180)(153,181)(154,182)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188), (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), (1,57,143)(2,58,144)(3,59,145)(4,60,146)(5,61,147)(6,62,148)(7,63,149)(8,64,150)(9,65,151)(10,66,152)(11,67,153)(12,68,154)(13,69,155)(14,70,156)(15,71,157)(16,72,158)(17,73,159)(18,74,160)(19,75,121)(20,76,122)(21,77,123)(22,78,124)(23,79,125)(24,80,126)(25,41,127)(26,42,128)(27,43,129)(28,44,130)(29,45,131)(30,46,132)(31,47,133)(32,48,134)(33,49,135)(34,50,136)(35,51,137)(36,52,138)(37,53,139)(38,54,140)(39,55,141)(40,56,142)(81,239,170)(82,240,171)(83,201,172)(84,202,173)(85,203,174)(86,204,175)(87,205,176)(88,206,177)(89,207,178)(90,208,179)(91,209,180)(92,210,181)(93,211,182)(94,212,183)(95,213,184)(96,214,185)(97,215,186)(98,216,187)(99,217,188)(100,218,189)(101,219,190)(102,220,191)(103,221,192)(104,222,193)(105,223,194)(106,224,195)(107,225,196)(108,226,197)(109,227,198)(110,228,199)(111,229,200)(112,230,161)(113,231,162)(114,232,163)(115,233,164)(116,234,165)(117,235,166)(118,236,167)(119,237,168)(120,238,169), (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)(41,159)(42,148)(43,137)(44,126)(45,155)(46,144)(47,133)(48,122)(49,151)(50,140)(51,129)(52,158)(53,147)(54,136)(55,125)(56,154)(57,143)(58,132)(59,121)(60,150)(61,139)(62,128)(63,157)(64,146)(65,135)(66,124)(67,153)(68,142)(69,131)(70,160)(71,149)(72,138)(73,127)(74,156)(75,145)(76,134)(77,123)(78,152)(79,141)(80,130)(81,93)(83,111)(84,100)(85,89)(86,118)(87,107)(88,96)(90,114)(91,103)(94,110)(95,99)(97,117)(98,106)(101,113)(104,120)(105,109)(108,116)(115,119)(161,230)(162,219)(163,208)(164,237)(165,226)(166,215)(167,204)(168,233)(169,222)(170,211)(171,240)(172,229)(173,218)(174,207)(175,236)(176,225)(177,214)(178,203)(179,232)(180,221)(181,210)(182,239)(183,228)(184,217)(185,206)(186,235)(187,224)(188,213)(189,202)(190,231)(191,220)(192,209)(193,238)(194,227)(195,216)(196,205)(197,234)(198,223)(199,212)(200,201)>;
G:=Group( (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,81)(41,224)(42,225)(43,226)(44,227)(45,228)(46,229)(47,230)(48,231)(49,232)(50,233)(51,234)(52,235)(53,236)(54,237)(55,238)(56,239)(57,240)(58,201)(59,202)(60,203)(61,204)(62,205)(63,206)(64,207)(65,208)(66,209)(67,210)(68,211)(69,212)(70,213)(71,214)(72,215)(73,216)(74,217)(75,218)(76,219)(77,220)(78,221)(79,222)(80,223)(121,189)(122,190)(123,191)(124,192)(125,193)(126,194)(127,195)(128,196)(129,197)(130,198)(131,199)(132,200)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(141,169)(142,170)(143,171)(144,172)(145,173)(146,174)(147,175)(148,176)(149,177)(150,178)(151,179)(152,180)(153,181)(154,182)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188), (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), (1,57,143)(2,58,144)(3,59,145)(4,60,146)(5,61,147)(6,62,148)(7,63,149)(8,64,150)(9,65,151)(10,66,152)(11,67,153)(12,68,154)(13,69,155)(14,70,156)(15,71,157)(16,72,158)(17,73,159)(18,74,160)(19,75,121)(20,76,122)(21,77,123)(22,78,124)(23,79,125)(24,80,126)(25,41,127)(26,42,128)(27,43,129)(28,44,130)(29,45,131)(30,46,132)(31,47,133)(32,48,134)(33,49,135)(34,50,136)(35,51,137)(36,52,138)(37,53,139)(38,54,140)(39,55,141)(40,56,142)(81,239,170)(82,240,171)(83,201,172)(84,202,173)(85,203,174)(86,204,175)(87,205,176)(88,206,177)(89,207,178)(90,208,179)(91,209,180)(92,210,181)(93,211,182)(94,212,183)(95,213,184)(96,214,185)(97,215,186)(98,216,187)(99,217,188)(100,218,189)(101,219,190)(102,220,191)(103,221,192)(104,222,193)(105,223,194)(106,224,195)(107,225,196)(108,226,197)(109,227,198)(110,228,199)(111,229,200)(112,230,161)(113,231,162)(114,232,163)(115,233,164)(116,234,165)(117,235,166)(118,236,167)(119,237,168)(120,238,169), (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)(41,159)(42,148)(43,137)(44,126)(45,155)(46,144)(47,133)(48,122)(49,151)(50,140)(51,129)(52,158)(53,147)(54,136)(55,125)(56,154)(57,143)(58,132)(59,121)(60,150)(61,139)(62,128)(63,157)(64,146)(65,135)(66,124)(67,153)(68,142)(69,131)(70,160)(71,149)(72,138)(73,127)(74,156)(75,145)(76,134)(77,123)(78,152)(79,141)(80,130)(81,93)(83,111)(84,100)(85,89)(86,118)(87,107)(88,96)(90,114)(91,103)(94,110)(95,99)(97,117)(98,106)(101,113)(104,120)(105,109)(108,116)(115,119)(161,230)(162,219)(163,208)(164,237)(165,226)(166,215)(167,204)(168,233)(169,222)(170,211)(171,240)(172,229)(173,218)(174,207)(175,236)(176,225)(177,214)(178,203)(179,232)(180,221)(181,210)(182,239)(183,228)(184,217)(185,206)(186,235)(187,224)(188,213)(189,202)(190,231)(191,220)(192,209)(193,238)(194,227)(195,216)(196,205)(197,234)(198,223)(199,212)(200,201) );
G=PermutationGroup([(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,101),(21,102),(22,103),(23,104),(24,105),(25,106),(26,107),(27,108),(28,109),(29,110),(30,111),(31,112),(32,113),(33,114),(34,115),(35,116),(36,117),(37,118),(38,119),(39,120),(40,81),(41,224),(42,225),(43,226),(44,227),(45,228),(46,229),(47,230),(48,231),(49,232),(50,233),(51,234),(52,235),(53,236),(54,237),(55,238),(56,239),(57,240),(58,201),(59,202),(60,203),(61,204),(62,205),(63,206),(64,207),(65,208),(66,209),(67,210),(68,211),(69,212),(70,213),(71,214),(72,215),(73,216),(74,217),(75,218),(76,219),(77,220),(78,221),(79,222),(80,223),(121,189),(122,190),(123,191),(124,192),(125,193),(126,194),(127,195),(128,196),(129,197),(130,198),(131,199),(132,200),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(141,169),(142,170),(143,171),(144,172),(145,173),(146,174),(147,175),(148,176),(149,177),(150,178),(151,179),(152,180),(153,181),(154,182),(155,183),(156,184),(157,185),(158,186),(159,187),(160,188)], [(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)], [(1,57,143),(2,58,144),(3,59,145),(4,60,146),(5,61,147),(6,62,148),(7,63,149),(8,64,150),(9,65,151),(10,66,152),(11,67,153),(12,68,154),(13,69,155),(14,70,156),(15,71,157),(16,72,158),(17,73,159),(18,74,160),(19,75,121),(20,76,122),(21,77,123),(22,78,124),(23,79,125),(24,80,126),(25,41,127),(26,42,128),(27,43,129),(28,44,130),(29,45,131),(30,46,132),(31,47,133),(32,48,134),(33,49,135),(34,50,136),(35,51,137),(36,52,138),(37,53,139),(38,54,140),(39,55,141),(40,56,142),(81,239,170),(82,240,171),(83,201,172),(84,202,173),(85,203,174),(86,204,175),(87,205,176),(88,206,177),(89,207,178),(90,208,179),(91,209,180),(92,210,181),(93,211,182),(94,212,183),(95,213,184),(96,214,185),(97,215,186),(98,216,187),(99,217,188),(100,218,189),(101,219,190),(102,220,191),(103,221,192),(104,222,193),(105,223,194),(106,224,195),(107,225,196),(108,226,197),(109,227,198),(110,228,199),(111,229,200),(112,230,161),(113,231,162),(114,232,163),(115,233,164),(116,234,165),(117,235,166),(118,236,167),(119,237,168),(120,238,169)], [(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),(41,159),(42,148),(43,137),(44,126),(45,155),(46,144),(47,133),(48,122),(49,151),(50,140),(51,129),(52,158),(53,147),(54,136),(55,125),(56,154),(57,143),(58,132),(59,121),(60,150),(61,139),(62,128),(63,157),(64,146),(65,135),(66,124),(67,153),(68,142),(69,131),(70,160),(71,149),(72,138),(73,127),(74,156),(75,145),(76,134),(77,123),(78,152),(79,141),(80,130),(81,93),(83,111),(84,100),(85,89),(86,118),(87,107),(88,96),(90,114),(91,103),(94,110),(95,99),(97,117),(98,106),(101,113),(104,120),(105,109),(108,116),(115,119),(161,230),(162,219),(163,208),(164,237),(165,226),(166,215),(167,204),(168,233),(169,222),(170,211),(171,240),(172,229),(173,218),(174,207),(175,236),(176,225),(177,214),(178,203),(179,232),(180,221),(181,210),(182,239),(183,228),(184,217),(185,206),(186,235),(187,224),(188,213),(189,202),(190,231),(191,220),(192,209),(193,238),(194,227),(195,216),(196,205),(197,234),(198,223),(199,212),(200,201)])
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