direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C2×C40, C120⋊47C22, C60.283C23, C6⋊1(C2×C40), C30⋊12(C2×C8), C3⋊1(C22×C40), (C2×C24)⋊10C10, (C2×C120)⋊26C2, C24⋊11(C2×C10), (C4×S3).5C20, C4.23(S3×C20), C15⋊14(C22×C8), D6.8(C2×C20), (S3×C20).16C4, C20.125(C4×S3), C60.223(C2×C4), C12.27(C2×C20), (C2×C20).451D6, (C22×S3).5C20, C6.12(C22×C20), C22.13(S3×C20), (C2×Dic3).8C20, (S3×C20).67C22, (C2×C60).563C22, C30.203(C22×C4), C20.241(C22×S3), C12.35(C22×C10), (C10×Dic3).25C4, Dic3.10(C2×C20), C2.2(S3×C2×C20), (C10×C3⋊C8)⋊27C2, (C2×C3⋊C8)⋊13C10, C3⋊C8⋊13(C2×C10), C4.35(S3×C2×C10), (C5×C3⋊C8)⋊46C22, (S3×C2×C20).25C2, (S3×C2×C4).12C10, (S3×C2×C10).15C4, C10.139(S3×C2×C4), (C2×C4).98(S3×C10), (C2×C6).14(C2×C20), (C2×C10).85(C4×S3), (C4×S3).18(C2×C10), (S3×C10).44(C2×C4), (C2×C30).159(C2×C4), (C2×C12).116(C2×C10), (C5×Dic3).52(C2×C4), SmallGroup(480,778)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C40 |
Generators and relations for S3×C2×C40
G = < a,b,c,d | a2=b40=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 260 in 152 conjugacy classes, 98 normal (38 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C30, C22×C8, C40, C40, C2×C20, C2×C20, C22×C10, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C60, S3×C10, C2×C30, C2×C40, C2×C40, C22×C20, S3×C2×C8, C5×C3⋊C8, C120, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, C22×C40, S3×C40, C10×C3⋊C8, C2×C120, S3×C2×C20, S3×C2×C40
Quotients: C1, C2, C4, C22, C5, S3, C8, C2×C4, C23, C10, D6, C2×C8, C22×C4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C22×C8, C40, C2×C20, C22×C10, S3×C8, S3×C2×C4, S3×C10, C2×C40, C22×C20, S3×C2×C8, S3×C20, S3×C2×C10, C22×C40, S3×C40, S3×C2×C20, S3×C2×C40
►On 240 points
Generators in S240
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(81 227)(82 228)(83 229)(84 230)(85 231)(86 232)(87 233)(88 234)(89 235)(90 236)(91 237)(92 238)(93 239)(94 240)(95 201)(96 202)(97 203)(98 204)(99 205)(100 206)(101 207)(102 208)(103 209)(104 210)(105 211)(106 212)(107 213)(108 214)(109 215)(110 216)(111 217)(112 218)(113 219)(114 220)(115 221)(116 222)(117 223)(118 224)(119 225)(120 226)(121 173)(122 174)(123 175)(124 176)(125 177)(126 178)(127 179)(128 180)(129 181)(130 182)(131 183)(132 184)(133 185)(134 186)(135 187)(136 188)(137 189)(138 190)(139 191)(140 192)(141 193)(142 194)(143 195)(144 196)(145 197)(146 198)(147 199)(148 200)(149 161)(150 162)(151 163)(152 164)(153 165)(154 166)(155 167)(156 168)(157 169)(158 170)(159 171)(160 172)
(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)
(1 123 217)(2 124 218)(3 125 219)(4 126 220)(5 127 221)(6 128 222)(7 129 223)(8 130 224)(9 131 225)(10 132 226)(11 133 227)(12 134 228)(13 135 229)(14 136 230)(15 137 231)(16 138 232)(17 139 233)(18 140 234)(19 141 235)(20 142 236)(21 143 237)(22 144 238)(23 145 239)(24 146 240)(25 147 201)(26 148 202)(27 149 203)(28 150 204)(29 151 205)(30 152 206)(31 153 207)(32 154 208)(33 155 209)(34 156 210)(35 157 211)(36 158 212)(37 159 213)(38 160 214)(39 121 215)(40 122 216)(41 182 118)(42 183 119)(43 184 120)(44 185 81)(45 186 82)(46 187 83)(47 188 84)(48 189 85)(49 190 86)(50 191 87)(51 192 88)(52 193 89)(53 194 90)(54 195 91)(55 196 92)(56 197 93)(57 198 94)(58 199 95)(59 200 96)(60 161 97)(61 162 98)(62 163 99)(63 164 100)(64 165 101)(65 166 102)(66 167 103)(67 168 104)(68 169 105)(69 170 106)(70 171 107)(71 172 108)(72 173 109)(73 174 110)(74 175 111)(75 176 112)(76 177 113)(77 178 114)(78 179 115)(79 180 116)(80 181 117)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(40 53)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)(97 129)(98 130)(99 131)(100 132)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 141)(110 142)(111 143)(112 144)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)(161 223)(162 224)(163 225)(164 226)(165 227)(166 228)(167 229)(168 230)(169 231)(170 232)(171 233)(172 234)(173 235)(174 236)(175 237)(176 238)(177 239)(178 240)(179 201)(180 202)(181 203)(182 204)(183 205)(184 206)(185 207)(186 208)(187 209)(188 210)(189 211)(190 212)(191 213)(192 214)(193 215)(194 216)(195 217)(196 218)(197 219)(198 220)(199 221)(200 222)
G:=sub<Sym(240)| (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(81,227)(82,228)(83,229)(84,230)(85,231)(86,232)(87,233)(88,234)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,201)(96,202)(97,203)(98,204)(99,205)(100,206)(101,207)(102,208)(103,209)(104,210)(105,211)(106,212)(107,213)(108,214)(109,215)(110,216)(111,217)(112,218)(113,219)(114,220)(115,221)(116,222)(117,223)(118,224)(119,225)(120,226)(121,173)(122,174)(123,175)(124,176)(125,177)(126,178)(127,179)(128,180)(129,181)(130,182)(131,183)(132,184)(133,185)(134,186)(135,187)(136,188)(137,189)(138,190)(139,191)(140,192)(141,193)(142,194)(143,195)(144,196)(145,197)(146,198)(147,199)(148,200)(149,161)(150,162)(151,163)(152,164)(153,165)(154,166)(155,167)(156,168)(157,169)(158,170)(159,171)(160,172), (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), (1,123,217)(2,124,218)(3,125,219)(4,126,220)(5,127,221)(6,128,222)(7,129,223)(8,130,224)(9,131,225)(10,132,226)(11,133,227)(12,134,228)(13,135,229)(14,136,230)(15,137,231)(16,138,232)(17,139,233)(18,140,234)(19,141,235)(20,142,236)(21,143,237)(22,144,238)(23,145,239)(24,146,240)(25,147,201)(26,148,202)(27,149,203)(28,150,204)(29,151,205)(30,152,206)(31,153,207)(32,154,208)(33,155,209)(34,156,210)(35,157,211)(36,158,212)(37,159,213)(38,160,214)(39,121,215)(40,122,216)(41,182,118)(42,183,119)(43,184,120)(44,185,81)(45,186,82)(46,187,83)(47,188,84)(48,189,85)(49,190,86)(50,191,87)(51,192,88)(52,193,89)(53,194,90)(54,195,91)(55,196,92)(56,197,93)(57,198,94)(58,199,95)(59,200,96)(60,161,97)(61,162,98)(62,163,99)(63,164,100)(64,165,101)(65,166,102)(66,167,103)(67,168,104)(68,169,105)(69,170,106)(70,171,107)(71,172,108)(72,173,109)(73,174,110)(74,175,111)(75,176,112)(76,177,113)(77,178,114)(78,179,115)(79,180,116)(80,181,117), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(40,53)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)(99,131)(100,132)(101,133)(102,134)(103,135)(104,136)(105,137)(106,138)(107,139)(108,140)(109,141)(110,142)(111,143)(112,144)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(161,223)(162,224)(163,225)(164,226)(165,227)(166,228)(167,229)(168,230)(169,231)(170,232)(171,233)(172,234)(173,235)(174,236)(175,237)(176,238)(177,239)(178,240)(179,201)(180,202)(181,203)(182,204)(183,205)(184,206)(185,207)(186,208)(187,209)(188,210)(189,211)(190,212)(191,213)(192,214)(193,215)(194,216)(195,217)(196,218)(197,219)(198,220)(199,221)(200,222)>;
G:=Group( (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(81,227)(82,228)(83,229)(84,230)(85,231)(86,232)(87,233)(88,234)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,201)(96,202)(97,203)(98,204)(99,205)(100,206)(101,207)(102,208)(103,209)(104,210)(105,211)(106,212)(107,213)(108,214)(109,215)(110,216)(111,217)(112,218)(113,219)(114,220)(115,221)(116,222)(117,223)(118,224)(119,225)(120,226)(121,173)(122,174)(123,175)(124,176)(125,177)(126,178)(127,179)(128,180)(129,181)(130,182)(131,183)(132,184)(133,185)(134,186)(135,187)(136,188)(137,189)(138,190)(139,191)(140,192)(141,193)(142,194)(143,195)(144,196)(145,197)(146,198)(147,199)(148,200)(149,161)(150,162)(151,163)(152,164)(153,165)(154,166)(155,167)(156,168)(157,169)(158,170)(159,171)(160,172), (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), (1,123,217)(2,124,218)(3,125,219)(4,126,220)(5,127,221)(6,128,222)(7,129,223)(8,130,224)(9,131,225)(10,132,226)(11,133,227)(12,134,228)(13,135,229)(14,136,230)(15,137,231)(16,138,232)(17,139,233)(18,140,234)(19,141,235)(20,142,236)(21,143,237)(22,144,238)(23,145,239)(24,146,240)(25,147,201)(26,148,202)(27,149,203)(28,150,204)(29,151,205)(30,152,206)(31,153,207)(32,154,208)(33,155,209)(34,156,210)(35,157,211)(36,158,212)(37,159,213)(38,160,214)(39,121,215)(40,122,216)(41,182,118)(42,183,119)(43,184,120)(44,185,81)(45,186,82)(46,187,83)(47,188,84)(48,189,85)(49,190,86)(50,191,87)(51,192,88)(52,193,89)(53,194,90)(54,195,91)(55,196,92)(56,197,93)(57,198,94)(58,199,95)(59,200,96)(60,161,97)(61,162,98)(62,163,99)(63,164,100)(64,165,101)(65,166,102)(66,167,103)(67,168,104)(68,169,105)(69,170,106)(70,171,107)(71,172,108)(72,173,109)(73,174,110)(74,175,111)(75,176,112)(76,177,113)(77,178,114)(78,179,115)(79,180,116)(80,181,117), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(40,53)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)(99,131)(100,132)(101,133)(102,134)(103,135)(104,136)(105,137)(106,138)(107,139)(108,140)(109,141)(110,142)(111,143)(112,144)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(161,223)(162,224)(163,225)(164,226)(165,227)(166,228)(167,229)(168,230)(169,231)(170,232)(171,233)(172,234)(173,235)(174,236)(175,237)(176,238)(177,239)(178,240)(179,201)(180,202)(181,203)(182,204)(183,205)(184,206)(185,207)(186,208)(187,209)(188,210)(189,211)(190,212)(191,213)(192,214)(193,215)(194,216)(195,217)(196,218)(197,219)(198,220)(199,221)(200,222) );
G=PermutationGroup([[(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,41),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(81,227),(82,228),(83,229),(84,230),(85,231),(86,232),(87,233),(88,234),(89,235),(90,236),(91,237),(92,238),(93,239),(94,240),(95,201),(96,202),(97,203),(98,204),(99,205),(100,206),(101,207),(102,208),(103,209),(104,210),(105,211),(106,212),(107,213),(108,214),(109,215),(110,216),(111,217),(112,218),(113,219),(114,220),(115,221),(116,222),(117,223),(118,224),(119,225),(120,226),(121,173),(122,174),(123,175),(124,176),(125,177),(126,178),(127,179),(128,180),(129,181),(130,182),(131,183),(132,184),(133,185),(134,186),(135,187),(136,188),(137,189),(138,190),(139,191),(140,192),(141,193),(142,194),(143,195),(144,196),(145,197),(146,198),(147,199),(148,200),(149,161),(150,162),(151,163),(152,164),(153,165),(154,166),(155,167),(156,168),(157,169),(158,170),(159,171),(160,172)], [(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)], [(1,123,217),(2,124,218),(3,125,219),(4,126,220),(5,127,221),(6,128,222),(7,129,223),(8,130,224),(9,131,225),(10,132,226),(11,133,227),(12,134,228),(13,135,229),(14,136,230),(15,137,231),(16,138,232),(17,139,233),(18,140,234),(19,141,235),(20,142,236),(21,143,237),(22,144,238),(23,145,239),(24,146,240),(25,147,201),(26,148,202),(27,149,203),(28,150,204),(29,151,205),(30,152,206),(31,153,207),(32,154,208),(33,155,209),(34,156,210),(35,157,211),(36,158,212),(37,159,213),(38,160,214),(39,121,215),(40,122,216),(41,182,118),(42,183,119),(43,184,120),(44,185,81),(45,186,82),(46,187,83),(47,188,84),(48,189,85),(49,190,86),(50,191,87),(51,192,88),(52,193,89),(53,194,90),(54,195,91),(55,196,92),(56,197,93),(57,198,94),(58,199,95),(59,200,96),(60,161,97),(61,162,98),(62,163,99),(63,164,100),(64,165,101),(65,166,102),(66,167,103),(67,168,104),(68,169,105),(69,170,106),(70,171,107),(71,172,108),(72,173,109),(73,174,110),(74,175,111),(75,176,112),(76,177,113),(77,178,114),(78,179,115),(79,180,116),(80,181,117)], [(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52),(40,53),(81,153),(82,154),(83,155),(84,156),(85,157),(86,158),(87,159),(88,160),(89,121),(90,122),(91,123),(92,124),(93,125),(94,126),(95,127),(96,128),(97,129),(98,130),(99,131),(100,132),(101,133),(102,134),(103,135),(104,136),(105,137),(106,138),(107,139),(108,140),(109,141),(110,142),(111,143),(112,144),(113,145),(114,146),(115,147),(116,148),(117,149),(118,150),(119,151),(120,152),(161,223),(162,224),(163,225),(164,226),(165,227),(166,228),(167,229),(168,230),(169,231),(170,232),(171,233),(172,234),(173,235),(174,236),(175,237),(176,238),(177,239),(178,240),(179,201),(180,202),(181,203),(182,204),(183,205),(184,206),(185,207),(186,208),(187,209),(188,210),(189,211),(190,212),(191,213),(192,214),(193,215),(194,216),(195,217),(196,218),(197,219),(198,220),(199,221),(200,222)]])
240 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10L | 10M | ··· | 10AB | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20AF | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40AF | 40AG | ··· | 40BL | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
240 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C8 | C10 | C10 | C10 | C10 | C20 | C20 | C20 | C40 | S3 | D6 | D6 | C4×S3 | C4×S3 | C5×S3 | S3×C8 | S3×C10 | S3×C10 | S3×C20 | S3×C20 | S3×C40 |
| kernel | S3×C2×C40 | S3×C40 | C10×C3⋊C8 | C2×C120 | S3×C2×C20 | S3×C20 | C10×Dic3 | S3×C2×C10 | S3×C2×C8 | S3×C10 | S3×C8 | C2×C3⋊C8 | C2×C24 | S3×C2×C4 | C4×S3 | C2×Dic3 | C22×S3 | D6 | C2×C40 | C40 | C2×C20 | C20 | C2×C10 | C2×C8 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 16 | 16 | 4 | 4 | 4 | 16 | 8 | 8 | 64 | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 8 | 4 | 8 | 8 | 32 |
Matrix representation of S3×C2×C40 ►in GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 36 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 0 | 240 |
| 0 | 1 | 240 |
| 1 | 0 | 0 |
| 0 | 0 | 240 |
| 0 | 240 | 0 |
G:=sub<GL(3,GF(241))| [240,0,0,0,1,0,0,0,1],[36,0,0,0,8,0,0,0,8],[1,0,0,0,0,1,0,240,240],[1,0,0,0,0,240,0,240,0] >;
S3×C2×C40 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_{40} % in TeX
G:=Group("S3xC2xC40"); // GroupNames label
G:=SmallGroup(480,778);
// by ID
G=gap.SmallGroup(480,778);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,226,102,15686]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations