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## G = S3×C2×C40order 480 = 25·3·5

### Direct product of C2×C40 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C40
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — S3×C2×C20 — S3×C2×C40
 Lower central C3 — S3×C2×C40
 Upper central C1 — 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 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×6], C5, S3 [×4], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], D6 [×6], C2×C6, C15, C2×C8, C2×C8 [×5], C22×C4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C5×S3 [×4], C30, C30 [×2], C22×C8, C40 [×2], C40 [×2], C2×C20, C2×C20 [×5], C22×C10, S3×C8 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3 [×2], C60 [×2], S3×C10 [×6], C2×C30, C2×C40, C2×C40 [×5], C22×C20, S3×C2×C8, C5×C3⋊C8 [×2], C120 [×2], S3×C20 [×4], C10×Dic3, C2×C60, S3×C2×C10, C22×C40, S3×C40 [×4], C10×C3⋊C8, C2×C120, S3×C2×C20, S3×C2×C40
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C8 [×4], C2×C4 [×6], C23, C10 [×7], D6 [×3], C2×C8 [×6], C22×C4, C20 [×4], C2×C10 [×7], C4×S3 [×2], C22×S3, C5×S3, C22×C8, C40 [×4], C2×C20 [×6], C22×C10, S3×C8 [×2], S3×C2×C4, S3×C10 [×3], C2×C40 [×6], C22×C20, S3×C2×C8, S3×C20 [×2], S3×C2×C10, C22×C40, S3×C40 [×2], S3×C2×C20, S3×C2×C40

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

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

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

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

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

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