direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C80, C48⋊5C10, C240⋊13C2, D6.2C40, C40.80D6, Dic3.2C40, C120.107C22, C3⋊1(C2×C80), C3⋊C16⋊6C10, C3⋊C8.3C20, C15⋊12(C2×C16), C2.1(S3×C40), C6.1(C2×C40), (S3×C8).3C10, (S3×C40).6C2, (S3×C10).6C8, (C4×S3).4C20, C4.16(S3×C20), C8.18(S3×C10), C10.25(S3×C8), C30.53(C2×C8), (S3×C20).15C4, C20.118(C4×S3), C24.23(C2×C10), C12.21(C2×C20), C60.216(C2×C4), (C5×Dic3).6C8, (C5×C3⋊C16)⋊13C2, (C5×C3⋊C8).10C4, SmallGroup(480,116)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C80 |
Generators and relations for S3×C80
G = < a,b,c | a80=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C80
►On 240 points
Generators in S240
(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)
(1 209 96)(2 210 97)(3 211 98)(4 212 99)(5 213 100)(6 214 101)(7 215 102)(8 216 103)(9 217 104)(10 218 105)(11 219 106)(12 220 107)(13 221 108)(14 222 109)(15 223 110)(16 224 111)(17 225 112)(18 226 113)(19 227 114)(20 228 115)(21 229 116)(22 230 117)(23 231 118)(24 232 119)(25 233 120)(26 234 121)(27 235 122)(28 236 123)(29 237 124)(30 238 125)(31 239 126)(32 240 127)(33 161 128)(34 162 129)(35 163 130)(36 164 131)(37 165 132)(38 166 133)(39 167 134)(40 168 135)(41 169 136)(42 170 137)(43 171 138)(44 172 139)(45 173 140)(46 174 141)(47 175 142)(48 176 143)(49 177 144)(50 178 145)(51 179 146)(52 180 147)(53 181 148)(54 182 149)(55 183 150)(56 184 151)(57 185 152)(58 186 153)(59 187 154)(60 188 155)(61 189 156)(62 190 157)(63 191 158)(64 192 159)(65 193 160)(66 194 81)(67 195 82)(68 196 83)(69 197 84)(70 198 85)(71 199 86)(72 200 87)(73 201 88)(74 202 89)(75 203 90)(76 204 91)(77 205 92)(78 206 93)(79 207 94)(80 208 95)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(81 234)(82 235)(83 236)(84 237)(85 238)(86 239)(87 240)(88 161)(89 162)(90 163)(91 164)(92 165)(93 166)(94 167)(95 168)(96 169)(97 170)(98 171)(99 172)(100 173)(101 174)(102 175)(103 176)(104 177)(105 178)(106 179)(107 180)(108 181)(109 182)(110 183)(111 184)(112 185)(113 186)(114 187)(115 188)(116 189)(117 190)(118 191)(119 192)(120 193)(121 194)(122 195)(123 196)(124 197)(125 198)(126 199)(127 200)(128 201)(129 202)(130 203)(131 204)(132 205)(133 206)(134 207)(135 208)(136 209)(137 210)(138 211)(139 212)(140 213)(141 214)(142 215)(143 216)(144 217)(145 218)(146 219)(147 220)(148 221)(149 222)(150 223)(151 224)(152 225)(153 226)(154 227)(155 228)(156 229)(157 230)(158 231)(159 232)(160 233)
G:=sub<Sym(240)| (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), (1,209,96)(2,210,97)(3,211,98)(4,212,99)(5,213,100)(6,214,101)(7,215,102)(8,216,103)(9,217,104)(10,218,105)(11,219,106)(12,220,107)(13,221,108)(14,222,109)(15,223,110)(16,224,111)(17,225,112)(18,226,113)(19,227,114)(20,228,115)(21,229,116)(22,230,117)(23,231,118)(24,232,119)(25,233,120)(26,234,121)(27,235,122)(28,236,123)(29,237,124)(30,238,125)(31,239,126)(32,240,127)(33,161,128)(34,162,129)(35,163,130)(36,164,131)(37,165,132)(38,166,133)(39,167,134)(40,168,135)(41,169,136)(42,170,137)(43,171,138)(44,172,139)(45,173,140)(46,174,141)(47,175,142)(48,176,143)(49,177,144)(50,178,145)(51,179,146)(52,180,147)(53,181,148)(54,182,149)(55,183,150)(56,184,151)(57,185,152)(58,186,153)(59,187,154)(60,188,155)(61,189,156)(62,190,157)(63,191,158)(64,192,159)(65,193,160)(66,194,81)(67,195,82)(68,196,83)(69,197,84)(70,198,85)(71,199,86)(72,200,87)(73,201,88)(74,202,89)(75,203,90)(76,204,91)(77,205,92)(78,206,93)(79,207,94)(80,208,95), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(81,234)(82,235)(83,236)(84,237)(85,238)(86,239)(87,240)(88,161)(89,162)(90,163)(91,164)(92,165)(93,166)(94,167)(95,168)(96,169)(97,170)(98,171)(99,172)(100,173)(101,174)(102,175)(103,176)(104,177)(105,178)(106,179)(107,180)(108,181)(109,182)(110,183)(111,184)(112,185)(113,186)(114,187)(115,188)(116,189)(117,190)(118,191)(119,192)(120,193)(121,194)(122,195)(123,196)(124,197)(125,198)(126,199)(127,200)(128,201)(129,202)(130,203)(131,204)(132,205)(133,206)(134,207)(135,208)(136,209)(137,210)(138,211)(139,212)(140,213)(141,214)(142,215)(143,216)(144,217)(145,218)(146,219)(147,220)(148,221)(149,222)(150,223)(151,224)(152,225)(153,226)(154,227)(155,228)(156,229)(157,230)(158,231)(159,232)(160,233)>;
G:=Group( (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), (1,209,96)(2,210,97)(3,211,98)(4,212,99)(5,213,100)(6,214,101)(7,215,102)(8,216,103)(9,217,104)(10,218,105)(11,219,106)(12,220,107)(13,221,108)(14,222,109)(15,223,110)(16,224,111)(17,225,112)(18,226,113)(19,227,114)(20,228,115)(21,229,116)(22,230,117)(23,231,118)(24,232,119)(25,233,120)(26,234,121)(27,235,122)(28,236,123)(29,237,124)(30,238,125)(31,239,126)(32,240,127)(33,161,128)(34,162,129)(35,163,130)(36,164,131)(37,165,132)(38,166,133)(39,167,134)(40,168,135)(41,169,136)(42,170,137)(43,171,138)(44,172,139)(45,173,140)(46,174,141)(47,175,142)(48,176,143)(49,177,144)(50,178,145)(51,179,146)(52,180,147)(53,181,148)(54,182,149)(55,183,150)(56,184,151)(57,185,152)(58,186,153)(59,187,154)(60,188,155)(61,189,156)(62,190,157)(63,191,158)(64,192,159)(65,193,160)(66,194,81)(67,195,82)(68,196,83)(69,197,84)(70,198,85)(71,199,86)(72,200,87)(73,201,88)(74,202,89)(75,203,90)(76,204,91)(77,205,92)(78,206,93)(79,207,94)(80,208,95), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(81,234)(82,235)(83,236)(84,237)(85,238)(86,239)(87,240)(88,161)(89,162)(90,163)(91,164)(92,165)(93,166)(94,167)(95,168)(96,169)(97,170)(98,171)(99,172)(100,173)(101,174)(102,175)(103,176)(104,177)(105,178)(106,179)(107,180)(108,181)(109,182)(110,183)(111,184)(112,185)(113,186)(114,187)(115,188)(116,189)(117,190)(118,191)(119,192)(120,193)(121,194)(122,195)(123,196)(124,197)(125,198)(126,199)(127,200)(128,201)(129,202)(130,203)(131,204)(132,205)(133,206)(134,207)(135,208)(136,209)(137,210)(138,211)(139,212)(140,213)(141,214)(142,215)(143,216)(144,217)(145,218)(146,219)(147,220)(148,221)(149,222)(150,223)(151,224)(152,225)(153,226)(154,227)(155,228)(156,229)(157,230)(158,231)(159,232)(160,233) );
G=PermutationGroup([[(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)], [(1,209,96),(2,210,97),(3,211,98),(4,212,99),(5,213,100),(6,214,101),(7,215,102),(8,216,103),(9,217,104),(10,218,105),(11,219,106),(12,220,107),(13,221,108),(14,222,109),(15,223,110),(16,224,111),(17,225,112),(18,226,113),(19,227,114),(20,228,115),(21,229,116),(22,230,117),(23,231,118),(24,232,119),(25,233,120),(26,234,121),(27,235,122),(28,236,123),(29,237,124),(30,238,125),(31,239,126),(32,240,127),(33,161,128),(34,162,129),(35,163,130),(36,164,131),(37,165,132),(38,166,133),(39,167,134),(40,168,135),(41,169,136),(42,170,137),(43,171,138),(44,172,139),(45,173,140),(46,174,141),(47,175,142),(48,176,143),(49,177,144),(50,178,145),(51,179,146),(52,180,147),(53,181,148),(54,182,149),(55,183,150),(56,184,151),(57,185,152),(58,186,153),(59,187,154),(60,188,155),(61,189,156),(62,190,157),(63,191,158),(64,192,159),(65,193,160),(66,194,81),(67,195,82),(68,196,83),(69,197,84),(70,198,85),(71,199,86),(72,200,87),(73,201,88),(74,202,89),(75,203,90),(76,204,91),(77,205,92),(78,206,93),(79,207,94),(80,208,95)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(81,234),(82,235),(83,236),(84,237),(85,238),(86,239),(87,240),(88,161),(89,162),(90,163),(91,164),(92,165),(93,166),(94,167),(95,168),(96,169),(97,170),(98,171),(99,172),(100,173),(101,174),(102,175),(103,176),(104,177),(105,178),(106,179),(107,180),(108,181),(109,182),(110,183),(111,184),(112,185),(113,186),(114,187),(115,188),(116,189),(117,190),(118,191),(119,192),(120,193),(121,194),(122,195),(123,196),(124,197),(125,198),(126,199),(127,200),(128,201),(129,202),(130,203),(131,204),(132,205),(133,206),(134,207),(135,208),(136,209),(137,210),(138,211),(139,212),(140,213),(141,214),(142,215),(143,216),(144,217),(145,218),(146,219),(147,220),(148,221),(149,222),(150,223),(151,224),(152,225),(153,226),(154,227),(155,228),(156,229),(157,230),(158,231),(159,232),(160,233)]])
240 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 15A | 15B | 15C | 15D | 16A | ··· | 16H | 16I | ··· | 16P | 20A | ··· | 20H | 20I | ··· | 20P | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40P | 40Q | ··· | 40AF | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80AF | 80AG | ··· | 80BL | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 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 |
| type | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C8 | C10 | C10 | C10 | C16 | C20 | C20 | C40 | C40 | C80 | S3 | D6 | C4×S3 | C5×S3 | S3×C8 | S3×C10 | S3×C16 | S3×C20 | S3×C40 | S3×C80 |
| kernel | S3×C80 | C5×C3⋊C16 | C240 | S3×C40 | C5×C3⋊C8 | S3×C20 | S3×C16 | C5×Dic3 | S3×C10 | C3⋊C16 | C48 | S3×C8 | C5×S3 | C3⋊C8 | C4×S3 | Dic3 | D6 | S3 | C80 | C40 | C20 | C16 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 16 | 8 | 8 | 16 | 16 | 64 | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 16 | 32 |
Matrix representation of S3×C80 ►in GL2(𝔽241) generated by
| 208 | 0 |
| 0 | 208 |
| 0 | 240 |
| 1 | 240 |
| 0 | 240 |
| 240 | 0 |
G:=sub<GL(2,GF(241))| [208,0,0,208],[0,1,240,240],[0,240,240,0] >;
S3×C80 in GAP, Magma, Sage, TeX
S_3\times C_{80} % in TeX
G:=Group("S3xC80"); // GroupNames label
G:=SmallGroup(480,116);
// by ID
G=gap.SmallGroup(480,116);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,148,80,102,15686]);
// Polycyclic
G:=Group<a,b,c|a^80=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export