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## G = S3×C80order 480 = 25·3·5

### Direct product of C80 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C80
 Chief series C1 — C3 — C6 — C12 — C24 — C120 — S3×C40 — S3×C80
 Lower central C3 — S3×C80
 Upper central C1 — 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)
(1 195 104)(2 196 105)(3 197 106)(4 198 107)(5 199 108)(6 200 109)(7 201 110)(8 202 111)(9 203 112)(10 204 113)(11 205 114)(12 206 115)(13 207 116)(14 208 117)(15 209 118)(16 210 119)(17 211 120)(18 212 121)(19 213 122)(20 214 123)(21 215 124)(22 216 125)(23 217 126)(24 218 127)(25 219 128)(26 220 129)(27 221 130)(28 222 131)(29 223 132)(30 224 133)(31 225 134)(32 226 135)(33 227 136)(34 228 137)(35 229 138)(36 230 139)(37 231 140)(38 232 141)(39 233 142)(40 234 143)(41 235 144)(42 236 145)(43 237 146)(44 238 147)(45 239 148)(46 240 149)(47 161 150)(48 162 151)(49 163 152)(50 164 153)(51 165 154)(52 166 155)(53 167 156)(54 168 157)(55 169 158)(56 170 159)(57 171 160)(58 172 81)(59 173 82)(60 174 83)(61 175 84)(62 176 85)(63 177 86)(64 178 87)(65 179 88)(66 180 89)(67 181 90)(68 182 91)(69 183 92)(70 184 93)(71 185 94)(72 186 95)(73 187 96)(74 188 97)(75 189 98)(76 190 99)(77 191 100)(78 192 101)(79 193 102)(80 194 103)
(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)(81 212)(82 213)(83 214)(84 215)(85 216)(86 217)(87 218)(88 219)(89 220)(90 221)(91 222)(92 223)(93 224)(94 225)(95 226)(96 227)(97 228)(98 229)(99 230)(100 231)(101 232)(102 233)(103 234)(104 235)(105 236)(106 237)(107 238)(108 239)(109 240)(110 161)(111 162)(112 163)(113 164)(114 165)(115 166)(116 167)(117 168)(118 169)(119 170)(120 171)(121 172)(122 173)(123 174)(124 175)(125 176)(126 177)(127 178)(128 179)(129 180)(130 181)(131 182)(132 183)(133 184)(134 185)(135 186)(136 187)(137 188)(138 189)(139 190)(140 191)(141 192)(142 193)(143 194)(144 195)(145 196)(146 197)(147 198)(148 199)(149 200)(150 201)(151 202)(152 203)(153 204)(154 205)(155 206)(156 207)(157 208)(158 209)(159 210)(160 211)

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

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

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

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

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