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G = S3×C5⋊C16order 480 = 25·3·5

Direct product of S3 and C5⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — S3×C5⋊C16
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — C3×C5⋊C16 — S3×C5⋊C16
 Lower central C15 — S3×C5⋊C16
 Upper central C1 — C4

Generators and relations for S3×C5⋊C16
G = < a,b,c,d | a3=b2=c5=d16=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Smallest permutation representation of S3×C5⋊C16
On 240 points
Generators in S240
(1 221 192)(2 222 177)(3 223 178)(4 224 179)(5 209 180)(6 210 181)(7 211 182)(8 212 183)(9 213 184)(10 214 185)(11 215 186)(12 216 187)(13 217 188)(14 218 189)(15 219 190)(16 220 191)(17 161 227)(18 162 228)(19 163 229)(20 164 230)(21 165 231)(22 166 232)(23 167 233)(24 168 234)(25 169 235)(26 170 236)(27 171 237)(28 172 238)(29 173 239)(30 174 240)(31 175 225)(32 176 226)(33 138 92)(34 139 93)(35 140 94)(36 141 95)(37 142 96)(38 143 81)(39 144 82)(40 129 83)(41 130 84)(42 131 85)(43 132 86)(44 133 87)(45 134 88)(46 135 89)(47 136 90)(48 137 91)(49 204 74)(50 205 75)(51 206 76)(52 207 77)(53 208 78)(54 193 79)(55 194 80)(56 195 65)(57 196 66)(58 197 67)(59 198 68)(60 199 69)(61 200 70)(62 201 71)(63 202 72)(64 203 73)(97 125 152)(98 126 153)(99 127 154)(100 128 155)(101 113 156)(102 114 157)(103 115 158)(104 116 159)(105 117 160)(106 118 145)(107 119 146)(108 120 147)(109 121 148)(110 122 149)(111 123 150)(112 124 151)
(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 73)(81 143)(82 144)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 125)(98 126)(99 127)(100 128)(101 113)(102 114)(103 115)(104 116)(105 117)(106 118)(107 119)(108 120)(109 121)(110 122)(111 123)(112 124)(161 227)(162 228)(163 229)(164 230)(165 231)(166 232)(167 233)(168 234)(169 235)(170 236)(171 237)(172 238)(173 239)(174 240)(175 225)(176 226)(177 222)(178 223)(179 224)(180 209)(181 210)(182 211)(183 212)(184 213)(185 214)(186 215)(187 216)(188 217)(189 218)(190 219)(191 220)(192 221)
(1 42 203 21 154)(2 22 43 155 204)(3 156 23 205 44)(4 206 157 45 24)(5 46 207 25 158)(6 26 47 159 208)(7 160 27 193 48)(8 194 145 33 28)(9 34 195 29 146)(10 30 35 147 196)(11 148 31 197 36)(12 198 149 37 32)(13 38 199 17 150)(14 18 39 151 200)(15 152 19 201 40)(16 202 153 41 20)(49 177 232 86 128)(50 87 178 113 233)(51 114 88 234 179)(52 235 115 180 89)(53 181 236 90 116)(54 91 182 117 237)(55 118 92 238 183)(56 239 119 184 93)(57 185 240 94 120)(58 95 186 121 225)(59 122 96 226 187)(60 227 123 188 81)(61 189 228 82 124)(62 83 190 125 229)(63 126 84 230 191)(64 231 127 192 85)(65 173 107 213 139)(66 214 174 140 108)(67 141 215 109 175)(68 110 142 176 216)(69 161 111 217 143)(70 218 162 144 112)(71 129 219 97 163)(72 98 130 164 220)(73 165 99 221 131)(74 222 166 132 100)(75 133 223 101 167)(76 102 134 168 224)(77 169 103 209 135)(78 210 170 136 104)(79 137 211 105 171)(80 106 138 172 212)
(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)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5 6 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 12A 12B 15 16A ··· 16H 16I ··· 16P 20A 20B 20C 20D 24A 24B 24C 24D 30 48A ··· 48H 60A 60B order 1 2 2 2 3 4 4 4 4 5 6 8 8 8 8 8 8 8 8 10 10 10 12 12 15 16 ··· 16 16 ··· 16 20 20 20 20 24 24 24 24 30 48 ··· 48 60 60 size 1 1 3 3 2 1 1 3 3 4 2 5 5 5 5 15 15 15 15 4 12 12 2 2 8 5 ··· 5 15 ··· 15 4 4 12 12 10 10 10 10 8 10 ··· 10 8 8

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 8 8 8 type + + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 C8 C8 C16 S3 D6 C4×S3 S3×C8 S3×C16 F5 C5⋊C8 C2×F5 C5⋊C8 C5⋊C16 S3×F5 S3×C5⋊C8 S3×C5⋊C16 kernel S3×C5⋊C16 C3×C5⋊C16 C15⋊C16 S3×C5⋊2C8 C15⋊3C8 S3×C20 C5×Dic3 S3×C10 C5×S3 C5⋊C16 C5⋊2C8 C20 C10 C5 C4×S3 Dic3 C12 D6 S3 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 16 1 1 2 4 8 1 1 1 1 4 1 1 2

Matrix representation of S3×C5⋊C16 in GL6(𝔽241)

 0 240 0 0 0 0 1 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 240 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 240 0 0 1 0 0 240 0 0 0 1 0 240 0 0 0 0 1 240
,
 76 0 0 0 0 0 0 76 0 0 0 0 0 0 214 58 1 194 0 0 215 11 16 167 0 0 230 225 74 168 0 0 47 226 27 183

G:=sub<GL(6,GF(241))| [0,1,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[76,0,0,0,0,0,0,76,0,0,0,0,0,0,214,215,230,47,0,0,58,11,225,226,0,0,1,16,74,27,0,0,194,167,168,183] >;

S3×C5⋊C16 in GAP, Magma, Sage, TeX

S_3\times C_5\rtimes C_{16}
% in TeX

G:=Group("S3xC5:C16");
// GroupNames label

G:=SmallGroup(480,239);
// by ID

G=gap.SmallGroup(480,239);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,58,80,1356,9414,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^5=d^16=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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