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G = S3×C52C16order 480 = 25·3·5

Direct product of S3 and C52C16

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3×C52C16, C40.49D6, C24.56D10, C120.52C22, C55(S3×C16), C156(C2×C16), (C5×S3)⋊2C16, (S3×C8).3D5, C8.35(S3×D5), C3⋊C8.3Dic5, (S3×C40).2C2, (S3×C10).3C8, (S3×C20).8C4, C10.20(S3×C8), C30.20(C2×C8), C153C1610C2, C20.105(C4×S3), C60.136(C2×C4), D6.2(C52C8), (C5×Dic3).3C8, (C4×S3).4Dic5, C4.16(S3×Dic5), C12.21(C2×Dic5), Dic3.2(C52C8), C31(C2×C52C16), (C5×C3⋊C8).6C4, C6.1(C2×C52C8), C2.1(S3×C52C8), (C3×C52C16)⋊7C2, SmallGroup(480,8)

Series: Derived Chief Lower central Upper central

C1C15 — S3×C52C16
C1C5C15C30C60C120C3×C52C16 — S3×C52C16
C15 — S3×C52C16
C1C8

Generators and relations for S3×C52C16
 G = < a,b,c,d | a3=b2=c5=d16=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

3C2
3C2
3C22
3C4
3C10
3C10
3C8
3C2×C4
3C20
3C2×C10
3C2×C8
5C16
15C16
3C40
3C2×C20
15C2×C16
5C48
5C3⋊C16
3C2×C40
3C52C16
5S3×C16
3C2×C52C16

Smallest permutation representation of S3×C52C16
On 240 points
Generators in S240
(1 203 216)(2 204 217)(3 205 218)(4 206 219)(5 207 220)(6 208 221)(7 193 222)(8 194 223)(9 195 224)(10 196 209)(11 197 210)(12 198 211)(13 199 212)(14 200 213)(15 201 214)(16 202 215)(17 102 157)(18 103 158)(19 104 159)(20 105 160)(21 106 145)(22 107 146)(23 108 147)(24 109 148)(25 110 149)(26 111 150)(27 112 151)(28 97 152)(29 98 153)(30 99 154)(31 100 155)(32 101 156)(33 240 175)(34 225 176)(35 226 161)(36 227 162)(37 228 163)(38 229 164)(39 230 165)(40 231 166)(41 232 167)(42 233 168)(43 234 169)(44 235 170)(45 236 171)(46 237 172)(47 238 173)(48 239 174)(49 90 188)(50 91 189)(51 92 190)(52 93 191)(53 94 192)(54 95 177)(55 96 178)(56 81 179)(57 82 180)(58 83 181)(59 84 182)(60 85 183)(61 86 184)(62 87 185)(63 88 186)(64 89 187)(65 115 136)(66 116 137)(67 117 138)(68 118 139)(69 119 140)(70 120 141)(71 121 142)(72 122 143)(73 123 144)(74 124 129)(75 125 130)(76 126 131)(77 127 132)(78 128 133)(79 113 134)(80 114 135)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 167)(34 168)(35 169)(36 170)(37 171)(38 172)(39 173)(40 174)(41 175)(42 176)(43 161)(44 162)(45 163)(46 164)(47 165)(48 166)(49 180)(50 181)(51 182)(52 183)(53 184)(54 185)(55 186)(56 187)(57 188)(58 189)(59 190)(60 191)(61 192)(62 177)(63 178)(64 179)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(97 160)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)(105 152)(106 153)(107 154)(108 155)(109 156)(110 157)(111 158)(112 159)(113 142)(114 143)(115 144)(116 129)(117 130)(118 131)(119 132)(120 133)(121 134)(122 135)(123 136)(124 137)(125 138)(126 139)(127 140)(128 141)(193 214)(194 215)(195 216)(196 217)(197 218)(198 219)(199 220)(200 221)(201 222)(202 223)(203 224)(204 209)(205 210)(206 211)(207 212)(208 213)(225 233)(226 234)(227 235)(228 236)(229 237)(230 238)(231 239)(232 240)
(1 24 69 227 85)(2 86 228 70 25)(3 26 71 229 87)(4 88 230 72 27)(5 28 73 231 89)(6 90 232 74 29)(7 30 75 233 91)(8 92 234 76 31)(9 32 77 235 93)(10 94 236 78 17)(11 18 79 237 95)(12 96 238 80 19)(13 20 65 239 81)(14 82 240 66 21)(15 22 67 225 83)(16 84 226 68 23)(33 137 145 213 57)(34 58 214 146 138)(35 139 147 215 59)(36 60 216 148 140)(37 141 149 217 61)(38 62 218 150 142)(39 143 151 219 63)(40 64 220 152 144)(41 129 153 221 49)(42 50 222 154 130)(43 131 155 223 51)(44 52 224 156 132)(45 133 157 209 53)(46 54 210 158 134)(47 135 159 211 55)(48 56 212 160 136)(97 123 166 187 207)(98 208 188 167 124)(99 125 168 189 193)(100 194 190 169 126)(101 127 170 191 195)(102 196 192 171 128)(103 113 172 177 197)(104 198 178 173 114)(105 115 174 179 199)(106 200 180 175 116)(107 117 176 181 201)(108 202 182 161 118)(109 119 162 183 203)(110 204 184 163 120)(111 121 164 185 205)(112 206 186 165 122)
(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)| (1,203,216)(2,204,217)(3,205,218)(4,206,219)(5,207,220)(6,208,221)(7,193,222)(8,194,223)(9,195,224)(10,196,209)(11,197,210)(12,198,211)(13,199,212)(14,200,213)(15,201,214)(16,202,215)(17,102,157)(18,103,158)(19,104,159)(20,105,160)(21,106,145)(22,107,146)(23,108,147)(24,109,148)(25,110,149)(26,111,150)(27,112,151)(28,97,152)(29,98,153)(30,99,154)(31,100,155)(32,101,156)(33,240,175)(34,225,176)(35,226,161)(36,227,162)(37,228,163)(38,229,164)(39,230,165)(40,231,166)(41,232,167)(42,233,168)(43,234,169)(44,235,170)(45,236,171)(46,237,172)(47,238,173)(48,239,174)(49,90,188)(50,91,189)(51,92,190)(52,93,191)(53,94,192)(54,95,177)(55,96,178)(56,81,179)(57,82,180)(58,83,181)(59,84,182)(60,85,183)(61,86,184)(62,87,185)(63,88,186)(64,89,187)(65,115,136)(66,116,137)(67,117,138)(68,118,139)(69,119,140)(70,120,141)(71,121,142)(72,122,143)(73,123,144)(74,124,129)(75,125,130)(76,126,131)(77,127,132)(78,128,133)(79,113,134)(80,114,135), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,167)(34,168)(35,169)(36,170)(37,171)(38,172)(39,173)(40,174)(41,175)(42,176)(43,161)(44,162)(45,163)(46,164)(47,165)(48,166)(49,180)(50,181)(51,182)(52,183)(53,184)(54,185)(55,186)(56,187)(57,188)(58,189)(59,190)(60,191)(61,192)(62,177)(63,178)(64,179)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(97,160)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)(113,142)(114,143)(115,144)(116,129)(117,130)(118,131)(119,132)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,141)(193,214)(194,215)(195,216)(196,217)(197,218)(198,219)(199,220)(200,221)(201,222)(202,223)(203,224)(204,209)(205,210)(206,211)(207,212)(208,213)(225,233)(226,234)(227,235)(228,236)(229,237)(230,238)(231,239)(232,240), (1,24,69,227,85)(2,86,228,70,25)(3,26,71,229,87)(4,88,230,72,27)(5,28,73,231,89)(6,90,232,74,29)(7,30,75,233,91)(8,92,234,76,31)(9,32,77,235,93)(10,94,236,78,17)(11,18,79,237,95)(12,96,238,80,19)(13,20,65,239,81)(14,82,240,66,21)(15,22,67,225,83)(16,84,226,68,23)(33,137,145,213,57)(34,58,214,146,138)(35,139,147,215,59)(36,60,216,148,140)(37,141,149,217,61)(38,62,218,150,142)(39,143,151,219,63)(40,64,220,152,144)(41,129,153,221,49)(42,50,222,154,130)(43,131,155,223,51)(44,52,224,156,132)(45,133,157,209,53)(46,54,210,158,134)(47,135,159,211,55)(48,56,212,160,136)(97,123,166,187,207)(98,208,188,167,124)(99,125,168,189,193)(100,194,190,169,126)(101,127,170,191,195)(102,196,192,171,128)(103,113,172,177,197)(104,198,178,173,114)(105,115,174,179,199)(106,200,180,175,116)(107,117,176,181,201)(108,202,182,161,118)(109,119,162,183,203)(110,204,184,163,120)(111,121,164,185,205)(112,206,186,165,122), (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( (1,203,216)(2,204,217)(3,205,218)(4,206,219)(5,207,220)(6,208,221)(7,193,222)(8,194,223)(9,195,224)(10,196,209)(11,197,210)(12,198,211)(13,199,212)(14,200,213)(15,201,214)(16,202,215)(17,102,157)(18,103,158)(19,104,159)(20,105,160)(21,106,145)(22,107,146)(23,108,147)(24,109,148)(25,110,149)(26,111,150)(27,112,151)(28,97,152)(29,98,153)(30,99,154)(31,100,155)(32,101,156)(33,240,175)(34,225,176)(35,226,161)(36,227,162)(37,228,163)(38,229,164)(39,230,165)(40,231,166)(41,232,167)(42,233,168)(43,234,169)(44,235,170)(45,236,171)(46,237,172)(47,238,173)(48,239,174)(49,90,188)(50,91,189)(51,92,190)(52,93,191)(53,94,192)(54,95,177)(55,96,178)(56,81,179)(57,82,180)(58,83,181)(59,84,182)(60,85,183)(61,86,184)(62,87,185)(63,88,186)(64,89,187)(65,115,136)(66,116,137)(67,117,138)(68,118,139)(69,119,140)(70,120,141)(71,121,142)(72,122,143)(73,123,144)(74,124,129)(75,125,130)(76,126,131)(77,127,132)(78,128,133)(79,113,134)(80,114,135), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,167)(34,168)(35,169)(36,170)(37,171)(38,172)(39,173)(40,174)(41,175)(42,176)(43,161)(44,162)(45,163)(46,164)(47,165)(48,166)(49,180)(50,181)(51,182)(52,183)(53,184)(54,185)(55,186)(56,187)(57,188)(58,189)(59,190)(60,191)(61,192)(62,177)(63,178)(64,179)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(97,160)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)(113,142)(114,143)(115,144)(116,129)(117,130)(118,131)(119,132)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,141)(193,214)(194,215)(195,216)(196,217)(197,218)(198,219)(199,220)(200,221)(201,222)(202,223)(203,224)(204,209)(205,210)(206,211)(207,212)(208,213)(225,233)(226,234)(227,235)(228,236)(229,237)(230,238)(231,239)(232,240), (1,24,69,227,85)(2,86,228,70,25)(3,26,71,229,87)(4,88,230,72,27)(5,28,73,231,89)(6,90,232,74,29)(7,30,75,233,91)(8,92,234,76,31)(9,32,77,235,93)(10,94,236,78,17)(11,18,79,237,95)(12,96,238,80,19)(13,20,65,239,81)(14,82,240,66,21)(15,22,67,225,83)(16,84,226,68,23)(33,137,145,213,57)(34,58,214,146,138)(35,139,147,215,59)(36,60,216,148,140)(37,141,149,217,61)(38,62,218,150,142)(39,143,151,219,63)(40,64,220,152,144)(41,129,153,221,49)(42,50,222,154,130)(43,131,155,223,51)(44,52,224,156,132)(45,133,157,209,53)(46,54,210,158,134)(47,135,159,211,55)(48,56,212,160,136)(97,123,166,187,207)(98,208,188,167,124)(99,125,168,189,193)(100,194,190,169,126)(101,127,170,191,195)(102,196,192,171,128)(103,113,172,177,197)(104,198,178,173,114)(105,115,174,179,199)(106,200,180,175,116)(107,117,176,181,201)(108,202,182,161,118)(109,119,162,183,203)(110,204,184,163,120)(111,121,164,185,205)(112,206,186,165,122), (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([(1,203,216),(2,204,217),(3,205,218),(4,206,219),(5,207,220),(6,208,221),(7,193,222),(8,194,223),(9,195,224),(10,196,209),(11,197,210),(12,198,211),(13,199,212),(14,200,213),(15,201,214),(16,202,215),(17,102,157),(18,103,158),(19,104,159),(20,105,160),(21,106,145),(22,107,146),(23,108,147),(24,109,148),(25,110,149),(26,111,150),(27,112,151),(28,97,152),(29,98,153),(30,99,154),(31,100,155),(32,101,156),(33,240,175),(34,225,176),(35,226,161),(36,227,162),(37,228,163),(38,229,164),(39,230,165),(40,231,166),(41,232,167),(42,233,168),(43,234,169),(44,235,170),(45,236,171),(46,237,172),(47,238,173),(48,239,174),(49,90,188),(50,91,189),(51,92,190),(52,93,191),(53,94,192),(54,95,177),(55,96,178),(56,81,179),(57,82,180),(58,83,181),(59,84,182),(60,85,183),(61,86,184),(62,87,185),(63,88,186),(64,89,187),(65,115,136),(66,116,137),(67,117,138),(68,118,139),(69,119,140),(70,120,141),(71,121,142),(72,122,143),(73,123,144),(74,124,129),(75,125,130),(76,126,131),(77,127,132),(78,128,133),(79,113,134),(80,114,135)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,167),(34,168),(35,169),(36,170),(37,171),(38,172),(39,173),(40,174),(41,175),(42,176),(43,161),(44,162),(45,163),(46,164),(47,165),(48,166),(49,180),(50,181),(51,182),(52,183),(53,184),(54,185),(55,186),(56,187),(57,188),(58,189),(59,190),(60,191),(61,192),(62,177),(63,178),(64,179),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80),(81,89),(82,90),(83,91),(84,92),(85,93),(86,94),(87,95),(88,96),(97,160),(98,145),(99,146),(100,147),(101,148),(102,149),(103,150),(104,151),(105,152),(106,153),(107,154),(108,155),(109,156),(110,157),(111,158),(112,159),(113,142),(114,143),(115,144),(116,129),(117,130),(118,131),(119,132),(120,133),(121,134),(122,135),(123,136),(124,137),(125,138),(126,139),(127,140),(128,141),(193,214),(194,215),(195,216),(196,217),(197,218),(198,219),(199,220),(200,221),(201,222),(202,223),(203,224),(204,209),(205,210),(206,211),(207,212),(208,213),(225,233),(226,234),(227,235),(228,236),(229,237),(230,238),(231,239),(232,240)], [(1,24,69,227,85),(2,86,228,70,25),(3,26,71,229,87),(4,88,230,72,27),(5,28,73,231,89),(6,90,232,74,29),(7,30,75,233,91),(8,92,234,76,31),(9,32,77,235,93),(10,94,236,78,17),(11,18,79,237,95),(12,96,238,80,19),(13,20,65,239,81),(14,82,240,66,21),(15,22,67,225,83),(16,84,226,68,23),(33,137,145,213,57),(34,58,214,146,138),(35,139,147,215,59),(36,60,216,148,140),(37,141,149,217,61),(38,62,218,150,142),(39,143,151,219,63),(40,64,220,152,144),(41,129,153,221,49),(42,50,222,154,130),(43,131,155,223,51),(44,52,224,156,132),(45,133,157,209,53),(46,54,210,158,134),(47,135,159,211,55),(48,56,212,160,136),(97,123,166,187,207),(98,208,188,167,124),(99,125,168,189,193),(100,194,190,169,126),(101,127,170,191,195),(102,196,192,171,128),(103,113,172,177,197),(104,198,178,173,114),(105,115,174,179,199),(106,200,180,175,116),(107,117,176,181,201),(108,202,182,161,118),(109,119,162,183,203),(110,204,184,163,120),(111,121,164,185,205),(112,206,186,165,122)], [(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)])

96 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B 6 8A8B8C8D8E8F8G8H10A10B10C10D10E10F12A12B15A15B16A···16H16I···16P20A20B20C20D20E20F20G20H24A24B24C24D30A30B40A···40H40I···40P48A···48H60A60B60C60D120A···120H
order122234444556888888881010101010101212151516···1616···16202020202020202024242424303040···4040···4048···4860606060120···120
size1133211332221111333322666622445···515···15222266662222442···26···610···1044444···4

96 irreducible representations

dim1111111112222222222224444
type+++++++-+-+-
imageC1C2C2C2C4C4C8C8C16S3D5D6Dic5D10Dic5C4×S3C52C8C52C8S3×C8C52C16S3×C16S3×D5S3×Dic5S3×C52C8S3×C52C16
kernelS3×C52C16C3×C52C16C153C16S3×C40C5×C3⋊C8S3×C20C5×Dic3S3×C10C5×S3C52C16S3×C8C40C3⋊C8C24C4×S3C20Dic3D6C10S3C5C8C4C2C1
# reps111122441612122224441682248

Matrix representation of S3×C52C16 in GL4(𝔽241) generated by

1000
0100
00240240
0010
,
1000
0100
000240
002400
,
5124000
1000
0010
0001
,
4817900
21719300
001260
000126
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,1,0,0,240,0],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,240,0],[51,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[48,217,0,0,179,193,0,0,0,0,126,0,0,0,0,126] >;

S3×C52C16 in GAP, Magma, Sage, TeX

S_3\times C_5\rtimes_2C_{16}
% in TeX

G:=Group("S3xC5:2C16");
// GroupNames label

G:=SmallGroup(480,8);
// by ID

G=gap.SmallGroup(480,8);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,58,80,1356,18822]);
// 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^-1>;
// generators/relations

Export

Subgroup lattice of S3×C52C16 in TeX

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