direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C5⋊2C16, C40.49D6, C24.56D10, C120.52C22, C5⋊5(S3×C16), C15⋊6(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), C15⋊3C16⋊10C2, C20.105(C4×S3), C60.136(C2×C4), D6.2(C5⋊2C8), (C5×Dic3).3C8, (C4×S3).4Dic5, C4.16(S3×Dic5), C12.21(C2×Dic5), Dic3.2(C5⋊2C8), C3⋊1(C2×C5⋊2C16), (C5×C3⋊C8).6C4, C6.1(C2×C5⋊2C8), C2.1(S3×C5⋊2C8), (C3×C5⋊2C16)⋊7C2, SmallGroup(480,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — S3×C5⋊2C16 |
Generators and relations for S3×C5⋊2C16
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 >
Smallest permutation representation of S3×C5⋊2C16
►On 240 points
Generators in S240
(1 206 234)(2 207 235)(3 208 236)(4 193 237)(5 194 238)(6 195 239)(7 196 240)(8 197 225)(9 198 226)(10 199 227)(11 200 228)(12 201 229)(13 202 230)(14 203 231)(15 204 232)(16 205 233)(17 147 87)(18 148 88)(19 149 89)(20 150 90)(21 151 91)(22 152 92)(23 153 93)(24 154 94)(25 155 95)(26 156 96)(27 157 81)(28 158 82)(29 159 83)(30 160 84)(31 145 85)(32 146 86)(33 161 50)(34 162 51)(35 163 52)(36 164 53)(37 165 54)(38 166 55)(39 167 56)(40 168 57)(41 169 58)(42 170 59)(43 171 60)(44 172 61)(45 173 62)(46 174 63)(47 175 64)(48 176 49)(65 118 140)(66 119 141)(67 120 142)(68 121 143)(69 122 144)(70 123 129)(71 124 130)(72 125 131)(73 126 132)(74 127 133)(75 128 134)(76 113 135)(77 114 136)(78 115 137)(79 116 138)(80 117 139)(97 212 181)(98 213 182)(99 214 183)(100 215 184)(101 216 185)(102 217 186)(103 218 187)(104 219 188)(105 220 189)(106 221 190)(107 222 191)(108 223 192)(109 224 177)(110 209 178)(111 210 179)(112 211 180)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 95)(18 96)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 169)(34 170)(35 171)(36 172)(37 173)(38 174)(39 175)(40 176)(41 161)(42 162)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(97 220)(98 221)(99 222)(100 223)(101 224)(102 209)(103 210)(104 211)(105 212)(106 213)(107 214)(108 215)(109 216)(110 217)(111 218)(112 219)(113 143)(114 144)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(127 141)(128 142)(145 153)(146 154)(147 155)(148 156)(149 157)(150 158)(151 159)(152 160)(177 185)(178 186)(179 187)(180 188)(181 189)(182 190)(183 191)(184 192)(193 229)(194 230)(195 231)(196 232)(197 233)(198 234)(199 235)(200 236)(201 237)(202 238)(203 239)(204 240)(205 225)(206 226)(207 227)(208 228)
(1 70 50 192 146)(2 147 177 51 71)(3 72 52 178 148)(4 149 179 53 73)(5 74 54 180 150)(6 151 181 55 75)(7 76 56 182 152)(8 153 183 57 77)(9 78 58 184 154)(10 155 185 59 79)(11 80 60 186 156)(12 157 187 61 65)(13 66 62 188 158)(14 159 189 63 67)(15 68 64 190 160)(16 145 191 49 69)(17 224 162 130 235)(18 236 131 163 209)(19 210 164 132 237)(20 238 133 165 211)(21 212 166 134 239)(22 240 135 167 213)(23 214 168 136 225)(24 226 137 169 215)(25 216 170 138 227)(26 228 139 171 217)(27 218 172 140 229)(28 230 141 173 219)(29 220 174 142 231)(30 232 143 175 221)(31 222 176 144 233)(32 234 129 161 223)(33 108 86 206 123)(34 124 207 87 109)(35 110 88 208 125)(36 126 193 89 111)(37 112 90 194 127)(38 128 195 91 97)(39 98 92 196 113)(40 114 197 93 99)(41 100 94 198 115)(42 116 199 95 101)(43 102 96 200 117)(44 118 201 81 103)(45 104 82 202 119)(46 120 203 83 105)(47 106 84 204 121)(48 122 205 85 107)
(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)| (1,206,234)(2,207,235)(3,208,236)(4,193,237)(5,194,238)(6,195,239)(7,196,240)(8,197,225)(9,198,226)(10,199,227)(11,200,228)(12,201,229)(13,202,230)(14,203,231)(15,204,232)(16,205,233)(17,147,87)(18,148,88)(19,149,89)(20,150,90)(21,151,91)(22,152,92)(23,153,93)(24,154,94)(25,155,95)(26,156,96)(27,157,81)(28,158,82)(29,159,83)(30,160,84)(31,145,85)(32,146,86)(33,161,50)(34,162,51)(35,163,52)(36,164,53)(37,165,54)(38,166,55)(39,167,56)(40,168,57)(41,169,58)(42,170,59)(43,171,60)(44,172,61)(45,173,62)(46,174,63)(47,175,64)(48,176,49)(65,118,140)(66,119,141)(67,120,142)(68,121,143)(69,122,144)(70,123,129)(71,124,130)(72,125,131)(73,126,132)(74,127,133)(75,128,134)(76,113,135)(77,114,136)(78,115,137)(79,116,138)(80,117,139)(97,212,181)(98,213,182)(99,214,183)(100,215,184)(101,216,185)(102,217,186)(103,218,187)(104,219,188)(105,220,189)(106,221,190)(107,222,191)(108,223,192)(109,224,177)(110,209,178)(111,210,179)(112,211,180), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,95)(18,96)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,169)(34,170)(35,171)(36,172)(37,173)(38,174)(39,175)(40,176)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(97,220)(98,221)(99,222)(100,223)(101,224)(102,209)(103,210)(104,211)(105,212)(106,213)(107,214)(108,215)(109,216)(110,217)(111,218)(112,219)(113,143)(114,144)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(127,141)(128,142)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(177,185)(178,186)(179,187)(180,188)(181,189)(182,190)(183,191)(184,192)(193,229)(194,230)(195,231)(196,232)(197,233)(198,234)(199,235)(200,236)(201,237)(202,238)(203,239)(204,240)(205,225)(206,226)(207,227)(208,228), (1,70,50,192,146)(2,147,177,51,71)(3,72,52,178,148)(4,149,179,53,73)(5,74,54,180,150)(6,151,181,55,75)(7,76,56,182,152)(8,153,183,57,77)(9,78,58,184,154)(10,155,185,59,79)(11,80,60,186,156)(12,157,187,61,65)(13,66,62,188,158)(14,159,189,63,67)(15,68,64,190,160)(16,145,191,49,69)(17,224,162,130,235)(18,236,131,163,209)(19,210,164,132,237)(20,238,133,165,211)(21,212,166,134,239)(22,240,135,167,213)(23,214,168,136,225)(24,226,137,169,215)(25,216,170,138,227)(26,228,139,171,217)(27,218,172,140,229)(28,230,141,173,219)(29,220,174,142,231)(30,232,143,175,221)(31,222,176,144,233)(32,234,129,161,223)(33,108,86,206,123)(34,124,207,87,109)(35,110,88,208,125)(36,126,193,89,111)(37,112,90,194,127)(38,128,195,91,97)(39,98,92,196,113)(40,114,197,93,99)(41,100,94,198,115)(42,116,199,95,101)(43,102,96,200,117)(44,118,201,81,103)(45,104,82,202,119)(46,120,203,83,105)(47,106,84,204,121)(48,122,205,85,107), (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( (1,206,234)(2,207,235)(3,208,236)(4,193,237)(5,194,238)(6,195,239)(7,196,240)(8,197,225)(9,198,226)(10,199,227)(11,200,228)(12,201,229)(13,202,230)(14,203,231)(15,204,232)(16,205,233)(17,147,87)(18,148,88)(19,149,89)(20,150,90)(21,151,91)(22,152,92)(23,153,93)(24,154,94)(25,155,95)(26,156,96)(27,157,81)(28,158,82)(29,159,83)(30,160,84)(31,145,85)(32,146,86)(33,161,50)(34,162,51)(35,163,52)(36,164,53)(37,165,54)(38,166,55)(39,167,56)(40,168,57)(41,169,58)(42,170,59)(43,171,60)(44,172,61)(45,173,62)(46,174,63)(47,175,64)(48,176,49)(65,118,140)(66,119,141)(67,120,142)(68,121,143)(69,122,144)(70,123,129)(71,124,130)(72,125,131)(73,126,132)(74,127,133)(75,128,134)(76,113,135)(77,114,136)(78,115,137)(79,116,138)(80,117,139)(97,212,181)(98,213,182)(99,214,183)(100,215,184)(101,216,185)(102,217,186)(103,218,187)(104,219,188)(105,220,189)(106,221,190)(107,222,191)(108,223,192)(109,224,177)(110,209,178)(111,210,179)(112,211,180), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,95)(18,96)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,169)(34,170)(35,171)(36,172)(37,173)(38,174)(39,175)(40,176)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(97,220)(98,221)(99,222)(100,223)(101,224)(102,209)(103,210)(104,211)(105,212)(106,213)(107,214)(108,215)(109,216)(110,217)(111,218)(112,219)(113,143)(114,144)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(127,141)(128,142)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(177,185)(178,186)(179,187)(180,188)(181,189)(182,190)(183,191)(184,192)(193,229)(194,230)(195,231)(196,232)(197,233)(198,234)(199,235)(200,236)(201,237)(202,238)(203,239)(204,240)(205,225)(206,226)(207,227)(208,228), (1,70,50,192,146)(2,147,177,51,71)(3,72,52,178,148)(4,149,179,53,73)(5,74,54,180,150)(6,151,181,55,75)(7,76,56,182,152)(8,153,183,57,77)(9,78,58,184,154)(10,155,185,59,79)(11,80,60,186,156)(12,157,187,61,65)(13,66,62,188,158)(14,159,189,63,67)(15,68,64,190,160)(16,145,191,49,69)(17,224,162,130,235)(18,236,131,163,209)(19,210,164,132,237)(20,238,133,165,211)(21,212,166,134,239)(22,240,135,167,213)(23,214,168,136,225)(24,226,137,169,215)(25,216,170,138,227)(26,228,139,171,217)(27,218,172,140,229)(28,230,141,173,219)(29,220,174,142,231)(30,232,143,175,221)(31,222,176,144,233)(32,234,129,161,223)(33,108,86,206,123)(34,124,207,87,109)(35,110,88,208,125)(36,126,193,89,111)(37,112,90,194,127)(38,128,195,91,97)(39,98,92,196,113)(40,114,197,93,99)(41,100,94,198,115)(42,116,199,95,101)(43,102,96,200,117)(44,118,201,81,103)(45,104,82,202,119)(46,120,203,83,105)(47,106,84,204,121)(48,122,205,85,107), (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([[(1,206,234),(2,207,235),(3,208,236),(4,193,237),(5,194,238),(6,195,239),(7,196,240),(8,197,225),(9,198,226),(10,199,227),(11,200,228),(12,201,229),(13,202,230),(14,203,231),(15,204,232),(16,205,233),(17,147,87),(18,148,88),(19,149,89),(20,150,90),(21,151,91),(22,152,92),(23,153,93),(24,154,94),(25,155,95),(26,156,96),(27,157,81),(28,158,82),(29,159,83),(30,160,84),(31,145,85),(32,146,86),(33,161,50),(34,162,51),(35,163,52),(36,164,53),(37,165,54),(38,166,55),(39,167,56),(40,168,57),(41,169,58),(42,170,59),(43,171,60),(44,172,61),(45,173,62),(46,174,63),(47,175,64),(48,176,49),(65,118,140),(66,119,141),(67,120,142),(68,121,143),(69,122,144),(70,123,129),(71,124,130),(72,125,131),(73,126,132),(74,127,133),(75,128,134),(76,113,135),(77,114,136),(78,115,137),(79,116,138),(80,117,139),(97,212,181),(98,213,182),(99,214,183),(100,215,184),(101,216,185),(102,217,186),(103,218,187),(104,219,188),(105,220,189),(106,221,190),(107,222,191),(108,223,192),(109,224,177),(110,209,178),(111,210,179),(112,211,180)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,95),(18,96),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(25,87),(26,88),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(33,169),(34,170),(35,171),(36,172),(37,173),(38,174),(39,175),(40,176),(41,161),(42,162),(43,163),(44,164),(45,165),(46,166),(47,167),(48,168),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80),(97,220),(98,221),(99,222),(100,223),(101,224),(102,209),(103,210),(104,211),(105,212),(106,213),(107,214),(108,215),(109,216),(110,217),(111,218),(112,219),(113,143),(114,144),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(127,141),(128,142),(145,153),(146,154),(147,155),(148,156),(149,157),(150,158),(151,159),(152,160),(177,185),(178,186),(179,187),(180,188),(181,189),(182,190),(183,191),(184,192),(193,229),(194,230),(195,231),(196,232),(197,233),(198,234),(199,235),(200,236),(201,237),(202,238),(203,239),(204,240),(205,225),(206,226),(207,227),(208,228)], [(1,70,50,192,146),(2,147,177,51,71),(3,72,52,178,148),(4,149,179,53,73),(5,74,54,180,150),(6,151,181,55,75),(7,76,56,182,152),(8,153,183,57,77),(9,78,58,184,154),(10,155,185,59,79),(11,80,60,186,156),(12,157,187,61,65),(13,66,62,188,158),(14,159,189,63,67),(15,68,64,190,160),(16,145,191,49,69),(17,224,162,130,235),(18,236,131,163,209),(19,210,164,132,237),(20,238,133,165,211),(21,212,166,134,239),(22,240,135,167,213),(23,214,168,136,225),(24,226,137,169,215),(25,216,170,138,227),(26,228,139,171,217),(27,218,172,140,229),(28,230,141,173,219),(29,220,174,142,231),(30,232,143,175,221),(31,222,176,144,233),(32,234,129,161,223),(33,108,86,206,123),(34,124,207,87,109),(35,110,88,208,125),(36,126,193,89,111),(37,112,90,194,127),(38,128,195,91,97),(39,98,92,196,113),(40,114,197,93,99),(41,100,94,198,115),(42,116,199,95,101),(43,102,96,200,117),(44,118,201,81,103),(45,104,82,202,119),(46,120,203,83,105),(47,106,84,204,121),(48,122,205,85,107)], [(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 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 16A | ··· | 16H | 16I | ··· | 16P | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 24C | 24D | 30A | 30B | 40A | ··· | 40H | 40I | ··· | 40P | 48A | ··· | 48H | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 4 | 4 | 5 | ··· | 5 | 15 | ··· | 15 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | D5 | D6 | Dic5 | D10 | Dic5 | C4×S3 | C5⋊2C8 | C5⋊2C8 | S3×C8 | C5⋊2C16 | S3×C16 | S3×D5 | S3×Dic5 | S3×C5⋊2C8 | S3×C5⋊2C16 |
| kernel | S3×C5⋊2C16 | C3×C5⋊2C16 | C15⋊3C16 | S3×C40 | C5×C3⋊C8 | S3×C20 | C5×Dic3 | S3×C10 | C5×S3 | C5⋊2C16 | S3×C8 | C40 | C3⋊C8 | C24 | C4×S3 | C20 | Dic3 | D6 | C10 | S3 | C5 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 16 | 8 | 2 | 2 | 4 | 8 |
Matrix representation of S3×C5⋊2C16 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 240 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 240 | 0 |
| 51 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 48 | 179 | 0 | 0 |
| 217 | 193 | 0 | 0 |
| 0 | 0 | 126 | 0 |
| 0 | 0 | 0 | 126 |
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×C5⋊2C16 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
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