direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C5⋊Q16, C15⋊8Q16, C30.43D4, C12.39D10, C60.39C22, Dic10.2C6, C5⋊2(C3×Q16), C4.4(C6×D5), C5⋊2C8.1C6, C20.4(C2×C6), (C5×Q8).3C6, (C3×Q8).2D5, Q8.2(C3×D5), C10.10(C3×D4), (Q8×C15).2C2, C6.26(C5⋊D4), (C3×Dic10).4C2, C2.7(C3×C5⋊D4), (C3×C5⋊2C8).2C2, SmallGroup(240,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C5⋊Q16
G = < a,b,c,d | a3=b5=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Smallest permutation representation of C3×C5⋊Q16
►Regular action on 240 points
Generators in S240
(1 87 184)(2 88 177)(3 81 178)(4 82 179)(5 83 180)(6 84 181)(7 85 182)(8 86 183)(9 73 231)(10 74 232)(11 75 225)(12 76 226)(13 77 227)(14 78 228)(15 79 229)(16 80 230)(17 218 100)(18 219 101)(19 220 102)(20 221 103)(21 222 104)(22 223 97)(23 224 98)(24 217 99)(25 147 113)(26 148 114)(27 149 115)(28 150 116)(29 151 117)(30 152 118)(31 145 119)(32 146 120)(33 95 111)(34 96 112)(35 89 105)(36 90 106)(37 91 107)(38 92 108)(39 93 109)(40 94 110)(41 165 205)(42 166 206)(43 167 207)(44 168 208)(45 161 201)(46 162 202)(47 163 203)(48 164 204)(49 198 72)(50 199 65)(51 200 66)(52 193 67)(53 194 68)(54 195 69)(55 196 70)(56 197 71)(57 188 132)(58 189 133)(59 190 134)(60 191 135)(61 192 136)(62 185 129)(63 186 130)(64 187 131)(121 139 238)(122 140 239)(123 141 240)(124 142 233)(125 143 234)(126 144 235)(127 137 236)(128 138 237)(153 169 214)(154 170 215)(155 171 216)(156 172 209)(157 173 210)(158 174 211)(159 175 212)(160 176 213)
(1 54 128 226 63)(2 64 227 121 55)(3 56 122 228 57)(4 58 229 123 49)(5 50 124 230 59)(6 60 231 125 51)(7 52 126 232 61)(8 62 225 127 53)(9 143 200 84 191)(10 192 85 193 144)(11 137 194 86 185)(12 186 87 195 138)(13 139 196 88 187)(14 188 81 197 140)(15 141 198 82 189)(16 190 83 199 142)(17 92 207 153 145)(18 146 154 208 93)(19 94 201 155 147)(20 148 156 202 95)(21 96 203 157 149)(22 150 158 204 89)(23 90 205 159 151)(24 152 160 206 91)(25 102 40 161 216)(26 209 162 33 103)(27 104 34 163 210)(28 211 164 35 97)(29 98 36 165 212)(30 213 166 37 99)(31 100 38 167 214)(32 215 168 39 101)(41 175 117 224 106)(42 107 217 118 176)(43 169 119 218 108)(44 109 219 120 170)(45 171 113 220 110)(46 111 221 114 172)(47 173 115 222 112)(48 105 223 116 174)(65 233 80 134 180)(66 181 135 73 234)(67 235 74 136 182)(68 183 129 75 236)(69 237 76 130 184)(70 177 131 77 238)(71 239 78 132 178)(72 179 133 79 240)
(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)
(1 208 5 204)(2 207 6 203)(3 206 7 202)(4 205 8 201)(9 115 13 119)(10 114 14 118)(11 113 15 117)(12 120 16 116)(17 125 21 121)(18 124 22 128)(19 123 23 127)(20 122 24 126)(25 79 29 75)(26 78 30 74)(27 77 31 73)(28 76 32 80)(33 71 37 67)(34 70 38 66)(35 69 39 65)(36 68 40 72)(41 86 45 82)(42 85 46 81)(43 84 47 88)(44 83 48 87)(49 90 53 94)(50 89 54 93)(51 96 55 92)(52 95 56 91)(57 160 61 156)(58 159 62 155)(59 158 63 154)(60 157 64 153)(97 237 101 233)(98 236 102 240)(99 235 103 239)(100 234 104 238)(105 195 109 199)(106 194 110 198)(107 193 111 197)(108 200 112 196)(129 216 133 212)(130 215 134 211)(131 214 135 210)(132 213 136 209)(137 220 141 224)(138 219 142 223)(139 218 143 222)(140 217 144 221)(145 231 149 227)(146 230 150 226)(147 229 151 225)(148 228 152 232)(161 179 165 183)(162 178 166 182)(163 177 167 181)(164 184 168 180)(169 191 173 187)(170 190 174 186)(171 189 175 185)(172 188 176 192)
G:=sub<Sym(240)| (1,87,184)(2,88,177)(3,81,178)(4,82,179)(5,83,180)(6,84,181)(7,85,182)(8,86,183)(9,73,231)(10,74,232)(11,75,225)(12,76,226)(13,77,227)(14,78,228)(15,79,229)(16,80,230)(17,218,100)(18,219,101)(19,220,102)(20,221,103)(21,222,104)(22,223,97)(23,224,98)(24,217,99)(25,147,113)(26,148,114)(27,149,115)(28,150,116)(29,151,117)(30,152,118)(31,145,119)(32,146,120)(33,95,111)(34,96,112)(35,89,105)(36,90,106)(37,91,107)(38,92,108)(39,93,109)(40,94,110)(41,165,205)(42,166,206)(43,167,207)(44,168,208)(45,161,201)(46,162,202)(47,163,203)(48,164,204)(49,198,72)(50,199,65)(51,200,66)(52,193,67)(53,194,68)(54,195,69)(55,196,70)(56,197,71)(57,188,132)(58,189,133)(59,190,134)(60,191,135)(61,192,136)(62,185,129)(63,186,130)(64,187,131)(121,139,238)(122,140,239)(123,141,240)(124,142,233)(125,143,234)(126,144,235)(127,137,236)(128,138,237)(153,169,214)(154,170,215)(155,171,216)(156,172,209)(157,173,210)(158,174,211)(159,175,212)(160,176,213), (1,54,128,226,63)(2,64,227,121,55)(3,56,122,228,57)(4,58,229,123,49)(5,50,124,230,59)(6,60,231,125,51)(7,52,126,232,61)(8,62,225,127,53)(9,143,200,84,191)(10,192,85,193,144)(11,137,194,86,185)(12,186,87,195,138)(13,139,196,88,187)(14,188,81,197,140)(15,141,198,82,189)(16,190,83,199,142)(17,92,207,153,145)(18,146,154,208,93)(19,94,201,155,147)(20,148,156,202,95)(21,96,203,157,149)(22,150,158,204,89)(23,90,205,159,151)(24,152,160,206,91)(25,102,40,161,216)(26,209,162,33,103)(27,104,34,163,210)(28,211,164,35,97)(29,98,36,165,212)(30,213,166,37,99)(31,100,38,167,214)(32,215,168,39,101)(41,175,117,224,106)(42,107,217,118,176)(43,169,119,218,108)(44,109,219,120,170)(45,171,113,220,110)(46,111,221,114,172)(47,173,115,222,112)(48,105,223,116,174)(65,233,80,134,180)(66,181,135,73,234)(67,235,74,136,182)(68,183,129,75,236)(69,237,76,130,184)(70,177,131,77,238)(71,239,78,132,178)(72,179,133,79,240), (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), (1,208,5,204)(2,207,6,203)(3,206,7,202)(4,205,8,201)(9,115,13,119)(10,114,14,118)(11,113,15,117)(12,120,16,116)(17,125,21,121)(18,124,22,128)(19,123,23,127)(20,122,24,126)(25,79,29,75)(26,78,30,74)(27,77,31,73)(28,76,32,80)(33,71,37,67)(34,70,38,66)(35,69,39,65)(36,68,40,72)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87)(49,90,53,94)(50,89,54,93)(51,96,55,92)(52,95,56,91)(57,160,61,156)(58,159,62,155)(59,158,63,154)(60,157,64,153)(97,237,101,233)(98,236,102,240)(99,235,103,239)(100,234,104,238)(105,195,109,199)(106,194,110,198)(107,193,111,197)(108,200,112,196)(129,216,133,212)(130,215,134,211)(131,214,135,210)(132,213,136,209)(137,220,141,224)(138,219,142,223)(139,218,143,222)(140,217,144,221)(145,231,149,227)(146,230,150,226)(147,229,151,225)(148,228,152,232)(161,179,165,183)(162,178,166,182)(163,177,167,181)(164,184,168,180)(169,191,173,187)(170,190,174,186)(171,189,175,185)(172,188,176,192)>;
G:=Group( (1,87,184)(2,88,177)(3,81,178)(4,82,179)(5,83,180)(6,84,181)(7,85,182)(8,86,183)(9,73,231)(10,74,232)(11,75,225)(12,76,226)(13,77,227)(14,78,228)(15,79,229)(16,80,230)(17,218,100)(18,219,101)(19,220,102)(20,221,103)(21,222,104)(22,223,97)(23,224,98)(24,217,99)(25,147,113)(26,148,114)(27,149,115)(28,150,116)(29,151,117)(30,152,118)(31,145,119)(32,146,120)(33,95,111)(34,96,112)(35,89,105)(36,90,106)(37,91,107)(38,92,108)(39,93,109)(40,94,110)(41,165,205)(42,166,206)(43,167,207)(44,168,208)(45,161,201)(46,162,202)(47,163,203)(48,164,204)(49,198,72)(50,199,65)(51,200,66)(52,193,67)(53,194,68)(54,195,69)(55,196,70)(56,197,71)(57,188,132)(58,189,133)(59,190,134)(60,191,135)(61,192,136)(62,185,129)(63,186,130)(64,187,131)(121,139,238)(122,140,239)(123,141,240)(124,142,233)(125,143,234)(126,144,235)(127,137,236)(128,138,237)(153,169,214)(154,170,215)(155,171,216)(156,172,209)(157,173,210)(158,174,211)(159,175,212)(160,176,213), (1,54,128,226,63)(2,64,227,121,55)(3,56,122,228,57)(4,58,229,123,49)(5,50,124,230,59)(6,60,231,125,51)(7,52,126,232,61)(8,62,225,127,53)(9,143,200,84,191)(10,192,85,193,144)(11,137,194,86,185)(12,186,87,195,138)(13,139,196,88,187)(14,188,81,197,140)(15,141,198,82,189)(16,190,83,199,142)(17,92,207,153,145)(18,146,154,208,93)(19,94,201,155,147)(20,148,156,202,95)(21,96,203,157,149)(22,150,158,204,89)(23,90,205,159,151)(24,152,160,206,91)(25,102,40,161,216)(26,209,162,33,103)(27,104,34,163,210)(28,211,164,35,97)(29,98,36,165,212)(30,213,166,37,99)(31,100,38,167,214)(32,215,168,39,101)(41,175,117,224,106)(42,107,217,118,176)(43,169,119,218,108)(44,109,219,120,170)(45,171,113,220,110)(46,111,221,114,172)(47,173,115,222,112)(48,105,223,116,174)(65,233,80,134,180)(66,181,135,73,234)(67,235,74,136,182)(68,183,129,75,236)(69,237,76,130,184)(70,177,131,77,238)(71,239,78,132,178)(72,179,133,79,240), (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), (1,208,5,204)(2,207,6,203)(3,206,7,202)(4,205,8,201)(9,115,13,119)(10,114,14,118)(11,113,15,117)(12,120,16,116)(17,125,21,121)(18,124,22,128)(19,123,23,127)(20,122,24,126)(25,79,29,75)(26,78,30,74)(27,77,31,73)(28,76,32,80)(33,71,37,67)(34,70,38,66)(35,69,39,65)(36,68,40,72)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87)(49,90,53,94)(50,89,54,93)(51,96,55,92)(52,95,56,91)(57,160,61,156)(58,159,62,155)(59,158,63,154)(60,157,64,153)(97,237,101,233)(98,236,102,240)(99,235,103,239)(100,234,104,238)(105,195,109,199)(106,194,110,198)(107,193,111,197)(108,200,112,196)(129,216,133,212)(130,215,134,211)(131,214,135,210)(132,213,136,209)(137,220,141,224)(138,219,142,223)(139,218,143,222)(140,217,144,221)(145,231,149,227)(146,230,150,226)(147,229,151,225)(148,228,152,232)(161,179,165,183)(162,178,166,182)(163,177,167,181)(164,184,168,180)(169,191,173,187)(170,190,174,186)(171,189,175,185)(172,188,176,192) );
G=PermutationGroup([[(1,87,184),(2,88,177),(3,81,178),(4,82,179),(5,83,180),(6,84,181),(7,85,182),(8,86,183),(9,73,231),(10,74,232),(11,75,225),(12,76,226),(13,77,227),(14,78,228),(15,79,229),(16,80,230),(17,218,100),(18,219,101),(19,220,102),(20,221,103),(21,222,104),(22,223,97),(23,224,98),(24,217,99),(25,147,113),(26,148,114),(27,149,115),(28,150,116),(29,151,117),(30,152,118),(31,145,119),(32,146,120),(33,95,111),(34,96,112),(35,89,105),(36,90,106),(37,91,107),(38,92,108),(39,93,109),(40,94,110),(41,165,205),(42,166,206),(43,167,207),(44,168,208),(45,161,201),(46,162,202),(47,163,203),(48,164,204),(49,198,72),(50,199,65),(51,200,66),(52,193,67),(53,194,68),(54,195,69),(55,196,70),(56,197,71),(57,188,132),(58,189,133),(59,190,134),(60,191,135),(61,192,136),(62,185,129),(63,186,130),(64,187,131),(121,139,238),(122,140,239),(123,141,240),(124,142,233),(125,143,234),(126,144,235),(127,137,236),(128,138,237),(153,169,214),(154,170,215),(155,171,216),(156,172,209),(157,173,210),(158,174,211),(159,175,212),(160,176,213)], [(1,54,128,226,63),(2,64,227,121,55),(3,56,122,228,57),(4,58,229,123,49),(5,50,124,230,59),(6,60,231,125,51),(7,52,126,232,61),(8,62,225,127,53),(9,143,200,84,191),(10,192,85,193,144),(11,137,194,86,185),(12,186,87,195,138),(13,139,196,88,187),(14,188,81,197,140),(15,141,198,82,189),(16,190,83,199,142),(17,92,207,153,145),(18,146,154,208,93),(19,94,201,155,147),(20,148,156,202,95),(21,96,203,157,149),(22,150,158,204,89),(23,90,205,159,151),(24,152,160,206,91),(25,102,40,161,216),(26,209,162,33,103),(27,104,34,163,210),(28,211,164,35,97),(29,98,36,165,212),(30,213,166,37,99),(31,100,38,167,214),(32,215,168,39,101),(41,175,117,224,106),(42,107,217,118,176),(43,169,119,218,108),(44,109,219,120,170),(45,171,113,220,110),(46,111,221,114,172),(47,173,115,222,112),(48,105,223,116,174),(65,233,80,134,180),(66,181,135,73,234),(67,235,74,136,182),(68,183,129,75,236),(69,237,76,130,184),(70,177,131,77,238),(71,239,78,132,178),(72,179,133,79,240)], [(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)], [(1,208,5,204),(2,207,6,203),(3,206,7,202),(4,205,8,201),(9,115,13,119),(10,114,14,118),(11,113,15,117),(12,120,16,116),(17,125,21,121),(18,124,22,128),(19,123,23,127),(20,122,24,126),(25,79,29,75),(26,78,30,74),(27,77,31,73),(28,76,32,80),(33,71,37,67),(34,70,38,66),(35,69,39,65),(36,68,40,72),(41,86,45,82),(42,85,46,81),(43,84,47,88),(44,83,48,87),(49,90,53,94),(50,89,54,93),(51,96,55,92),(52,95,56,91),(57,160,61,156),(58,159,62,155),(59,158,63,154),(60,157,64,153),(97,237,101,233),(98,236,102,240),(99,235,103,239),(100,234,104,238),(105,195,109,199),(106,194,110,198),(107,193,111,197),(108,200,112,196),(129,216,133,212),(130,215,134,211),(131,214,135,210),(132,213,136,209),(137,220,141,224),(138,219,142,223),(139,218,143,222),(140,217,144,221),(145,231,149,227),(146,230,150,226),(147,229,151,225),(148,228,152,232),(161,179,165,183),(162,178,166,182),(163,177,167,181),(164,184,168,180),(169,191,173,187),(170,190,174,186),(171,189,175,185),(172,188,176,192)]])
C3×C5⋊Q16 is a maximal subgroup of
Dic10.26D6 D15⋊Q16 C60.C23 Dic10.27D6 C60.44C23 D12.D10 D30.44D4 C3×D5×Q16
51 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 60A | ··· | 60L |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 4 | 20 | 2 | 2 | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D5 | Q16 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C3×Q16 | C6×D5 | C3×C5⋊D4 | C5⋊Q16 | C3×C5⋊Q16 |
| kernel | C3×C5⋊Q16 | C3×C5⋊2C8 | C3×Dic10 | Q8×C15 | C5⋊Q16 | C5⋊2C8 | Dic10 | C5×Q8 | C30 | C3×Q8 | C15 | C12 | C10 | Q8 | C6 | C5 | C4 | C2 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 4 |
Matrix representation of C3×C5⋊Q16 ►in GL6(𝔽241)
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 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 | 240 | 240 | 0 | 0 |
| 0 | 0 | 53 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 26 | 54 | 0 | 0 | 0 | 0 |
| 224 | 215 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 93 | 0 | 0 |
| 0 | 0 | 127 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 230 | 11 |
| 0 | 0 | 0 | 0 | 230 | 230 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 124 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 31 |
| 0 | 0 | 0 | 0 | 31 | 219 |
G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,53,0,0,0,0,240,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,224,0,0,0,0,54,215,0,0,0,0,0,0,177,127,0,0,0,0,93,64,0,0,0,0,0,0,230,230,0,0,0,0,11,230],[1,124,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,22,31,0,0,0,0,31,219] >;
C3×C5⋊Q16 in GAP, Magma, Sage, TeX
C_3\times C_5\rtimes Q_{16} % in TeX
G:=Group("C3xC5:Q16"); // GroupNames label
G:=SmallGroup(240,47);
// by ID
G=gap.SmallGroup(240,47);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-5,144,169,151,867,441,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^5=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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