direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C16×D15, C80⋊4S3, C48⋊5D5, C240⋊6C2, D30.6C8, C40.70D6, C8.19D30, C24.75D10, Dic15.6C8, C120.88C22, C5⋊4(S3×C16), C3⋊2(D5×C16), C6.6(C8×D5), C15⋊10(C2×C16), C2.1(C8×D15), C20.86(C4×S3), C10.15(S3×C8), C30.43(C2×C8), (C8×D15).6C2, C15⋊3C16⋊13C2, C12.54(C4×D5), C4.16(C4×D15), C15⋊3C8.10C4, C60.191(C2×C4), (C4×D15).15C4, SmallGroup(480,157)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C16×D15 |
Generators and relations for C16×D15
G = < a,b,c | a16=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C16×D15
►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 61 238 214 86 116 134 157 165 79 98 35 195 23 178)(2 62 239 215 87 117 135 158 166 80 99 36 196 24 179)(3 63 240 216 88 118 136 159 167 65 100 37 197 25 180)(4 64 225 217 89 119 137 160 168 66 101 38 198 26 181)(5 49 226 218 90 120 138 145 169 67 102 39 199 27 182)(6 50 227 219 91 121 139 146 170 68 103 40 200 28 183)(7 51 228 220 92 122 140 147 171 69 104 41 201 29 184)(8 52 229 221 93 123 141 148 172 70 105 42 202 30 185)(9 53 230 222 94 124 142 149 173 71 106 43 203 31 186)(10 54 231 223 95 125 143 150 174 72 107 44 204 32 187)(11 55 232 224 96 126 144 151 175 73 108 45 205 17 188)(12 56 233 209 81 127 129 152 176 74 109 46 206 18 189)(13 57 234 210 82 128 130 153 161 75 110 47 207 19 190)(14 58 235 211 83 113 131 154 162 76 111 48 208 20 191)(15 59 236 212 84 114 132 155 163 77 112 33 193 21 192)(16 60 237 213 85 115 133 156 164 78 97 34 194 22 177)
(1 178)(2 179)(3 180)(4 181)(5 182)(6 183)(7 184)(8 185)(9 186)(10 187)(11 188)(12 189)(13 190)(14 191)(15 192)(16 177)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 212)(34 213)(35 214)(36 215)(37 216)(38 217)(39 218)(40 219)(41 220)(42 221)(43 222)(44 223)(45 224)(46 209)(47 210)(48 211)(65 118)(66 119)(67 120)(68 121)(69 122)(70 123)(71 124)(72 125)(73 126)(74 127)(75 128)(76 113)(77 114)(78 115)(79 116)(80 117)(81 109)(82 110)(83 111)(84 112)(85 97)(86 98)(87 99)(88 100)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(129 176)(130 161)(131 162)(132 163)(133 164)(134 165)(135 166)(136 167)(137 168)(138 169)(139 170)(140 171)(141 172)(142 173)(143 174)(144 175)(193 236)(194 237)(195 238)(196 239)(197 240)(198 225)(199 226)(200 227)(201 228)(202 229)(203 230)(204 231)(205 232)(206 233)(207 234)(208 235)
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), (1,61,238,214,86,116,134,157,165,79,98,35,195,23,178)(2,62,239,215,87,117,135,158,166,80,99,36,196,24,179)(3,63,240,216,88,118,136,159,167,65,100,37,197,25,180)(4,64,225,217,89,119,137,160,168,66,101,38,198,26,181)(5,49,226,218,90,120,138,145,169,67,102,39,199,27,182)(6,50,227,219,91,121,139,146,170,68,103,40,200,28,183)(7,51,228,220,92,122,140,147,171,69,104,41,201,29,184)(8,52,229,221,93,123,141,148,172,70,105,42,202,30,185)(9,53,230,222,94,124,142,149,173,71,106,43,203,31,186)(10,54,231,223,95,125,143,150,174,72,107,44,204,32,187)(11,55,232,224,96,126,144,151,175,73,108,45,205,17,188)(12,56,233,209,81,127,129,152,176,74,109,46,206,18,189)(13,57,234,210,82,128,130,153,161,75,110,47,207,19,190)(14,58,235,211,83,113,131,154,162,76,111,48,208,20,191)(15,59,236,212,84,114,132,155,163,77,112,33,193,21,192)(16,60,237,213,85,115,133,156,164,78,97,34,194,22,177), (1,178)(2,179)(3,180)(4,181)(5,182)(6,183)(7,184)(8,185)(9,186)(10,187)(11,188)(12,189)(13,190)(14,191)(15,192)(16,177)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,212)(34,213)(35,214)(36,215)(37,216)(38,217)(39,218)(40,219)(41,220)(42,221)(43,222)(44,223)(45,224)(46,209)(47,210)(48,211)(65,118)(66,119)(67,120)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,113)(77,114)(78,115)(79,116)(80,117)(81,109)(82,110)(83,111)(84,112)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(129,176)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,169)(139,170)(140,171)(141,172)(142,173)(143,174)(144,175)(193,236)(194,237)(195,238)(196,239)(197,240)(198,225)(199,226)(200,227)(201,228)(202,229)(203,230)(204,231)(205,232)(206,233)(207,234)(208,235)>;
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), (1,61,238,214,86,116,134,157,165,79,98,35,195,23,178)(2,62,239,215,87,117,135,158,166,80,99,36,196,24,179)(3,63,240,216,88,118,136,159,167,65,100,37,197,25,180)(4,64,225,217,89,119,137,160,168,66,101,38,198,26,181)(5,49,226,218,90,120,138,145,169,67,102,39,199,27,182)(6,50,227,219,91,121,139,146,170,68,103,40,200,28,183)(7,51,228,220,92,122,140,147,171,69,104,41,201,29,184)(8,52,229,221,93,123,141,148,172,70,105,42,202,30,185)(9,53,230,222,94,124,142,149,173,71,106,43,203,31,186)(10,54,231,223,95,125,143,150,174,72,107,44,204,32,187)(11,55,232,224,96,126,144,151,175,73,108,45,205,17,188)(12,56,233,209,81,127,129,152,176,74,109,46,206,18,189)(13,57,234,210,82,128,130,153,161,75,110,47,207,19,190)(14,58,235,211,83,113,131,154,162,76,111,48,208,20,191)(15,59,236,212,84,114,132,155,163,77,112,33,193,21,192)(16,60,237,213,85,115,133,156,164,78,97,34,194,22,177), (1,178)(2,179)(3,180)(4,181)(5,182)(6,183)(7,184)(8,185)(9,186)(10,187)(11,188)(12,189)(13,190)(14,191)(15,192)(16,177)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,212)(34,213)(35,214)(36,215)(37,216)(38,217)(39,218)(40,219)(41,220)(42,221)(43,222)(44,223)(45,224)(46,209)(47,210)(48,211)(65,118)(66,119)(67,120)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,113)(77,114)(78,115)(79,116)(80,117)(81,109)(82,110)(83,111)(84,112)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(129,176)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,169)(139,170)(140,171)(141,172)(142,173)(143,174)(144,175)(193,236)(194,237)(195,238)(196,239)(197,240)(198,225)(199,226)(200,227)(201,228)(202,229)(203,230)(204,231)(205,232)(206,233)(207,234)(208,235) );
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)], [(1,61,238,214,86,116,134,157,165,79,98,35,195,23,178),(2,62,239,215,87,117,135,158,166,80,99,36,196,24,179),(3,63,240,216,88,118,136,159,167,65,100,37,197,25,180),(4,64,225,217,89,119,137,160,168,66,101,38,198,26,181),(5,49,226,218,90,120,138,145,169,67,102,39,199,27,182),(6,50,227,219,91,121,139,146,170,68,103,40,200,28,183),(7,51,228,220,92,122,140,147,171,69,104,41,201,29,184),(8,52,229,221,93,123,141,148,172,70,105,42,202,30,185),(9,53,230,222,94,124,142,149,173,71,106,43,203,31,186),(10,54,231,223,95,125,143,150,174,72,107,44,204,32,187),(11,55,232,224,96,126,144,151,175,73,108,45,205,17,188),(12,56,233,209,81,127,129,152,176,74,109,46,206,18,189),(13,57,234,210,82,128,130,153,161,75,110,47,207,19,190),(14,58,235,211,83,113,131,154,162,76,111,48,208,20,191),(15,59,236,212,84,114,132,155,163,77,112,33,193,21,192),(16,60,237,213,85,115,133,156,164,78,97,34,194,22,177)], [(1,178),(2,179),(3,180),(4,181),(5,182),(6,183),(7,184),(8,185),(9,186),(10,187),(11,188),(12,189),(13,190),(14,191),(15,192),(16,177),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,212),(34,213),(35,214),(36,215),(37,216),(38,217),(39,218),(40,219),(41,220),(42,221),(43,222),(44,223),(45,224),(46,209),(47,210),(48,211),(65,118),(66,119),(67,120),(68,121),(69,122),(70,123),(71,124),(72,125),(73,126),(74,127),(75,128),(76,113),(77,114),(78,115),(79,116),(80,117),(81,109),(82,110),(83,111),(84,112),(85,97),(86,98),(87,99),(88,100),(89,101),(90,102),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(129,176),(130,161),(131,162),(132,163),(133,164),(134,165),(135,166),(136,167),(137,168),(138,169),(139,170),(140,171),(141,172),(142,173),(143,174),(144,175),(193,236),(194,237),(195,238),(196,239),(197,240),(198,225),(199,226),(200,227),(201,228),(202,229),(203,230),(204,231),(205,232),(206,233),(207,234),(208,235)]])
144 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 | 12A | 12B | 15A | 15B | 15C | 15D | 16A | ··· | 16H | 16I | ··· | 16P | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 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 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 15 | 15 | 2 | 1 | 1 | 15 | 15 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 15 | ··· | 15 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | D5 | D6 | D10 | C4×S3 | D15 | C4×D5 | S3×C8 | D30 | C8×D5 | S3×C16 | C4×D15 | D5×C16 | C8×D15 | C16×D15 |
| kernel | C16×D15 | C15⋊3C16 | C240 | C8×D15 | C15⋊3C8 | C4×D15 | Dic15 | D30 | D15 | C80 | C48 | C40 | C24 | C20 | C16 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C16×D15 ►in GL3(𝔽241) generated by
| 115 | 0 | 0 |
| 0 | 211 | 0 |
| 0 | 0 | 211 |
| 1 | 0 | 0 |
| 0 | 94 | 110 |
| 0 | 131 | 161 |
| 240 | 0 | 0 |
| 0 | 157 | 94 |
| 0 | 148 | 84 |
G:=sub<GL(3,GF(241))| [115,0,0,0,211,0,0,0,211],[1,0,0,0,94,131,0,110,161],[240,0,0,0,157,148,0,94,84] >;
C16×D15 in GAP, Magma, Sage, TeX
C_{16}\times D_{15} % in TeX
G:=Group("C16xD15"); // GroupNames label
G:=SmallGroup(480,157);
// by ID
G=gap.SmallGroup(480,157);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,36,58,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^16=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export