direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C15×D16, C80⋊3C6, C240⋊9C2, C48⋊3C10, C16⋊1C30, D8⋊1C30, C30.60D8, C60.193D4, C120.103C22, (C5×D8)⋊5C6, (C3×D8)⋊5C10, C8.2(C2×C30), C4.1(D4×C15), C2.3(C15×D8), C6.15(C5×D8), (C15×D8)⋊13C2, C40.24(C2×C6), C20.36(C3×D4), C10.15(C3×D8), C12.36(C5×D4), C24.19(C2×C10), SmallGroup(480,214)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×D16
G = < a,b,c | a15=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C15×D16
►On 240 points
Generators in S240
(1 156 226 45 178 161 142 101 127 59 83 196 80 19 211)(2 157 227 46 179 162 143 102 128 60 84 197 65 20 212)(3 158 228 47 180 163 144 103 113 61 85 198 66 21 213)(4 159 229 48 181 164 129 104 114 62 86 199 67 22 214)(5 160 230 33 182 165 130 105 115 63 87 200 68 23 215)(6 145 231 34 183 166 131 106 116 64 88 201 69 24 216)(7 146 232 35 184 167 132 107 117 49 89 202 70 25 217)(8 147 233 36 185 168 133 108 118 50 90 203 71 26 218)(9 148 234 37 186 169 134 109 119 51 91 204 72 27 219)(10 149 235 38 187 170 135 110 120 52 92 205 73 28 220)(11 150 236 39 188 171 136 111 121 53 93 206 74 29 221)(12 151 237 40 189 172 137 112 122 54 94 207 75 30 222)(13 152 238 41 190 173 138 97 123 55 95 208 76 31 223)(14 153 239 42 191 174 139 98 124 56 96 193 77 32 224)(15 154 240 43 192 175 140 99 125 57 81 194 78 17 209)(16 155 225 44 177 176 141 100 126 58 82 195 79 18 210)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 21)(18 20)(22 32)(23 31)(24 30)(25 29)(26 28)(33 41)(34 40)(35 39)(36 38)(42 48)(43 47)(44 46)(49 53)(50 52)(54 64)(55 63)(56 62)(57 61)(58 60)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(81 85)(82 84)(86 96)(87 95)(88 94)(89 93)(90 92)(97 105)(98 104)(99 103)(100 102)(106 112)(107 111)(108 110)(113 125)(114 124)(115 123)(116 122)(117 121)(118 120)(126 128)(129 139)(130 138)(131 137)(132 136)(133 135)(140 144)(141 143)(145 151)(146 150)(147 149)(152 160)(153 159)(154 158)(155 157)(162 176)(163 175)(164 174)(165 173)(166 172)(167 171)(168 170)(177 179)(180 192)(181 191)(182 190)(183 189)(184 188)(185 187)(193 199)(194 198)(195 197)(200 208)(201 207)(202 206)(203 205)(209 213)(210 212)(214 224)(215 223)(216 222)(217 221)(218 220)(225 227)(228 240)(229 239)(230 238)(231 237)(232 236)(233 235)
G:=sub<Sym(240)| (1,156,226,45,178,161,142,101,127,59,83,196,80,19,211)(2,157,227,46,179,162,143,102,128,60,84,197,65,20,212)(3,158,228,47,180,163,144,103,113,61,85,198,66,21,213)(4,159,229,48,181,164,129,104,114,62,86,199,67,22,214)(5,160,230,33,182,165,130,105,115,63,87,200,68,23,215)(6,145,231,34,183,166,131,106,116,64,88,201,69,24,216)(7,146,232,35,184,167,132,107,117,49,89,202,70,25,217)(8,147,233,36,185,168,133,108,118,50,90,203,71,26,218)(9,148,234,37,186,169,134,109,119,51,91,204,72,27,219)(10,149,235,38,187,170,135,110,120,52,92,205,73,28,220)(11,150,236,39,188,171,136,111,121,53,93,206,74,29,221)(12,151,237,40,189,172,137,112,122,54,94,207,75,30,222)(13,152,238,41,190,173,138,97,123,55,95,208,76,31,223)(14,153,239,42,191,174,139,98,124,56,96,193,77,32,224)(15,154,240,43,192,175,140,99,125,57,81,194,78,17,209)(16,155,225,44,177,176,141,100,126,58,82,195,79,18,210), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,21)(18,20)(22,32)(23,31)(24,30)(25,29)(26,28)(33,41)(34,40)(35,39)(36,38)(42,48)(43,47)(44,46)(49,53)(50,52)(54,64)(55,63)(56,62)(57,61)(58,60)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,85)(82,84)(86,96)(87,95)(88,94)(89,93)(90,92)(97,105)(98,104)(99,103)(100,102)(106,112)(107,111)(108,110)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(126,128)(129,139)(130,138)(131,137)(132,136)(133,135)(140,144)(141,143)(145,151)(146,150)(147,149)(152,160)(153,159)(154,158)(155,157)(162,176)(163,175)(164,174)(165,173)(166,172)(167,171)(168,170)(177,179)(180,192)(181,191)(182,190)(183,189)(184,188)(185,187)(193,199)(194,198)(195,197)(200,208)(201,207)(202,206)(203,205)(209,213)(210,212)(214,224)(215,223)(216,222)(217,221)(218,220)(225,227)(228,240)(229,239)(230,238)(231,237)(232,236)(233,235)>;
G:=Group( (1,156,226,45,178,161,142,101,127,59,83,196,80,19,211)(2,157,227,46,179,162,143,102,128,60,84,197,65,20,212)(3,158,228,47,180,163,144,103,113,61,85,198,66,21,213)(4,159,229,48,181,164,129,104,114,62,86,199,67,22,214)(5,160,230,33,182,165,130,105,115,63,87,200,68,23,215)(6,145,231,34,183,166,131,106,116,64,88,201,69,24,216)(7,146,232,35,184,167,132,107,117,49,89,202,70,25,217)(8,147,233,36,185,168,133,108,118,50,90,203,71,26,218)(9,148,234,37,186,169,134,109,119,51,91,204,72,27,219)(10,149,235,38,187,170,135,110,120,52,92,205,73,28,220)(11,150,236,39,188,171,136,111,121,53,93,206,74,29,221)(12,151,237,40,189,172,137,112,122,54,94,207,75,30,222)(13,152,238,41,190,173,138,97,123,55,95,208,76,31,223)(14,153,239,42,191,174,139,98,124,56,96,193,77,32,224)(15,154,240,43,192,175,140,99,125,57,81,194,78,17,209)(16,155,225,44,177,176,141,100,126,58,82,195,79,18,210), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,21)(18,20)(22,32)(23,31)(24,30)(25,29)(26,28)(33,41)(34,40)(35,39)(36,38)(42,48)(43,47)(44,46)(49,53)(50,52)(54,64)(55,63)(56,62)(57,61)(58,60)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,85)(82,84)(86,96)(87,95)(88,94)(89,93)(90,92)(97,105)(98,104)(99,103)(100,102)(106,112)(107,111)(108,110)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(126,128)(129,139)(130,138)(131,137)(132,136)(133,135)(140,144)(141,143)(145,151)(146,150)(147,149)(152,160)(153,159)(154,158)(155,157)(162,176)(163,175)(164,174)(165,173)(166,172)(167,171)(168,170)(177,179)(180,192)(181,191)(182,190)(183,189)(184,188)(185,187)(193,199)(194,198)(195,197)(200,208)(201,207)(202,206)(203,205)(209,213)(210,212)(214,224)(215,223)(216,222)(217,221)(218,220)(225,227)(228,240)(229,239)(230,238)(231,237)(232,236)(233,235) );
G=PermutationGroup([[(1,156,226,45,178,161,142,101,127,59,83,196,80,19,211),(2,157,227,46,179,162,143,102,128,60,84,197,65,20,212),(3,158,228,47,180,163,144,103,113,61,85,198,66,21,213),(4,159,229,48,181,164,129,104,114,62,86,199,67,22,214),(5,160,230,33,182,165,130,105,115,63,87,200,68,23,215),(6,145,231,34,183,166,131,106,116,64,88,201,69,24,216),(7,146,232,35,184,167,132,107,117,49,89,202,70,25,217),(8,147,233,36,185,168,133,108,118,50,90,203,71,26,218),(9,148,234,37,186,169,134,109,119,51,91,204,72,27,219),(10,149,235,38,187,170,135,110,120,52,92,205,73,28,220),(11,150,236,39,188,171,136,111,121,53,93,206,74,29,221),(12,151,237,40,189,172,137,112,122,54,94,207,75,30,222),(13,152,238,41,190,173,138,97,123,55,95,208,76,31,223),(14,153,239,42,191,174,139,98,124,56,96,193,77,32,224),(15,154,240,43,192,175,140,99,125,57,81,194,78,17,209),(16,155,225,44,177,176,141,100,126,58,82,195,79,18,210)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,21),(18,20),(22,32),(23,31),(24,30),(25,29),(26,28),(33,41),(34,40),(35,39),(36,38),(42,48),(43,47),(44,46),(49,53),(50,52),(54,64),(55,63),(56,62),(57,61),(58,60),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(81,85),(82,84),(86,96),(87,95),(88,94),(89,93),(90,92),(97,105),(98,104),(99,103),(100,102),(106,112),(107,111),(108,110),(113,125),(114,124),(115,123),(116,122),(117,121),(118,120),(126,128),(129,139),(130,138),(131,137),(132,136),(133,135),(140,144),(141,143),(145,151),(146,150),(147,149),(152,160),(153,159),(154,158),(155,157),(162,176),(163,175),(164,174),(165,173),(166,172),(167,171),(168,170),(177,179),(180,192),(181,191),(182,190),(183,189),(184,188),(185,187),(193,199),(194,198),(195,197),(200,208),(201,207),(202,206),(203,205),(209,213),(210,212),(214,224),(215,223),(216,222),(217,221),(218,220),(225,227),(228,240),(229,239),(230,238),(231,237),(232,236),(233,235)]])
165 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 15A | ··· | 15H | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | ··· | 30H | 30I | ··· | 30X | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 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 | 8 | 8 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 8 | ··· | 8 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | D4 | D8 | C3×D4 | D16 | C5×D4 | C3×D8 | C5×D8 | C3×D16 | D4×C15 | C5×D16 | C15×D8 | C15×D16 |
| kernel | C15×D16 | C240 | C15×D8 | C5×D16 | C3×D16 | C80 | C5×D8 | C48 | C3×D8 | D16 | C16 | D8 | C60 | C30 | C20 | C15 | C12 | C10 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C15×D16 ►in GL2(𝔽31) generated by
| 28 | 0 |
| 0 | 28 |
| 0 | 20 |
| 17 | 14 |
| 14 | 6 |
| 14 | 17 |
G:=sub<GL(2,GF(31))| [28,0,0,28],[0,17,20,14],[14,14,6,17] >;
C15×D16 in GAP, Magma, Sage, TeX
C_{15}\times D_{16} % in TeX
G:=Group("C15xD16"); // GroupNames label
G:=SmallGroup(480,214);
// by ID
G=gap.SmallGroup(480,214);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,869,6304,3161,242,15125,7572,124]);
// Polycyclic
G:=Group<a,b,c|a^15=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export