metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D15⋊C16, D30.1C8, Dic15.1C8, C5⋊C16⋊3S3, C5⋊1(S3×C16), C3⋊C8.4F5, C15⋊2(C2×C16), C3⋊1(D5⋊C16), C10.1(S3×C8), C30.2(C2×C8), C15⋊C16⋊2C2, C4.27(S3×F5), C20.27(C4×S3), C60.27(C2×C4), C6.1(D5⋊C8), (C4×D15).3C4, C5⋊2C8.28D6, C12.34(C2×F5), C2.1(D15⋊C8), D15⋊2C8.2C2, (C3×C5⋊C16)⋊2C2, (C5×C3⋊C8).3C4, (C3×C5⋊2C8).28C22, SmallGroup(480,240)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — D15⋊C16 |
Generators and relations for D15⋊C16
G = < a,b,c | a15=b2=c16=1, bab=a-1, cac-1=a13, cbc-1=a12b >
15C2
15C2
15C4
15C22
5S3
5S3
3D5
3D5
3C8
5C8
15C2×C4
5Dic3
5D6
3Dic5
3D10
5C16
15C2×C8
15C16
5C24
3C40
15C2×C16
5C48
Smallest permutation representation of D15⋊C16
►On 240 points
Generators in S240
(1 196 59 137 222 110 36 91 172 18 150 120 235 192 78)(2 177 121 19 92 111 138 197 79 236 151 173 37 223 60)(3 224 174 237 198 112 20 178 61 38 152 80 139 93 122)(4 94 65 39 179 97 238 209 123 140 153 62 21 199 175)(5 200 63 141 210 98 40 95 176 22 154 124 239 180 66)(6 181 125 23 96 99 142 201 67 240 155 161 41 211 64)(7 212 162 225 202 100 24 182 49 42 156 68 143 81 126)(8 82 69 43 183 101 226 213 127 144 157 50 25 203 163)(9 204 51 129 214 102 44 83 164 26 158 128 227 184 70)(10 185 113 27 84 103 130 205 71 228 159 165 45 215 52)(11 216 166 229 206 104 28 186 53 46 160 72 131 85 114)(12 86 73 47 187 105 230 217 115 132 145 54 29 207 167)(13 208 55 133 218 106 48 87 168 30 146 116 231 188 74)(14 189 117 31 88 107 134 193 75 232 147 169 33 219 56)(15 220 170 233 194 108 32 190 57 34 148 76 135 89 118)(16 90 77 35 191 109 234 221 119 136 149 58 17 195 171)
(1 78)(2 60)(3 122)(4 175)(5 66)(6 64)(7 126)(8 163)(9 70)(10 52)(11 114)(12 167)(13 74)(14 56)(15 118)(16 171)(17 77)(18 110)(19 173)(20 61)(21 65)(22 98)(23 161)(24 49)(25 69)(26 102)(27 165)(28 53)(29 73)(30 106)(31 169)(32 57)(33 117)(34 108)(35 58)(36 172)(37 121)(38 112)(39 62)(40 176)(41 125)(42 100)(43 50)(44 164)(45 113)(46 104)(47 54)(48 168)(51 227)(55 231)(59 235)(63 239)(67 142)(68 225)(71 130)(72 229)(75 134)(76 233)(79 138)(80 237)(81 212)(82 203)(84 159)(85 216)(86 207)(88 147)(89 220)(90 195)(92 151)(93 224)(94 199)(96 155)(97 140)(99 240)(101 144)(103 228)(105 132)(107 232)(109 136)(111 236)(115 230)(116 133)(119 234)(120 137)(123 238)(124 141)(127 226)(128 129)(131 166)(135 170)(139 174)(143 162)(145 187)(146 218)(148 194)(149 191)(150 222)(152 198)(153 179)(154 210)(156 202)(157 183)(158 214)(160 206)(177 223)(180 200)(181 211)(184 204)(185 215)(188 208)(189 219)(192 196)
(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)| (1,196,59,137,222,110,36,91,172,18,150,120,235,192,78)(2,177,121,19,92,111,138,197,79,236,151,173,37,223,60)(3,224,174,237,198,112,20,178,61,38,152,80,139,93,122)(4,94,65,39,179,97,238,209,123,140,153,62,21,199,175)(5,200,63,141,210,98,40,95,176,22,154,124,239,180,66)(6,181,125,23,96,99,142,201,67,240,155,161,41,211,64)(7,212,162,225,202,100,24,182,49,42,156,68,143,81,126)(8,82,69,43,183,101,226,213,127,144,157,50,25,203,163)(9,204,51,129,214,102,44,83,164,26,158,128,227,184,70)(10,185,113,27,84,103,130,205,71,228,159,165,45,215,52)(11,216,166,229,206,104,28,186,53,46,160,72,131,85,114)(12,86,73,47,187,105,230,217,115,132,145,54,29,207,167)(13,208,55,133,218,106,48,87,168,30,146,116,231,188,74)(14,189,117,31,88,107,134,193,75,232,147,169,33,219,56)(15,220,170,233,194,108,32,190,57,34,148,76,135,89,118)(16,90,77,35,191,109,234,221,119,136,149,58,17,195,171), (1,78)(2,60)(3,122)(4,175)(5,66)(6,64)(7,126)(8,163)(9,70)(10,52)(11,114)(12,167)(13,74)(14,56)(15,118)(16,171)(17,77)(18,110)(19,173)(20,61)(21,65)(22,98)(23,161)(24,49)(25,69)(26,102)(27,165)(28,53)(29,73)(30,106)(31,169)(32,57)(33,117)(34,108)(35,58)(36,172)(37,121)(38,112)(39,62)(40,176)(41,125)(42,100)(43,50)(44,164)(45,113)(46,104)(47,54)(48,168)(51,227)(55,231)(59,235)(63,239)(67,142)(68,225)(71,130)(72,229)(75,134)(76,233)(79,138)(80,237)(81,212)(82,203)(84,159)(85,216)(86,207)(88,147)(89,220)(90,195)(92,151)(93,224)(94,199)(96,155)(97,140)(99,240)(101,144)(103,228)(105,132)(107,232)(109,136)(111,236)(115,230)(116,133)(119,234)(120,137)(123,238)(124,141)(127,226)(128,129)(131,166)(135,170)(139,174)(143,162)(145,187)(146,218)(148,194)(149,191)(150,222)(152,198)(153,179)(154,210)(156,202)(157,183)(158,214)(160,206)(177,223)(180,200)(181,211)(184,204)(185,215)(188,208)(189,219)(192,196), (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( (1,196,59,137,222,110,36,91,172,18,150,120,235,192,78)(2,177,121,19,92,111,138,197,79,236,151,173,37,223,60)(3,224,174,237,198,112,20,178,61,38,152,80,139,93,122)(4,94,65,39,179,97,238,209,123,140,153,62,21,199,175)(5,200,63,141,210,98,40,95,176,22,154,124,239,180,66)(6,181,125,23,96,99,142,201,67,240,155,161,41,211,64)(7,212,162,225,202,100,24,182,49,42,156,68,143,81,126)(8,82,69,43,183,101,226,213,127,144,157,50,25,203,163)(9,204,51,129,214,102,44,83,164,26,158,128,227,184,70)(10,185,113,27,84,103,130,205,71,228,159,165,45,215,52)(11,216,166,229,206,104,28,186,53,46,160,72,131,85,114)(12,86,73,47,187,105,230,217,115,132,145,54,29,207,167)(13,208,55,133,218,106,48,87,168,30,146,116,231,188,74)(14,189,117,31,88,107,134,193,75,232,147,169,33,219,56)(15,220,170,233,194,108,32,190,57,34,148,76,135,89,118)(16,90,77,35,191,109,234,221,119,136,149,58,17,195,171), (1,78)(2,60)(3,122)(4,175)(5,66)(6,64)(7,126)(8,163)(9,70)(10,52)(11,114)(12,167)(13,74)(14,56)(15,118)(16,171)(17,77)(18,110)(19,173)(20,61)(21,65)(22,98)(23,161)(24,49)(25,69)(26,102)(27,165)(28,53)(29,73)(30,106)(31,169)(32,57)(33,117)(34,108)(35,58)(36,172)(37,121)(38,112)(39,62)(40,176)(41,125)(42,100)(43,50)(44,164)(45,113)(46,104)(47,54)(48,168)(51,227)(55,231)(59,235)(63,239)(67,142)(68,225)(71,130)(72,229)(75,134)(76,233)(79,138)(80,237)(81,212)(82,203)(84,159)(85,216)(86,207)(88,147)(89,220)(90,195)(92,151)(93,224)(94,199)(96,155)(97,140)(99,240)(101,144)(103,228)(105,132)(107,232)(109,136)(111,236)(115,230)(116,133)(119,234)(120,137)(123,238)(124,141)(127,226)(128,129)(131,166)(135,170)(139,174)(143,162)(145,187)(146,218)(148,194)(149,191)(150,222)(152,198)(153,179)(154,210)(156,202)(157,183)(158,214)(160,206)(177,223)(180,200)(181,211)(184,204)(185,215)(188,208)(189,219)(192,196), (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([[(1,196,59,137,222,110,36,91,172,18,150,120,235,192,78),(2,177,121,19,92,111,138,197,79,236,151,173,37,223,60),(3,224,174,237,198,112,20,178,61,38,152,80,139,93,122),(4,94,65,39,179,97,238,209,123,140,153,62,21,199,175),(5,200,63,141,210,98,40,95,176,22,154,124,239,180,66),(6,181,125,23,96,99,142,201,67,240,155,161,41,211,64),(7,212,162,225,202,100,24,182,49,42,156,68,143,81,126),(8,82,69,43,183,101,226,213,127,144,157,50,25,203,163),(9,204,51,129,214,102,44,83,164,26,158,128,227,184,70),(10,185,113,27,84,103,130,205,71,228,159,165,45,215,52),(11,216,166,229,206,104,28,186,53,46,160,72,131,85,114),(12,86,73,47,187,105,230,217,115,132,145,54,29,207,167),(13,208,55,133,218,106,48,87,168,30,146,116,231,188,74),(14,189,117,31,88,107,134,193,75,232,147,169,33,219,56),(15,220,170,233,194,108,32,190,57,34,148,76,135,89,118),(16,90,77,35,191,109,234,221,119,136,149,58,17,195,171)], [(1,78),(2,60),(3,122),(4,175),(5,66),(6,64),(7,126),(8,163),(9,70),(10,52),(11,114),(12,167),(13,74),(14,56),(15,118),(16,171),(17,77),(18,110),(19,173),(20,61),(21,65),(22,98),(23,161),(24,49),(25,69),(26,102),(27,165),(28,53),(29,73),(30,106),(31,169),(32,57),(33,117),(34,108),(35,58),(36,172),(37,121),(38,112),(39,62),(40,176),(41,125),(42,100),(43,50),(44,164),(45,113),(46,104),(47,54),(48,168),(51,227),(55,231),(59,235),(63,239),(67,142),(68,225),(71,130),(72,229),(75,134),(76,233),(79,138),(80,237),(81,212),(82,203),(84,159),(85,216),(86,207),(88,147),(89,220),(90,195),(92,151),(93,224),(94,199),(96,155),(97,140),(99,240),(101,144),(103,228),(105,132),(107,232),(109,136),(111,236),(115,230),(116,133),(119,234),(120,137),(123,238),(124,141),(127,226),(128,129),(131,166),(135,170),(139,174),(143,162),(145,187),(146,218),(148,194),(149,191),(150,222),(152,198),(153,179),(154,210),(156,202),(157,183),(158,214),(160,206),(177,223),(180,200),(181,211),(184,204),(185,215),(188,208),(189,219),(192,196)], [(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)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5 | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 12A | 12B | 15 | 16A | ··· | 16H | 16I | ··· | 16P | 20A | 20B | 24A | 24B | 24C | 24D | 30 | 40A | 40B | 40C | 40D | 48A | ··· | 48H | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 15 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 40 | 40 | 40 | 40 | 48 | ··· | 48 | 60 | 60 |
| size | 1 | 1 | 15 | 15 | 2 | 1 | 1 | 15 | 15 | 4 | 2 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 4 | 2 | 2 | 8 | 5 | ··· | 5 | 15 | ··· | 15 | 4 | 4 | 10 | 10 | 10 | 10 | 8 | 12 | 12 | 12 | 12 | 10 | ··· | 10 | 8 | 8 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | D6 | C4×S3 | S3×C8 | S3×C16 | F5 | C2×F5 | D5⋊C8 | D5⋊C16 | S3×F5 | D15⋊C8 | D15⋊C16 |
| kernel | D15⋊C16 | C3×C5⋊C16 | C15⋊C16 | D15⋊2C8 | C5×C3⋊C8 | C4×D15 | Dic15 | D30 | D15 | C5⋊C16 | C5⋊2C8 | C20 | C10 | C5 | C3⋊C8 | C12 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 2 | 4 | 8 | 1 | 1 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of D15⋊C16 ►in GL6(𝔽241)
| 240 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 1 | 0 |
| 0 | 0 | 0 | 240 | 0 | 1 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 1 |
| 0 | 0 | 0 | 240 | 1 | 0 |
| 111 | 0 | 0 | 0 | 0 | 0 |
| 0 | 111 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 218 | 138 | 0 |
| 0 | 0 | 161 | 218 | 0 | 23 |
| 0 | 0 | 23 | 0 | 218 | 161 |
| 0 | 0 | 0 | 138 | 218 | 23 |
G:=sub<GL(6,GF(241))| [240,240,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0],[240,240,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,240,240,240,240,0,0,0,0,0,1,0,0,0,0,1,0],[111,0,0,0,0,0,0,111,0,0,0,0,0,0,23,161,23,0,0,0,218,218,0,138,0,0,138,0,218,218,0,0,0,23,161,23] >;
D15⋊C16 in GAP, Magma, Sage, TeX
D_{15}\rtimes C_{16} % in TeX
G:=Group("D15:C16"); // GroupNames label
G:=SmallGroup(480,240);
// by ID
G=gap.SmallGroup(480,240);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,58,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c|a^15=b^2=c^16=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^12*b>;
// generators/relations
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