metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊1D16, D40⋊3S3, D24⋊3D5, C40.4D6, C30.1D8, C24.4D10, C60.72D4, C120.32C22, (C3×D40)⋊6C2, (C5×D24)⋊5C2, C5⋊2(C3⋊D16), C3⋊2(C5⋊D16), C15⋊3C16⋊4C2, C8.25(S3×D5), C6.8(D4⋊D5), C10.8(D4⋊S3), C12.1(C5⋊D4), C4.1(C15⋊D4), C2.4(C15⋊D8), C20.1(C3⋊D4), SmallGroup(480,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊D16
G = < a,b,c | a15=b16=c2=1, bab-1=a-1, cac=a4, cbc=b-1 >
24C2
40C2
12C22
20C22
8S3
40C6
8D5
24C10
6D4
10D4
4D6
20C2×C6
4D10
12C2×C10
3D8
5D8
15C16
2D12
10C3×D4
2D20
15D16
Smallest permutation representation of C15⋊D16
►On 240 points
Generators in S240
(1 234 107 207 63 216 78 180 48 83 20 128 168 139 158)(2 159 140 169 113 21 84 33 181 79 217 64 208 108 235)(3 236 109 193 49 218 80 182 34 85 22 114 170 141 160)(4 145 142 171 115 23 86 35 183 65 219 50 194 110 237)(5 238 111 195 51 220 66 184 36 87 24 116 172 143 146)(6 147 144 173 117 25 88 37 185 67 221 52 196 112 239)(7 240 97 197 53 222 68 186 38 89 26 118 174 129 148)(8 149 130 175 119 27 90 39 187 69 223 54 198 98 225)(9 226 99 199 55 224 70 188 40 91 28 120 176 131 150)(10 151 132 161 121 29 92 41 189 71 209 56 200 100 227)(11 228 101 201 57 210 72 190 42 93 30 122 162 133 152)(12 153 134 163 123 31 94 43 191 73 211 58 202 102 229)(13 230 103 203 59 212 74 192 44 95 32 124 164 135 154)(14 155 136 165 125 17 96 45 177 75 213 60 204 104 231)(15 232 105 205 61 214 76 178 46 81 18 126 166 137 156)(16 157 138 167 127 19 82 47 179 77 215 62 206 106 233)
(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 23)(18 22)(19 21)(24 32)(25 31)(26 30)(27 29)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 100)(40 99)(41 98)(42 97)(43 112)(44 111)(45 110)(46 109)(47 108)(48 107)(49 232)(50 231)(51 230)(52 229)(53 228)(54 227)(55 226)(56 225)(57 240)(58 239)(59 238)(60 237)(61 236)(62 235)(63 234)(64 233)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(113 157)(114 156)(115 155)(116 154)(117 153)(118 152)(119 151)(120 150)(121 149)(122 148)(123 147)(124 146)(125 145)(126 160)(127 159)(128 158)(129 190)(130 189)(131 188)(132 187)(133 186)(134 185)(135 184)(136 183)(137 182)(138 181)(139 180)(140 179)(141 178)(142 177)(143 192)(144 191)(161 198)(162 197)(163 196)(164 195)(165 194)(166 193)(167 208)(168 207)(169 206)(170 205)(171 204)(172 203)(173 202)(174 201)(175 200)(176 199)(209 223)(210 222)(211 221)(212 220)(213 219)(214 218)(215 217)
G:=sub<Sym(240)| (1,234,107,207,63,216,78,180,48,83,20,128,168,139,158)(2,159,140,169,113,21,84,33,181,79,217,64,208,108,235)(3,236,109,193,49,218,80,182,34,85,22,114,170,141,160)(4,145,142,171,115,23,86,35,183,65,219,50,194,110,237)(5,238,111,195,51,220,66,184,36,87,24,116,172,143,146)(6,147,144,173,117,25,88,37,185,67,221,52,196,112,239)(7,240,97,197,53,222,68,186,38,89,26,118,174,129,148)(8,149,130,175,119,27,90,39,187,69,223,54,198,98,225)(9,226,99,199,55,224,70,188,40,91,28,120,176,131,150)(10,151,132,161,121,29,92,41,189,71,209,56,200,100,227)(11,228,101,201,57,210,72,190,42,93,30,122,162,133,152)(12,153,134,163,123,31,94,43,191,73,211,58,202,102,229)(13,230,103,203,59,212,74,192,44,95,32,124,164,135,154)(14,155,136,165,125,17,96,45,177,75,213,60,204,104,231)(15,232,105,205,61,214,76,178,46,81,18,126,166,137,156)(16,157,138,167,127,19,82,47,179,77,215,62,206,106,233), (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)(17,23)(18,22)(19,21)(24,32)(25,31)(26,30)(27,29)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,232)(50,231)(51,230)(52,229)(53,228)(54,227)(55,226)(56,225)(57,240)(58,239)(59,238)(60,237)(61,236)(62,235)(63,234)(64,233)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,160)(127,159)(128,158)(129,190)(130,189)(131,188)(132,187)(133,186)(134,185)(135,184)(136,183)(137,182)(138,181)(139,180)(140,179)(141,178)(142,177)(143,192)(144,191)(161,198)(162,197)(163,196)(164,195)(165,194)(166,193)(167,208)(168,207)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(209,223)(210,222)(211,221)(212,220)(213,219)(214,218)(215,217)>;
G:=Group( (1,234,107,207,63,216,78,180,48,83,20,128,168,139,158)(2,159,140,169,113,21,84,33,181,79,217,64,208,108,235)(3,236,109,193,49,218,80,182,34,85,22,114,170,141,160)(4,145,142,171,115,23,86,35,183,65,219,50,194,110,237)(5,238,111,195,51,220,66,184,36,87,24,116,172,143,146)(6,147,144,173,117,25,88,37,185,67,221,52,196,112,239)(7,240,97,197,53,222,68,186,38,89,26,118,174,129,148)(8,149,130,175,119,27,90,39,187,69,223,54,198,98,225)(9,226,99,199,55,224,70,188,40,91,28,120,176,131,150)(10,151,132,161,121,29,92,41,189,71,209,56,200,100,227)(11,228,101,201,57,210,72,190,42,93,30,122,162,133,152)(12,153,134,163,123,31,94,43,191,73,211,58,202,102,229)(13,230,103,203,59,212,74,192,44,95,32,124,164,135,154)(14,155,136,165,125,17,96,45,177,75,213,60,204,104,231)(15,232,105,205,61,214,76,178,46,81,18,126,166,137,156)(16,157,138,167,127,19,82,47,179,77,215,62,206,106,233), (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)(17,23)(18,22)(19,21)(24,32)(25,31)(26,30)(27,29)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,232)(50,231)(51,230)(52,229)(53,228)(54,227)(55,226)(56,225)(57,240)(58,239)(59,238)(60,237)(61,236)(62,235)(63,234)(64,233)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,160)(127,159)(128,158)(129,190)(130,189)(131,188)(132,187)(133,186)(134,185)(135,184)(136,183)(137,182)(138,181)(139,180)(140,179)(141,178)(142,177)(143,192)(144,191)(161,198)(162,197)(163,196)(164,195)(165,194)(166,193)(167,208)(168,207)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(209,223)(210,222)(211,221)(212,220)(213,219)(214,218)(215,217) );
G=PermutationGroup([[(1,234,107,207,63,216,78,180,48,83,20,128,168,139,158),(2,159,140,169,113,21,84,33,181,79,217,64,208,108,235),(3,236,109,193,49,218,80,182,34,85,22,114,170,141,160),(4,145,142,171,115,23,86,35,183,65,219,50,194,110,237),(5,238,111,195,51,220,66,184,36,87,24,116,172,143,146),(6,147,144,173,117,25,88,37,185,67,221,52,196,112,239),(7,240,97,197,53,222,68,186,38,89,26,118,174,129,148),(8,149,130,175,119,27,90,39,187,69,223,54,198,98,225),(9,226,99,199,55,224,70,188,40,91,28,120,176,131,150),(10,151,132,161,121,29,92,41,189,71,209,56,200,100,227),(11,228,101,201,57,210,72,190,42,93,30,122,162,133,152),(12,153,134,163,123,31,94,43,191,73,211,58,202,102,229),(13,230,103,203,59,212,74,192,44,95,32,124,164,135,154),(14,155,136,165,125,17,96,45,177,75,213,60,204,104,231),(15,232,105,205,61,214,76,178,46,81,18,126,166,137,156),(16,157,138,167,127,19,82,47,179,77,215,62,206,106,233)], [(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),(17,23),(18,22),(19,21),(24,32),(25,31),(26,30),(27,29),(33,106),(34,105),(35,104),(36,103),(37,102),(38,101),(39,100),(40,99),(41,98),(42,97),(43,112),(44,111),(45,110),(46,109),(47,108),(48,107),(49,232),(50,231),(51,230),(52,229),(53,228),(54,227),(55,226),(56,225),(57,240),(58,239),(59,238),(60,237),(61,236),(62,235),(63,234),(64,233),(65,96),(66,95),(67,94),(68,93),(69,92),(70,91),(71,90),(72,89),(73,88),(74,87),(75,86),(76,85),(77,84),(78,83),(79,82),(80,81),(113,157),(114,156),(115,155),(116,154),(117,153),(118,152),(119,151),(120,150),(121,149),(122,148),(123,147),(124,146),(125,145),(126,160),(127,159),(128,158),(129,190),(130,189),(131,188),(132,187),(133,186),(134,185),(135,184),(136,183),(137,182),(138,181),(139,180),(140,179),(141,178),(142,177),(143,192),(144,191),(161,198),(162,197),(163,196),(164,195),(165,194),(166,193),(167,208),(168,207),(169,206),(170,205),(171,204),(172,203),(173,202),(174,201),(175,200),(176,199),(209,223),(210,222),(211,221),(212,220),(213,219),(214,218),(215,217)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 16A | 16B | 16C | 16D | 20A | 20B | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 24 | 40 | 2 | 2 | 2 | 2 | 2 | 40 | 40 | 2 | 2 | 2 | 2 | 24 | 24 | 24 | 24 | 4 | 4 | 4 | 30 | 30 | 30 | 30 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | C3⋊D4 | D16 | C5⋊D4 | D4⋊S3 | S3×D5 | D4⋊D5 | C3⋊D16 | C15⋊D4 | C5⋊D16 | C15⋊D8 | C15⋊D16 |
| kernel | C15⋊D16 | C15⋊3C16 | C3×D40 | C5×D24 | D40 | C60 | D24 | C40 | C30 | C24 | C20 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C15⋊D16 ►in GL6(𝔽241)
| 0 | 189 | 0 | 0 | 0 | 0 |
| 51 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 162 | 0 | 0 |
| 0 | 0 | 0 | 225 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 172 | 145 | 0 | 0 | 0 | 0 |
| 145 | 69 | 0 | 0 | 0 | 0 |
| 0 | 0 | 201 | 40 | 0 | 0 |
| 0 | 0 | 207 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 172 | 204 |
| 0 | 0 | 0 | 0 | 28 | 15 |
| 51 | 52 | 0 | 0 | 0 | 0 |
| 191 | 190 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 26 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 94 |
| 0 | 0 | 0 | 0 | 194 | 232 |
G:=sub<GL(6,GF(241))| [0,51,0,0,0,0,189,51,0,0,0,0,0,0,15,0,0,0,0,0,162,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[172,145,0,0,0,0,145,69,0,0,0,0,0,0,201,207,0,0,0,0,40,40,0,0,0,0,0,0,172,28,0,0,0,0,204,15],[51,191,0,0,0,0,52,190,0,0,0,0,0,0,1,0,0,0,0,0,26,240,0,0,0,0,0,0,9,194,0,0,0,0,94,232] >;
C15⋊D16 in GAP, Magma, Sage, TeX
C_{15}\rtimes D_{16} % in TeX
G:=Group("C15:D16"); // GroupNames label
G:=SmallGroup(480,13);
// by ID
G=gap.SmallGroup(480,13);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,254,135,142,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^15=b^16=c^2=1,b*a*b^-1=a^-1,c*a*c=a^4,c*b*c=b^-1>;
// generators/relations
Export