metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D15⋊2C16, D30.4C8, C40.50D6, C24.57D10, Dic15.4C8, C120.53C22, C5⋊3(S3×C16), C3⋊C16⋊6D5, C3⋊1(D5×C16), C15⋊7(C2×C16), C6.1(C8×D5), C5⋊2C16⋊6S3, C8.36(S3×D5), C15⋊3C8.8C4, C20.68(C4×S3), C10.10(S3×C8), C30.21(C2×C8), (C8×D15).4C2, C12.36(C4×D5), C60.137(C2×C4), (C4×D15).11C4, C2.1(D15⋊2C8), C4.16(D30.C2), (C5×C3⋊C16)⋊5C2, (C3×C5⋊2C16)⋊8C2, SmallGroup(480,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — D15⋊2C16 |
Generators and relations for D15⋊2C16
G = < a,b,c | a15=b2=c16=1, bab=a-1, cac-1=a11, cbc-1=a10b >
15C2
15C2
15C4
15C22
5S3
5S3
3D5
3D5
15C2×C4
15C8
5Dic3
5D6
3Dic5
3D10
3C16
5C16
15C2×C8
15C2×C16
5C48
3C80
Smallest permutation representation of D15⋊2C16
►On 240 points
Generators in S240
(1 185 108 92 29 119 65 238 148 224 197 132 38 176 49)(2 133 239 93 50 198 66 109 161 209 120 186 39 149 30)(3 187 110 94 31 121 67 240 150 210 199 134 40 162 51)(4 135 225 95 52 200 68 111 163 211 122 188 41 151 32)(5 189 112 96 17 123 69 226 152 212 201 136 42 164 53)(6 137 227 81 54 202 70 97 165 213 124 190 43 153 18)(7 191 98 82 19 125 71 228 154 214 203 138 44 166 55)(8 139 229 83 56 204 72 99 167 215 126 192 45 155 20)(9 177 100 84 21 127 73 230 156 216 205 140 46 168 57)(10 141 231 85 58 206 74 101 169 217 128 178 47 157 22)(11 179 102 86 23 113 75 232 158 218 207 142 48 170 59)(12 143 233 87 60 208 76 103 171 219 114 180 33 159 24)(13 181 104 88 25 115 77 234 160 220 193 144 34 172 61)(14 129 235 89 62 194 78 105 173 221 116 182 35 145 26)(15 183 106 90 27 117 79 236 146 222 195 130 36 174 63)(16 131 237 91 64 196 80 107 175 223 118 184 37 147 28)
(1 57)(2 22)(3 59)(4 24)(5 61)(6 26)(7 63)(8 28)(9 49)(10 30)(11 51)(12 32)(13 53)(14 18)(15 55)(16 20)(17 193)(19 195)(21 197)(23 199)(25 201)(27 203)(29 205)(31 207)(33 225)(34 112)(35 227)(36 98)(37 229)(38 100)(39 231)(40 102)(41 233)(42 104)(43 235)(44 106)(45 237)(46 108)(47 239)(48 110)(50 128)(52 114)(54 116)(56 118)(58 120)(60 122)(62 124)(64 126)(65 156)(66 169)(67 158)(68 171)(69 160)(70 173)(71 146)(72 175)(73 148)(74 161)(75 150)(76 163)(77 152)(78 165)(79 154)(80 167)(81 182)(82 130)(83 184)(84 132)(85 186)(86 134)(87 188)(88 136)(89 190)(90 138)(91 192)(92 140)(93 178)(94 142)(95 180)(96 144)(97 105)(99 107)(101 109)(103 111)(113 210)(115 212)(117 214)(119 216)(121 218)(123 220)(125 222)(127 224)(129 153)(131 155)(133 157)(135 159)(137 145)(139 147)(141 149)(143 151)(162 179)(164 181)(166 183)(168 185)(170 187)(172 189)(174 191)(176 177)(194 213)(196 215)(198 217)(200 219)(202 221)(204 223)(206 209)(208 211)(226 234)(228 236)(230 238)(232 240)
(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)
G:=sub<Sym(240)| (1,185,108,92,29,119,65,238,148,224,197,132,38,176,49)(2,133,239,93,50,198,66,109,161,209,120,186,39,149,30)(3,187,110,94,31,121,67,240,150,210,199,134,40,162,51)(4,135,225,95,52,200,68,111,163,211,122,188,41,151,32)(5,189,112,96,17,123,69,226,152,212,201,136,42,164,53)(6,137,227,81,54,202,70,97,165,213,124,190,43,153,18)(7,191,98,82,19,125,71,228,154,214,203,138,44,166,55)(8,139,229,83,56,204,72,99,167,215,126,192,45,155,20)(9,177,100,84,21,127,73,230,156,216,205,140,46,168,57)(10,141,231,85,58,206,74,101,169,217,128,178,47,157,22)(11,179,102,86,23,113,75,232,158,218,207,142,48,170,59)(12,143,233,87,60,208,76,103,171,219,114,180,33,159,24)(13,181,104,88,25,115,77,234,160,220,193,144,34,172,61)(14,129,235,89,62,194,78,105,173,221,116,182,35,145,26)(15,183,106,90,27,117,79,236,146,222,195,130,36,174,63)(16,131,237,91,64,196,80,107,175,223,118,184,37,147,28), (1,57)(2,22)(3,59)(4,24)(5,61)(6,26)(7,63)(8,28)(9,49)(10,30)(11,51)(12,32)(13,53)(14,18)(15,55)(16,20)(17,193)(19,195)(21,197)(23,199)(25,201)(27,203)(29,205)(31,207)(33,225)(34,112)(35,227)(36,98)(37,229)(38,100)(39,231)(40,102)(41,233)(42,104)(43,235)(44,106)(45,237)(46,108)(47,239)(48,110)(50,128)(52,114)(54,116)(56,118)(58,120)(60,122)(62,124)(64,126)(65,156)(66,169)(67,158)(68,171)(69,160)(70,173)(71,146)(72,175)(73,148)(74,161)(75,150)(76,163)(77,152)(78,165)(79,154)(80,167)(81,182)(82,130)(83,184)(84,132)(85,186)(86,134)(87,188)(88,136)(89,190)(90,138)(91,192)(92,140)(93,178)(94,142)(95,180)(96,144)(97,105)(99,107)(101,109)(103,111)(113,210)(115,212)(117,214)(119,216)(121,218)(123,220)(125,222)(127,224)(129,153)(131,155)(133,157)(135,159)(137,145)(139,147)(141,149)(143,151)(162,179)(164,181)(166,183)(168,185)(170,187)(172,189)(174,191)(176,177)(194,213)(196,215)(198,217)(200,219)(202,221)(204,223)(206,209)(208,211)(226,234)(228,236)(230,238)(232,240), (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)>;
G:=Group( (1,185,108,92,29,119,65,238,148,224,197,132,38,176,49)(2,133,239,93,50,198,66,109,161,209,120,186,39,149,30)(3,187,110,94,31,121,67,240,150,210,199,134,40,162,51)(4,135,225,95,52,200,68,111,163,211,122,188,41,151,32)(5,189,112,96,17,123,69,226,152,212,201,136,42,164,53)(6,137,227,81,54,202,70,97,165,213,124,190,43,153,18)(7,191,98,82,19,125,71,228,154,214,203,138,44,166,55)(8,139,229,83,56,204,72,99,167,215,126,192,45,155,20)(9,177,100,84,21,127,73,230,156,216,205,140,46,168,57)(10,141,231,85,58,206,74,101,169,217,128,178,47,157,22)(11,179,102,86,23,113,75,232,158,218,207,142,48,170,59)(12,143,233,87,60,208,76,103,171,219,114,180,33,159,24)(13,181,104,88,25,115,77,234,160,220,193,144,34,172,61)(14,129,235,89,62,194,78,105,173,221,116,182,35,145,26)(15,183,106,90,27,117,79,236,146,222,195,130,36,174,63)(16,131,237,91,64,196,80,107,175,223,118,184,37,147,28), (1,57)(2,22)(3,59)(4,24)(5,61)(6,26)(7,63)(8,28)(9,49)(10,30)(11,51)(12,32)(13,53)(14,18)(15,55)(16,20)(17,193)(19,195)(21,197)(23,199)(25,201)(27,203)(29,205)(31,207)(33,225)(34,112)(35,227)(36,98)(37,229)(38,100)(39,231)(40,102)(41,233)(42,104)(43,235)(44,106)(45,237)(46,108)(47,239)(48,110)(50,128)(52,114)(54,116)(56,118)(58,120)(60,122)(62,124)(64,126)(65,156)(66,169)(67,158)(68,171)(69,160)(70,173)(71,146)(72,175)(73,148)(74,161)(75,150)(76,163)(77,152)(78,165)(79,154)(80,167)(81,182)(82,130)(83,184)(84,132)(85,186)(86,134)(87,188)(88,136)(89,190)(90,138)(91,192)(92,140)(93,178)(94,142)(95,180)(96,144)(97,105)(99,107)(101,109)(103,111)(113,210)(115,212)(117,214)(119,216)(121,218)(123,220)(125,222)(127,224)(129,153)(131,155)(133,157)(135,159)(137,145)(139,147)(141,149)(143,151)(162,179)(164,181)(166,183)(168,185)(170,187)(172,189)(174,191)(176,177)(194,213)(196,215)(198,217)(200,219)(202,221)(204,223)(206,209)(208,211)(226,234)(228,236)(230,238)(232,240), (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) );
G=PermutationGroup([[(1,185,108,92,29,119,65,238,148,224,197,132,38,176,49),(2,133,239,93,50,198,66,109,161,209,120,186,39,149,30),(3,187,110,94,31,121,67,240,150,210,199,134,40,162,51),(4,135,225,95,52,200,68,111,163,211,122,188,41,151,32),(5,189,112,96,17,123,69,226,152,212,201,136,42,164,53),(6,137,227,81,54,202,70,97,165,213,124,190,43,153,18),(7,191,98,82,19,125,71,228,154,214,203,138,44,166,55),(8,139,229,83,56,204,72,99,167,215,126,192,45,155,20),(9,177,100,84,21,127,73,230,156,216,205,140,46,168,57),(10,141,231,85,58,206,74,101,169,217,128,178,47,157,22),(11,179,102,86,23,113,75,232,158,218,207,142,48,170,59),(12,143,233,87,60,208,76,103,171,219,114,180,33,159,24),(13,181,104,88,25,115,77,234,160,220,193,144,34,172,61),(14,129,235,89,62,194,78,105,173,221,116,182,35,145,26),(15,183,106,90,27,117,79,236,146,222,195,130,36,174,63),(16,131,237,91,64,196,80,107,175,223,118,184,37,147,28)], [(1,57),(2,22),(3,59),(4,24),(5,61),(6,26),(7,63),(8,28),(9,49),(10,30),(11,51),(12,32),(13,53),(14,18),(15,55),(16,20),(17,193),(19,195),(21,197),(23,199),(25,201),(27,203),(29,205),(31,207),(33,225),(34,112),(35,227),(36,98),(37,229),(38,100),(39,231),(40,102),(41,233),(42,104),(43,235),(44,106),(45,237),(46,108),(47,239),(48,110),(50,128),(52,114),(54,116),(56,118),(58,120),(60,122),(62,124),(64,126),(65,156),(66,169),(67,158),(68,171),(69,160),(70,173),(71,146),(72,175),(73,148),(74,161),(75,150),(76,163),(77,152),(78,165),(79,154),(80,167),(81,182),(82,130),(83,184),(84,132),(85,186),(86,134),(87,188),(88,136),(89,190),(90,138),(91,192),(92,140),(93,178),(94,142),(95,180),(96,144),(97,105),(99,107),(101,109),(103,111),(113,210),(115,212),(117,214),(119,216),(121,218),(123,220),(125,222),(127,224),(129,153),(131,155),(133,157),(135,159),(137,145),(139,147),(141,149),(143,151),(162,179),(164,181),(166,183),(168,185),(170,187),(172,189),(174,191),(176,177),(194,213),(196,215),(198,217),(200,219),(202,221),(204,223),(206,209),(208,211),(226,234),(228,236),(230,238),(232,240)], [(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)]])
96 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 | 16A | ··· | 16H | 16I | ··· | 16P | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 40A | ··· | 40H | 48A | ··· | 48H | 60A | 60B | 60C | 60D | 80A | ··· | 80P | 120A | ··· | 120H |
| 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 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | 60 | 60 | 60 | 80 | ··· | 80 | 120 | ··· | 120 |
| 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 | 4 | 4 | 3 | ··· | 3 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | D5 | D6 | D10 | C4×S3 | C4×D5 | S3×C8 | C8×D5 | S3×C16 | D5×C16 | S3×D5 | D30.C2 | D15⋊2C8 | D15⋊2C16 |
| kernel | D15⋊2C16 | C5×C3⋊C16 | C3×C5⋊2C16 | C8×D15 | C15⋊3C8 | C4×D15 | Dic15 | D30 | D15 | C5⋊2C16 | C3⋊C16 | C40 | C24 | C20 | C12 | C10 | C6 | C5 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 2 | 2 | 4 | 8 |
Matrix representation of D15⋊2C16 ►in GL4(𝔽241) generated by
| 1 | 54 | 0 | 0 |
| 174 | 239 | 0 | 0 |
| 0 | 0 | 0 | 52 |
| 0 | 0 | 190 | 189 |
| 240 | 187 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 1 | 1 |
| 130 | 0 | 0 | 0 |
| 207 | 111 | 0 | 0 |
| 0 | 0 | 130 | 0 |
| 0 | 0 | 0 | 130 |
G:=sub<GL(4,GF(241))| [1,174,0,0,54,239,0,0,0,0,0,190,0,0,52,189],[240,0,0,0,187,1,0,0,0,0,240,1,0,0,0,1],[130,207,0,0,0,111,0,0,0,0,130,0,0,0,0,130] >;
D15⋊2C16 in GAP, Magma, Sage, TeX
D_{15}\rtimes_2C_{16} % in TeX
G:=Group("D15:2C16"); // GroupNames label
G:=SmallGroup(480,9);
// by ID
G=gap.SmallGroup(480,9);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,36,58,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^15=b^2=c^16=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b>;
// generators/relations
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