metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D16.D7, C7⋊2SD64, C28.6D8, C14.9D16, C16.5D14, C56.10D4, Dic56⋊3C2, C112.3C22, C7⋊C32⋊2C2, C4.2(D4⋊D7), (C7×D16).1C2, C8.10(C7⋊D4), C2.5(C7⋊D16), SmallGroup(448,77)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D16.D7
G = < a,b,c,d | a16=b2=c7=1, d2=a8, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a13b, dcd-1=c-1 >
Smallest permutation representation of D16.D7
►On 224 points
Generators in S224
(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)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(30 32)(33 34)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(49 52)(50 51)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 80)(81 83)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(97 109)(98 108)(99 107)(100 106)(101 105)(102 104)(110 112)(113 124)(114 123)(115 122)(116 121)(117 120)(118 119)(125 128)(126 127)(129 144)(130 143)(131 142)(132 141)(133 140)(134 139)(135 138)(136 137)(145 155)(146 154)(147 153)(148 152)(149 151)(156 160)(157 159)(161 166)(162 165)(163 164)(167 176)(168 175)(169 174)(170 173)(171 172)(177 183)(178 182)(179 181)(184 192)(185 191)(186 190)(187 189)(194 208)(195 207)(196 206)(197 205)(198 204)(199 203)(200 202)(209 215)(210 214)(211 213)(216 224)(217 223)(218 222)(219 221)
(1 137 119 80 51 42 172)(2 138 120 65 52 43 173)(3 139 121 66 53 44 174)(4 140 122 67 54 45 175)(5 141 123 68 55 46 176)(6 142 124 69 56 47 161)(7 143 125 70 57 48 162)(8 144 126 71 58 33 163)(9 129 127 72 59 34 164)(10 130 128 73 60 35 165)(11 131 113 74 61 36 166)(12 132 114 75 62 37 167)(13 133 115 76 63 38 168)(14 134 116 77 64 39 169)(15 135 117 78 49 40 170)(16 136 118 79 50 41 171)(17 195 160 182 97 214 84)(18 196 145 183 98 215 85)(19 197 146 184 99 216 86)(20 198 147 185 100 217 87)(21 199 148 186 101 218 88)(22 200 149 187 102 219 89)(23 201 150 188 103 220 90)(24 202 151 189 104 221 91)(25 203 152 190 105 222 92)(26 204 153 191 106 223 93)(27 205 154 192 107 224 94)(28 206 155 177 108 209 95)(29 207 156 178 109 210 96)(30 208 157 179 110 211 81)(31 193 158 180 111 212 82)(32 194 159 181 112 213 83)
(1 91 9 83)(2 90 10 82)(3 89 11 81)(4 88 12 96)(5 87 13 95)(6 86 14 94)(7 85 15 93)(8 84 16 92)(17 171 25 163)(18 170 26 162)(19 169 27 161)(20 168 28 176)(21 167 29 175)(22 166 30 174)(23 165 31 173)(24 164 32 172)(33 195 41 203)(34 194 42 202)(35 193 43 201)(36 208 44 200)(37 207 45 199)(38 206 46 198)(39 205 47 197)(40 204 48 196)(49 153 57 145)(50 152 58 160)(51 151 59 159)(52 150 60 158)(53 149 61 157)(54 148 62 156)(55 147 63 155)(56 146 64 154)(65 188 73 180)(66 187 74 179)(67 186 75 178)(68 185 76 177)(69 184 77 192)(70 183 78 191)(71 182 79 190)(72 181 80 189)(97 118 105 126)(98 117 106 125)(99 116 107 124)(100 115 108 123)(101 114 109 122)(102 113 110 121)(103 128 111 120)(104 127 112 119)(129 213 137 221)(130 212 138 220)(131 211 139 219)(132 210 140 218)(133 209 141 217)(134 224 142 216)(135 223 143 215)(136 222 144 214)
G:=sub<Sym(224)| (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(30,32)(33,34)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(49,52)(50,51)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,80)(81,83)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(110,112)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(125,128)(126,127)(129,144)(130,143)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)(145,155)(146,154)(147,153)(148,152)(149,151)(156,160)(157,159)(161,166)(162,165)(163,164)(167,176)(168,175)(169,174)(170,173)(171,172)(177,183)(178,182)(179,181)(184,192)(185,191)(186,190)(187,189)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202)(209,215)(210,214)(211,213)(216,224)(217,223)(218,222)(219,221), (1,137,119,80,51,42,172)(2,138,120,65,52,43,173)(3,139,121,66,53,44,174)(4,140,122,67,54,45,175)(5,141,123,68,55,46,176)(6,142,124,69,56,47,161)(7,143,125,70,57,48,162)(8,144,126,71,58,33,163)(9,129,127,72,59,34,164)(10,130,128,73,60,35,165)(11,131,113,74,61,36,166)(12,132,114,75,62,37,167)(13,133,115,76,63,38,168)(14,134,116,77,64,39,169)(15,135,117,78,49,40,170)(16,136,118,79,50,41,171)(17,195,160,182,97,214,84)(18,196,145,183,98,215,85)(19,197,146,184,99,216,86)(20,198,147,185,100,217,87)(21,199,148,186,101,218,88)(22,200,149,187,102,219,89)(23,201,150,188,103,220,90)(24,202,151,189,104,221,91)(25,203,152,190,105,222,92)(26,204,153,191,106,223,93)(27,205,154,192,107,224,94)(28,206,155,177,108,209,95)(29,207,156,178,109,210,96)(30,208,157,179,110,211,81)(31,193,158,180,111,212,82)(32,194,159,181,112,213,83), (1,91,9,83)(2,90,10,82)(3,89,11,81)(4,88,12,96)(5,87,13,95)(6,86,14,94)(7,85,15,93)(8,84,16,92)(17,171,25,163)(18,170,26,162)(19,169,27,161)(20,168,28,176)(21,167,29,175)(22,166,30,174)(23,165,31,173)(24,164,32,172)(33,195,41,203)(34,194,42,202)(35,193,43,201)(36,208,44,200)(37,207,45,199)(38,206,46,198)(39,205,47,197)(40,204,48,196)(49,153,57,145)(50,152,58,160)(51,151,59,159)(52,150,60,158)(53,149,61,157)(54,148,62,156)(55,147,63,155)(56,146,64,154)(65,188,73,180)(66,187,74,179)(67,186,75,178)(68,185,76,177)(69,184,77,192)(70,183,78,191)(71,182,79,190)(72,181,80,189)(97,118,105,126)(98,117,106,125)(99,116,107,124)(100,115,108,123)(101,114,109,122)(102,113,110,121)(103,128,111,120)(104,127,112,119)(129,213,137,221)(130,212,138,220)(131,211,139,219)(132,210,140,218)(133,209,141,217)(134,224,142,216)(135,223,143,215)(136,222,144,214)>;
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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(30,32)(33,34)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(49,52)(50,51)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,80)(81,83)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(110,112)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(125,128)(126,127)(129,144)(130,143)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)(145,155)(146,154)(147,153)(148,152)(149,151)(156,160)(157,159)(161,166)(162,165)(163,164)(167,176)(168,175)(169,174)(170,173)(171,172)(177,183)(178,182)(179,181)(184,192)(185,191)(186,190)(187,189)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202)(209,215)(210,214)(211,213)(216,224)(217,223)(218,222)(219,221), (1,137,119,80,51,42,172)(2,138,120,65,52,43,173)(3,139,121,66,53,44,174)(4,140,122,67,54,45,175)(5,141,123,68,55,46,176)(6,142,124,69,56,47,161)(7,143,125,70,57,48,162)(8,144,126,71,58,33,163)(9,129,127,72,59,34,164)(10,130,128,73,60,35,165)(11,131,113,74,61,36,166)(12,132,114,75,62,37,167)(13,133,115,76,63,38,168)(14,134,116,77,64,39,169)(15,135,117,78,49,40,170)(16,136,118,79,50,41,171)(17,195,160,182,97,214,84)(18,196,145,183,98,215,85)(19,197,146,184,99,216,86)(20,198,147,185,100,217,87)(21,199,148,186,101,218,88)(22,200,149,187,102,219,89)(23,201,150,188,103,220,90)(24,202,151,189,104,221,91)(25,203,152,190,105,222,92)(26,204,153,191,106,223,93)(27,205,154,192,107,224,94)(28,206,155,177,108,209,95)(29,207,156,178,109,210,96)(30,208,157,179,110,211,81)(31,193,158,180,111,212,82)(32,194,159,181,112,213,83), (1,91,9,83)(2,90,10,82)(3,89,11,81)(4,88,12,96)(5,87,13,95)(6,86,14,94)(7,85,15,93)(8,84,16,92)(17,171,25,163)(18,170,26,162)(19,169,27,161)(20,168,28,176)(21,167,29,175)(22,166,30,174)(23,165,31,173)(24,164,32,172)(33,195,41,203)(34,194,42,202)(35,193,43,201)(36,208,44,200)(37,207,45,199)(38,206,46,198)(39,205,47,197)(40,204,48,196)(49,153,57,145)(50,152,58,160)(51,151,59,159)(52,150,60,158)(53,149,61,157)(54,148,62,156)(55,147,63,155)(56,146,64,154)(65,188,73,180)(66,187,74,179)(67,186,75,178)(68,185,76,177)(69,184,77,192)(70,183,78,191)(71,182,79,190)(72,181,80,189)(97,118,105,126)(98,117,106,125)(99,116,107,124)(100,115,108,123)(101,114,109,122)(102,113,110,121)(103,128,111,120)(104,127,112,119)(129,213,137,221)(130,212,138,220)(131,211,139,219)(132,210,140,218)(133,209,141,217)(134,224,142,216)(135,223,143,215)(136,222,144,214) );
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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(30,32),(33,34),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42),(49,52),(50,51),(53,64),(54,63),(55,62),(56,61),(57,60),(58,59),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(79,80),(81,83),(84,96),(85,95),(86,94),(87,93),(88,92),(89,91),(97,109),(98,108),(99,107),(100,106),(101,105),(102,104),(110,112),(113,124),(114,123),(115,122),(116,121),(117,120),(118,119),(125,128),(126,127),(129,144),(130,143),(131,142),(132,141),(133,140),(134,139),(135,138),(136,137),(145,155),(146,154),(147,153),(148,152),(149,151),(156,160),(157,159),(161,166),(162,165),(163,164),(167,176),(168,175),(169,174),(170,173),(171,172),(177,183),(178,182),(179,181),(184,192),(185,191),(186,190),(187,189),(194,208),(195,207),(196,206),(197,205),(198,204),(199,203),(200,202),(209,215),(210,214),(211,213),(216,224),(217,223),(218,222),(219,221)], [(1,137,119,80,51,42,172),(2,138,120,65,52,43,173),(3,139,121,66,53,44,174),(4,140,122,67,54,45,175),(5,141,123,68,55,46,176),(6,142,124,69,56,47,161),(7,143,125,70,57,48,162),(8,144,126,71,58,33,163),(9,129,127,72,59,34,164),(10,130,128,73,60,35,165),(11,131,113,74,61,36,166),(12,132,114,75,62,37,167),(13,133,115,76,63,38,168),(14,134,116,77,64,39,169),(15,135,117,78,49,40,170),(16,136,118,79,50,41,171),(17,195,160,182,97,214,84),(18,196,145,183,98,215,85),(19,197,146,184,99,216,86),(20,198,147,185,100,217,87),(21,199,148,186,101,218,88),(22,200,149,187,102,219,89),(23,201,150,188,103,220,90),(24,202,151,189,104,221,91),(25,203,152,190,105,222,92),(26,204,153,191,106,223,93),(27,205,154,192,107,224,94),(28,206,155,177,108,209,95),(29,207,156,178,109,210,96),(30,208,157,179,110,211,81),(31,193,158,180,111,212,82),(32,194,159,181,112,213,83)], [(1,91,9,83),(2,90,10,82),(3,89,11,81),(4,88,12,96),(5,87,13,95),(6,86,14,94),(7,85,15,93),(8,84,16,92),(17,171,25,163),(18,170,26,162),(19,169,27,161),(20,168,28,176),(21,167,29,175),(22,166,30,174),(23,165,31,173),(24,164,32,172),(33,195,41,203),(34,194,42,202),(35,193,43,201),(36,208,44,200),(37,207,45,199),(38,206,46,198),(39,205,47,197),(40,204,48,196),(49,153,57,145),(50,152,58,160),(51,151,59,159),(52,150,60,158),(53,149,61,157),(54,148,62,156),(55,147,63,155),(56,146,64,154),(65,188,73,180),(66,187,74,179),(67,186,75,178),(68,185,76,177),(69,184,77,192),(70,183,78,191),(71,182,79,190),(72,181,80,189),(97,118,105,126),(98,117,106,125),(99,116,107,124),(100,115,108,123),(101,114,109,122),(102,113,110,121),(103,128,111,120),(104,127,112,119),(129,213,137,221),(130,212,138,220),(131,211,139,219),(132,210,140,218),(133,209,141,217),(134,224,142,216),(135,223,143,215),(136,222,144,214)]])
52 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | ··· | 14I | 16A | 16B | 16C | 16D | 28A | 28B | 28C | 32A | ··· | 32H | 56A | ··· | 56F | 112A | ··· | 112L |
| order | 1 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | 28 | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 16 | 2 | 112 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | ··· | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | ··· | 14 | 4 | ··· | 4 | 4 | ··· | 4 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | D4 | D7 | D8 | D14 | D16 | C7⋊D4 | SD64 | D4⋊D7 | C7⋊D16 | D16.D7 |
| kernel | D16.D7 | C7⋊C32 | Dic56 | C7×D16 | C56 | D16 | C28 | C16 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 8 | 3 | 6 | 12 |
Matrix representation of D16.D7 ►in GL4(𝔽449) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 123 |
| 0 | 0 | 270 | 333 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 333 | 424 |
| 0 | 0 | 179 | 116 |
| 0 | 1 | 0 | 0 |
| 448 | 51 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 412 | 19 | 0 | 0 |
| 377 | 37 | 0 | 0 |
| 0 | 0 | 373 | 277 |
| 0 | 0 | 386 | 76 |
G:=sub<GL(4,GF(449))| [1,0,0,0,0,1,0,0,0,0,4,270,0,0,123,333],[1,0,0,0,0,1,0,0,0,0,333,179,0,0,424,116],[0,448,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[412,377,0,0,19,37,0,0,0,0,373,386,0,0,277,76] >;
D16.D7 in GAP, Magma, Sage, TeX
D_{16}.D_7 % in TeX
G:=Group("D16.D7"); // GroupNames label
G:=SmallGroup(448,77);
// by ID
G=gap.SmallGroup(448,77);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,85,254,135,142,675,346,192,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^7=1,d^2=a^8,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^13*b,d*c*d^-1=c^-1>;
// generators/relations
Export