metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D16.D5, C5⋊2SD64, C20.6D8, C40.10D4, C10.9D16, C16.5D10, Dic40⋊3C2, C80.3C22, C5⋊2C32⋊2C2, C4.2(D4⋊D5), (C5×D16).1C2, C8.10(C5⋊D4), C2.5(C5⋊D16), SmallGroup(320,78)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D16.D5
G = < a,b,c,d | a16=b2=c5=1, d2=a8, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a13b, dcd-1=c-1 >
Smallest permutation representation of D16.D5
►On 160 points
Generators in S160
(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)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 22)(18 21)(19 20)(23 32)(24 31)(25 30)(26 29)(27 28)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 64)(62 63)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 96)(94 95)(97 105)(98 104)(99 103)(100 102)(106 112)(107 111)(108 110)(113 117)(114 116)(118 128)(119 127)(120 126)(121 125)(122 124)(129 135)(130 134)(131 133)(136 144)(137 143)(138 142)(139 141)(145 146)(147 160)(148 159)(149 158)(150 157)(151 156)(152 155)(153 154)
(1 154 28 63 95)(2 155 29 64 96)(3 156 30 49 81)(4 157 31 50 82)(5 158 32 51 83)(6 159 17 52 84)(7 160 18 53 85)(8 145 19 54 86)(9 146 20 55 87)(10 147 21 56 88)(11 148 22 57 89)(12 149 23 58 90)(13 150 24 59 91)(14 151 25 60 92)(15 152 26 61 93)(16 153 27 62 94)(33 65 141 110 124)(34 66 142 111 125)(35 67 143 112 126)(36 68 144 97 127)(37 69 129 98 128)(38 70 130 99 113)(39 71 131 100 114)(40 72 132 101 115)(41 73 133 102 116)(42 74 134 103 117)(43 75 135 104 118)(44 76 136 105 119)(45 77 137 106 120)(46 78 138 107 121)(47 79 139 108 122)(48 80 140 109 123)
(1 102 9 110)(2 101 10 109)(3 100 11 108)(4 99 12 107)(5 98 13 106)(6 97 14 105)(7 112 15 104)(8 111 16 103)(17 68 25 76)(18 67 26 75)(19 66 27 74)(20 65 28 73)(21 80 29 72)(22 79 30 71)(23 78 31 70)(24 77 32 69)(33 63 41 55)(34 62 42 54)(35 61 43 53)(36 60 44 52)(37 59 45 51)(38 58 46 50)(39 57 47 49)(40 56 48 64)(81 114 89 122)(82 113 90 121)(83 128 91 120)(84 127 92 119)(85 126 93 118)(86 125 94 117)(87 124 95 116)(88 123 96 115)(129 150 137 158)(130 149 138 157)(131 148 139 156)(132 147 140 155)(133 146 141 154)(134 145 142 153)(135 160 143 152)(136 159 144 151)
G:=sub<Sym(160)| (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,22)(18,21)(19,20)(23,32)(24,31)(25,30)(26,29)(27,28)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,64)(62,63)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,96)(94,95)(97,105)(98,104)(99,103)(100,102)(106,112)(107,111)(108,110)(113,117)(114,116)(118,128)(119,127)(120,126)(121,125)(122,124)(129,135)(130,134)(131,133)(136,144)(137,143)(138,142)(139,141)(145,146)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154), (1,154,28,63,95)(2,155,29,64,96)(3,156,30,49,81)(4,157,31,50,82)(5,158,32,51,83)(6,159,17,52,84)(7,160,18,53,85)(8,145,19,54,86)(9,146,20,55,87)(10,147,21,56,88)(11,148,22,57,89)(12,149,23,58,90)(13,150,24,59,91)(14,151,25,60,92)(15,152,26,61,93)(16,153,27,62,94)(33,65,141,110,124)(34,66,142,111,125)(35,67,143,112,126)(36,68,144,97,127)(37,69,129,98,128)(38,70,130,99,113)(39,71,131,100,114)(40,72,132,101,115)(41,73,133,102,116)(42,74,134,103,117)(43,75,135,104,118)(44,76,136,105,119)(45,77,137,106,120)(46,78,138,107,121)(47,79,139,108,122)(48,80,140,109,123), (1,102,9,110)(2,101,10,109)(3,100,11,108)(4,99,12,107)(5,98,13,106)(6,97,14,105)(7,112,15,104)(8,111,16,103)(17,68,25,76)(18,67,26,75)(19,66,27,74)(20,65,28,73)(21,80,29,72)(22,79,30,71)(23,78,31,70)(24,77,32,69)(33,63,41,55)(34,62,42,54)(35,61,43,53)(36,60,44,52)(37,59,45,51)(38,58,46,50)(39,57,47,49)(40,56,48,64)(81,114,89,122)(82,113,90,121)(83,128,91,120)(84,127,92,119)(85,126,93,118)(86,125,94,117)(87,124,95,116)(88,123,96,115)(129,150,137,158)(130,149,138,157)(131,148,139,156)(132,147,140,155)(133,146,141,154)(134,145,142,153)(135,160,143,152)(136,159,144,151)>;
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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,22)(18,21)(19,20)(23,32)(24,31)(25,30)(26,29)(27,28)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,64)(62,63)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,96)(94,95)(97,105)(98,104)(99,103)(100,102)(106,112)(107,111)(108,110)(113,117)(114,116)(118,128)(119,127)(120,126)(121,125)(122,124)(129,135)(130,134)(131,133)(136,144)(137,143)(138,142)(139,141)(145,146)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154), (1,154,28,63,95)(2,155,29,64,96)(3,156,30,49,81)(4,157,31,50,82)(5,158,32,51,83)(6,159,17,52,84)(7,160,18,53,85)(8,145,19,54,86)(9,146,20,55,87)(10,147,21,56,88)(11,148,22,57,89)(12,149,23,58,90)(13,150,24,59,91)(14,151,25,60,92)(15,152,26,61,93)(16,153,27,62,94)(33,65,141,110,124)(34,66,142,111,125)(35,67,143,112,126)(36,68,144,97,127)(37,69,129,98,128)(38,70,130,99,113)(39,71,131,100,114)(40,72,132,101,115)(41,73,133,102,116)(42,74,134,103,117)(43,75,135,104,118)(44,76,136,105,119)(45,77,137,106,120)(46,78,138,107,121)(47,79,139,108,122)(48,80,140,109,123), (1,102,9,110)(2,101,10,109)(3,100,11,108)(4,99,12,107)(5,98,13,106)(6,97,14,105)(7,112,15,104)(8,111,16,103)(17,68,25,76)(18,67,26,75)(19,66,27,74)(20,65,28,73)(21,80,29,72)(22,79,30,71)(23,78,31,70)(24,77,32,69)(33,63,41,55)(34,62,42,54)(35,61,43,53)(36,60,44,52)(37,59,45,51)(38,58,46,50)(39,57,47,49)(40,56,48,64)(81,114,89,122)(82,113,90,121)(83,128,91,120)(84,127,92,119)(85,126,93,118)(86,125,94,117)(87,124,95,116)(88,123,96,115)(129,150,137,158)(130,149,138,157)(131,148,139,156)(132,147,140,155)(133,146,141,154)(134,145,142,153)(135,160,143,152)(136,159,144,151) );
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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,22),(18,21),(19,20),(23,32),(24,31),(25,30),(26,29),(27,28),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,64),(62,63),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(93,96),(94,95),(97,105),(98,104),(99,103),(100,102),(106,112),(107,111),(108,110),(113,117),(114,116),(118,128),(119,127),(120,126),(121,125),(122,124),(129,135),(130,134),(131,133),(136,144),(137,143),(138,142),(139,141),(145,146),(147,160),(148,159),(149,158),(150,157),(151,156),(152,155),(153,154)], [(1,154,28,63,95),(2,155,29,64,96),(3,156,30,49,81),(4,157,31,50,82),(5,158,32,51,83),(6,159,17,52,84),(7,160,18,53,85),(8,145,19,54,86),(9,146,20,55,87),(10,147,21,56,88),(11,148,22,57,89),(12,149,23,58,90),(13,150,24,59,91),(14,151,25,60,92),(15,152,26,61,93),(16,153,27,62,94),(33,65,141,110,124),(34,66,142,111,125),(35,67,143,112,126),(36,68,144,97,127),(37,69,129,98,128),(38,70,130,99,113),(39,71,131,100,114),(40,72,132,101,115),(41,73,133,102,116),(42,74,134,103,117),(43,75,135,104,118),(44,76,136,105,119),(45,77,137,106,120),(46,78,138,107,121),(47,79,139,108,122),(48,80,140,109,123)], [(1,102,9,110),(2,101,10,109),(3,100,11,108),(4,99,12,107),(5,98,13,106),(6,97,14,105),(7,112,15,104),(8,111,16,103),(17,68,25,76),(18,67,26,75),(19,66,27,74),(20,65,28,73),(21,80,29,72),(22,79,30,71),(23,78,31,70),(24,77,32,69),(33,63,41,55),(34,62,42,54),(35,61,43,53),(36,60,44,52),(37,59,45,51),(38,58,46,50),(39,57,47,49),(40,56,48,64),(81,114,89,122),(82,113,90,121),(83,128,91,120),(84,127,92,119),(85,126,93,118),(86,125,94,117),(87,124,95,116),(88,123,96,115),(129,150,137,158),(130,149,138,157),(131,148,139,156),(132,147,140,155),(133,146,141,154),(134,145,142,153),(135,160,143,152),(136,159,144,151)]])
41 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 16A | 16B | 16C | 16D | 20A | 20B | 32A | ··· | 32H | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 32 | ··· | 32 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 16 | 2 | 80 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | 16 | 16 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | D4 | D5 | D8 | D10 | D16 | C5⋊D4 | SD64 | D4⋊D5 | C5⋊D16 | D16.D5 |
| kernel | D16.D5 | C5⋊2C32 | Dic40 | C5×D16 | C40 | D16 | C20 | C16 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 4 | 8 |
Matrix representation of D16.D5 ►in GL4(𝔽641) generated by
| 12 | 466 | 0 | 0 |
| 175 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 466 | 0 | 0 |
| 466 | 629 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 278 | 1 |
| 0 | 0 | 640 | 0 |
| 82 | 167 | 0 | 0 |
| 167 | 559 | 0 | 0 |
| 0 | 0 | 8 | 347 |
| 0 | 0 | 46 | 633 |
G:=sub<GL(4,GF(641))| [12,175,0,0,466,12,0,0,0,0,1,0,0,0,0,1],[12,466,0,0,466,629,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,278,640,0,0,1,0],[82,167,0,0,167,559,0,0,0,0,8,46,0,0,347,633] >;
D16.D5 in GAP, Magma, Sage, TeX
D_{16}.D_5 % in TeX
G:=Group("D16.D5"); // GroupNames label
G:=SmallGroup(320,78);
// by ID
G=gap.SmallGroup(320,78);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,85,254,135,142,675,346,192,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^5=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
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