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G = D16.D5order 320 = 26·5

The non-split extension by D16 of D5 acting via D5/C5=C2

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), (C5xD16).1C2, C8.10(C5:D4), C2.5(C5:D16), SmallGroup(320,78)

Series: Derived Chief Lower central Upper central

C1C80 — D16.D5
C1C5C10C20C40C80Dic40 — D16.D5
C5C10C20C40C80 — D16.D5
C1C2C4C8C16D16

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 >

Subgroups: 174 in 32 conjugacy classes, 15 normal (all characteristic)
Quotients: C1, C2, C22, D4, D5, D8, D10, D16, C5:D4, SD64, D4:D5, C5:D16, D16.D5
16C2
8C22
40C4
16C10
4D4
20Q8
8Dic5
8C2xC10
2D8
10Q16
4Dic10
4C5xD4
5C32
5Q32
2Dic20
2C5xD8
5SD64

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 2A2B4A4B5A5B8A8B10A10B10C10D10E10F16A16B16C16D20A20B32A···32H40A40B40C40D80A···80H
order12244558810101010101016161616202032···324040404080···80
size11162802222221616161622224410···1044444···4

41 irreducible representations

dim11112222222444
type+++++++++++-
imageC1C2C2C2D4D5D8D10D16C5:D4SD64D4:D5C5:D16D16.D5
kernelD16.D5C5:2C32Dic40C5xD16C40D16C20C16C10C8C5C4C2C1
# reps11111222448248

Matrix representation of D16.D5 in GL4(F641) generated by

1246600
1751200
0010
0001
,
1246600
46662900
0010
0001
,
1000
0100
002781
006400
,
8216700
16755900
008347
0046633
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

Export

Subgroup lattice of D16.D5 in TeX

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