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G = D163D5order 320 = 26·5

The semidirect product of D16 and D5 acting through Inn(D16)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D163D5, D8.1D10, D10.5D8, C16.8D10, Dic404C2, C80.6C22, C40.15C23, Dic5.24D8, Dic20.2C22, C4.3(D4×D5), (D5×C16)⋊2C2, (C5×D16)⋊3C2, C52(C4○D16), D8.D52C2, C2.18(D5×D8), C20.9(C2×D4), D83D54C2, (C4×D5).58D4, C10.34(C2×D8), C52C8.24D4, (C5×D8).1C22, C8.21(C22×D5), C52C16.5C22, (C8×D5).39C22, SmallGroup(320,539)

Series: Derived Chief Lower central Upper central

C1C40 — D163D5
C1C5C10C20C40C8×D5D83D5 — D163D5
C5C10C20C40 — D163D5
C1C2C4C8D16

Generators and relations for D163D5
 G = < a,b,c,d | a16=b2=c5=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a8b, dcd=c-1 >

Subgroups: 390 in 84 conjugacy classes, 31 normal (21 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C5, C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], D5, C10, C10 [×2], C16, C16, C2×C8, D8 [×2], SD16 [×2], Q16 [×2], C4○D4 [×2], Dic5, Dic5 [×2], C20, D10, C2×C10 [×2], C2×C16, D16, SD32 [×2], Q32, C4○D8 [×2], C52C8, C40, Dic10 [×2], C4×D5, C2×Dic5 [×2], C5⋊D4 [×2], C5×D4 [×2], C4○D16, C52C16, C80, C8×D5, Dic20 [×2], D4.D5 [×2], C5×D8 [×2], D42D5 [×2], D5×C16, Dic40, D8.D5 [×2], C5×D16, D83D5 [×2], D163D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, C22×D5, C4○D16, D4×D5, D5×D8, D163D5

Smallest permutation representation of D163D5
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 66)(2 65)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 72)(12 71)(13 70)(14 69)(15 68)(16 67)(17 35)(18 34)(19 33)(20 48)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 36)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 96)(58 95)(59 94)(60 93)(61 92)(62 91)(63 90)(64 89)(97 127)(98 126)(99 125)(100 124)(101 123)(102 122)(103 121)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(112 128)(129 156)(130 155)(131 154)(132 153)(133 152)(134 151)(135 150)(136 149)(137 148)(138 147)(139 146)(140 145)(141 160)(142 159)(143 158)(144 157)
(1 94 132 42 127)(2 95 133 43 128)(3 96 134 44 113)(4 81 135 45 114)(5 82 136 46 115)(6 83 137 47 116)(7 84 138 48 117)(8 85 139 33 118)(9 86 140 34 119)(10 87 141 35 120)(11 88 142 36 121)(12 89 143 37 122)(13 90 144 38 123)(14 91 129 39 124)(15 92 130 40 125)(16 93 131 41 126)(17 104 73 50 160)(18 105 74 51 145)(19 106 75 52 146)(20 107 76 53 147)(21 108 77 54 148)(22 109 78 55 149)(23 110 79 56 150)(24 111 80 57 151)(25 112 65 58 152)(26 97 66 59 153)(27 98 67 60 154)(28 99 68 61 155)(29 100 69 62 156)(30 101 70 63 157)(31 102 71 64 158)(32 103 72 49 159)
(1 127)(2 128)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 121)(12 122)(13 123)(14 124)(15 125)(16 126)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 81)(46 82)(47 83)(48 84)(65 104)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)(145 153)(146 154)(147 155)(148 156)(149 157)(150 158)(151 159)(152 160)

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), (1,66)(2,65)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,35)(18,34)(19,33)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(97,127)(98,126)(99,125)(100,124)(101,123)(102,122)(103,121)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(112,128)(129,156)(130,155)(131,154)(132,153)(133,152)(134,151)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,160)(142,159)(143,158)(144,157), (1,94,132,42,127)(2,95,133,43,128)(3,96,134,44,113)(4,81,135,45,114)(5,82,136,46,115)(6,83,137,47,116)(7,84,138,48,117)(8,85,139,33,118)(9,86,140,34,119)(10,87,141,35,120)(11,88,142,36,121)(12,89,143,37,122)(13,90,144,38,123)(14,91,129,39,124)(15,92,130,40,125)(16,93,131,41,126)(17,104,73,50,160)(18,105,74,51,145)(19,106,75,52,146)(20,107,76,53,147)(21,108,77,54,148)(22,109,78,55,149)(23,110,79,56,150)(24,111,80,57,151)(25,112,65,58,152)(26,97,66,59,153)(27,98,67,60,154)(28,99,68,61,155)(29,100,69,62,156)(30,101,70,63,157)(31,102,71,64,158)(32,103,72,49,159), (1,127)(2,128)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,121)(12,122)(13,123)(14,124)(15,125)(16,126)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,81)(46,82)(47,83)(48,84)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)>;

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), (1,66)(2,65)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,35)(18,34)(19,33)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(97,127)(98,126)(99,125)(100,124)(101,123)(102,122)(103,121)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(112,128)(129,156)(130,155)(131,154)(132,153)(133,152)(134,151)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,160)(142,159)(143,158)(144,157), (1,94,132,42,127)(2,95,133,43,128)(3,96,134,44,113)(4,81,135,45,114)(5,82,136,46,115)(6,83,137,47,116)(7,84,138,48,117)(8,85,139,33,118)(9,86,140,34,119)(10,87,141,35,120)(11,88,142,36,121)(12,89,143,37,122)(13,90,144,38,123)(14,91,129,39,124)(15,92,130,40,125)(16,93,131,41,126)(17,104,73,50,160)(18,105,74,51,145)(19,106,75,52,146)(20,107,76,53,147)(21,108,77,54,148)(22,109,78,55,149)(23,110,79,56,150)(24,111,80,57,151)(25,112,65,58,152)(26,97,66,59,153)(27,98,67,60,154)(28,99,68,61,155)(29,100,69,62,156)(30,101,70,63,157)(31,102,71,64,158)(32,103,72,49,159), (1,127)(2,128)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,121)(12,122)(13,123)(14,124)(15,125)(16,126)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,81)(46,82)(47,83)(48,84)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160) );

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)], [(1,66),(2,65),(3,80),(4,79),(5,78),(6,77),(7,76),(8,75),(9,74),(10,73),(11,72),(12,71),(13,70),(14,69),(15,68),(16,67),(17,35),(18,34),(19,33),(20,48),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,40),(29,39),(30,38),(31,37),(32,36),(49,88),(50,87),(51,86),(52,85),(53,84),(54,83),(55,82),(56,81),(57,96),(58,95),(59,94),(60,93),(61,92),(62,91),(63,90),(64,89),(97,127),(98,126),(99,125),(100,124),(101,123),(102,122),(103,121),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113),(112,128),(129,156),(130,155),(131,154),(132,153),(133,152),(134,151),(135,150),(136,149),(137,148),(138,147),(139,146),(140,145),(141,160),(142,159),(143,158),(144,157)], [(1,94,132,42,127),(2,95,133,43,128),(3,96,134,44,113),(4,81,135,45,114),(5,82,136,46,115),(6,83,137,47,116),(7,84,138,48,117),(8,85,139,33,118),(9,86,140,34,119),(10,87,141,35,120),(11,88,142,36,121),(12,89,143,37,122),(13,90,144,38,123),(14,91,129,39,124),(15,92,130,40,125),(16,93,131,41,126),(17,104,73,50,160),(18,105,74,51,145),(19,106,75,52,146),(20,107,76,53,147),(21,108,77,54,148),(22,109,78,55,149),(23,110,79,56,150),(24,111,80,57,151),(25,112,65,58,152),(26,97,66,59,153),(27,98,67,60,154),(28,99,68,61,155),(29,100,69,62,156),(30,101,70,63,157),(31,102,71,64,158),(32,103,72,49,159)], [(1,127),(2,128),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,119),(10,120),(11,121),(12,122),(13,123),(14,124),(15,125),(16,126),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,57),(33,85),(34,86),(35,87),(36,88),(37,89),(38,90),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,81),(46,82),(47,83),(48,84),(65,104),(66,105),(67,106),(68,107),(69,108),(70,109),(71,110),(72,111),(73,112),(74,97),(75,98),(76,99),(77,100),(78,101),(79,102),(80,103),(145,153),(146,154),(147,155),(148,156),(149,157),(150,158),(151,159),(152,160)])

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B8A8B8C8D10A10B10C10D10E10F16A16B16C16D16E16F16G16H20A20B40A40B40C40D80A···80H
order1222244444558888101010101010161616161616161620204040404080···80
size11881025540402222101022161616162222101010104444444···4

44 irreducible representations

dim11111122222222444
type+++++++++++++++-
imageC1C2C2C2C2C2D4D4D5D8D8D10D10C4○D16D4×D5D5×D8D163D5
kernelD163D5D5×C16Dic40D8.D5C5×D16D83D5C52C8C4×D5D16Dic5D10C16D8C5C4C2C1
# reps11121211222248248

Matrix representation of D163D5 in GL4(𝔽241) generated by

165000
013000
002400
000240
,
013000
165000
0010
0001
,
1000
0100
002401
0050190
,
1000
024000
002400
00501
G:=sub<GL(4,GF(241))| [165,0,0,0,0,130,0,0,0,0,240,0,0,0,0,240],[0,165,0,0,130,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,240,50,0,0,1,190],[1,0,0,0,0,240,0,0,0,0,240,50,0,0,0,1] >;

D163D5 in GAP, Magma, Sage, TeX

D_{16}\rtimes_3D_5
% in TeX

G:=Group("D16:3D5");
// GroupNames label

G:=SmallGroup(320,539);
// by ID

G=gap.SmallGroup(320,539);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,758,135,346,185,192,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^16=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations

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