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G = D5xC2xC16order 320 = 26·5

Direct product of C2xC16 and D5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D5xC2xC16, C80:11C22, C40.63C23, (C2xC80):14C2, C10:3(C2xC16), C5:3(C22xC16), C8.43(C4xD5), C4.23(C8xD5), C20.62(C2xC8), (C4xD5).10C8, (C8xD5).14C4, C40.101(C2xC4), D10.21(C2xC8), (C2xC8).341D10, C22.13(C8xD5), C8.57(C22xD5), C5:2C16:13C22, C10.36(C22xC8), (C2xDic5).16C8, Dic5.22(C2xC8), (C22xD5).10C8, (C8xD5).67C22, (C2xC40).408C22, C20.187(C22xC4), C2.2(D5xC2xC8), (D5xC2xC8).36C2, (C2xC4xD5).49C4, C4.102(C2xC4xD5), (C2xC5:2C16):17C2, (C2xC5:2C8).39C4, (C2xC10).42(C2xC8), C5:2C8.57(C2xC4), (C2xC4).175(C4xD5), (C2xC20).421(C2xC4), (C4xD5).100(C2xC4), SmallGroup(320,526)

Series: Derived Chief Lower central Upper central

C1C5 — D5xC2xC16
C1C5C10C20C40C8xD5D5xC2xC8 — D5xC2xC16
C5 — D5xC2xC16
C1C2xC16

Generators and relations for D5xC2xC16
 G = < a,b,c,d | a2=b16=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 238 in 98 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, C23, D5, C10, C10, C16, C16, C2xC8, C2xC8, C22xC4, Dic5, C20, D10, C2xC10, C2xC16, C2xC16, C22xC8, C5:2C8, C40, C4xD5, C2xDic5, C2xC20, C22xD5, C22xC16, C5:2C16, C80, C8xD5, C2xC5:2C8, C2xC40, C2xC4xD5, D5xC16, C2xC5:2C16, C2xC80, D5xC2xC8, D5xC2xC16
Quotients: C1, C2, C4, C22, C8, C2xC4, C23, D5, C16, C2xC8, C22xC4, D10, C2xC16, C22xC8, C4xD5, C22xD5, C22xC16, C8xD5, C2xC4xD5, D5xC16, D5xC2xC8, D5xC2xC16

Smallest permutation representation of D5xC2xC16
On 160 points
Generators in S160
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 81)(14 82)(15 83)(16 84)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 148)(34 149)(35 150)(36 151)(37 152)(38 153)(39 154)(40 155)(41 156)(42 157)(43 158)(44 159)(45 160)(46 145)(47 146)(48 147)(49 139)(50 140)(51 141)(52 142)(53 143)(54 144)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 135)(62 136)(63 137)(64 138)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 111)(79 112)(80 97)
(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 54 26 152 75)(2 55 27 153 76)(3 56 28 154 77)(4 57 29 155 78)(5 58 30 156 79)(6 59 31 157 80)(7 60 32 158 65)(8 61 17 159 66)(9 62 18 160 67)(10 63 19 145 68)(11 64 20 146 69)(12 49 21 147 70)(13 50 22 148 71)(14 51 23 149 72)(15 52 24 150 73)(16 53 25 151 74)(33 104 81 140 118)(34 105 82 141 119)(35 106 83 142 120)(36 107 84 143 121)(37 108 85 144 122)(38 109 86 129 123)(39 110 87 130 124)(40 111 88 131 125)(41 112 89 132 126)(42 97 90 133 127)(43 98 91 134 128)(44 99 92 135 113)(45 100 93 136 114)(46 101 94 137 115)(47 102 95 138 116)(48 103 96 139 117)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 65)(16 66)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 132)(34 133)(35 134)(36 135)(37 136)(38 137)(39 138)(40 139)(41 140)(42 141)(43 142)(44 143)(45 144)(46 129)(47 130)(48 131)(49 155)(50 156)(51 157)(52 158)(53 159)(54 160)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 151)(62 152)(63 153)(64 154)(81 112)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)

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

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

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

128 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H5A5B8A···8H8I···8P10A···10F16A···16P16Q···16AF20A···20H40A···40P80A···80AF
order1222222244444444558···88···810···1016···1616···1620···2040···4080···80
size1111555511115555221···15···52···21···15···52···22···22···2

128 irreducible representations

dim11111111111122222222
type++++++++
imageC1C2C2C2C2C4C4C4C8C8C8C16D5D10D10C4xD5C4xD5C8xD5C8xD5D5xC16
kernelD5xC2xC16D5xC16C2xC5:2C16C2xC80D5xC2xC8C8xD5C2xC5:2C8C2xC4xD5C4xD5C2xDic5C22xD5D10C2xC16C16C2xC8C8C2xC4C4C22C2
# reps1411142284432242448832

Matrix representation of D5xC2xC16 in GL3(F241) generated by

24000
02400
00240
,
13000
080
008
,
100
01891
02400
,
100
01189
00240
G:=sub<GL(3,GF(241))| [240,0,0,0,240,0,0,0,240],[130,0,0,0,8,0,0,0,8],[1,0,0,0,189,240,0,1,0],[1,0,0,0,1,0,0,189,240] >;

D5xC2xC16 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_{16}
% in TeX

G:=Group("D5xC2xC16");
// GroupNames label

G:=SmallGroup(320,526);
// by ID

G=gap.SmallGroup(320,526);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,58,80,102,12550]);
// Polycyclic

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

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