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G = D5×C2×C16order 320 = 26·5

Direct product of C2×C16 and D5

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

Aliases: D5×C2×C16, C8011C22, C40.63C23, (C2×C80)⋊14C2, C103(C2×C16), C53(C22×C16), C8.43(C4×D5), C4.23(C8×D5), C20.62(C2×C8), (C4×D5).10C8, (C8×D5).14C4, C40.101(C2×C4), D10.21(C2×C8), (C2×C8).341D10, C22.13(C8×D5), C8.57(C22×D5), C52C1613C22, C10.36(C22×C8), (C2×Dic5).16C8, Dic5.22(C2×C8), (C22×D5).10C8, (C8×D5).67C22, (C2×C40).408C22, C20.187(C22×C4), C2.2(D5×C2×C8), (D5×C2×C8).36C2, (C2×C4×D5).49C4, C4.102(C2×C4×D5), (C2×C52C16)⋊17C2, (C2×C52C8).39C4, (C2×C10).42(C2×C8), C52C8.57(C2×C4), (C2×C4).175(C4×D5), (C2×C20).421(C2×C4), (C4×D5).100(C2×C4), SmallGroup(320,526)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C2×C16
C1C5C10C20C40C8×D5D5×C2×C8 — D5×C2×C16
C5 — D5×C2×C16
C1C2×C16

Generators and relations for D5×C2×C16
 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 [×2], C2 [×4], C4 [×2], C4 [×2], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×4], C10, C10 [×2], C16 [×2], C16 [×2], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], D10 [×6], C2×C10, C2×C16, C2×C16 [×5], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C22×C16, C52C16 [×2], C80 [×2], C8×D5 [×4], C2×C52C8, C2×C40, C2×C4×D5, D5×C16 [×4], C2×C52C16, C2×C80, D5×C2×C8, D5×C2×C16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, D5, C16 [×4], C2×C8 [×6], C22×C4, D10 [×3], C2×C16 [×6], C22×C8, C4×D5 [×2], C22×D5, C22×C16, C8×D5 [×2], C2×C4×D5, D5×C16 [×2], D5×C2×C8, D5×C2×C16

Smallest permutation representation of D5×C2×C16
On 160 points
Generators in S160
(1 107)(2 108)(3 109)(4 110)(5 111)(6 112)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 49)(81 130)(82 131)(83 132)(84 133)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 141)(93 142)(94 143)(95 144)(96 129)(113 154)(114 155)(115 156)(116 157)(117 158)(118 159)(119 160)(120 145)(121 146)(122 147)(123 148)(124 149)(125 150)(126 151)(127 152)(128 153)
(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 94 72 56 124)(2 95 73 57 125)(3 96 74 58 126)(4 81 75 59 127)(5 82 76 60 128)(6 83 77 61 113)(7 84 78 62 114)(8 85 79 63 115)(9 86 80 64 116)(10 87 65 49 117)(11 88 66 50 118)(12 89 67 51 119)(13 90 68 52 120)(14 91 69 53 121)(15 92 70 54 122)(16 93 71 55 123)(17 38 148 106 142)(18 39 149 107 143)(19 40 150 108 144)(20 41 151 109 129)(21 42 152 110 130)(22 43 153 111 131)(23 44 154 112 132)(24 45 155 97 133)(25 46 156 98 134)(26 47 157 99 135)(27 48 158 100 136)(28 33 159 101 137)(29 34 160 102 138)(30 35 145 103 139)(31 36 146 104 140)(32 37 147 105 141)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 113)(15 114)(16 115)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 95)(50 96)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(97 147)(98 148)(99 149)(100 150)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)(107 157)(108 158)(109 159)(110 160)(111 145)(112 146)

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

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

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

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++++++++
imageC1C2C2C2C2C4C4C4C8C8C8C16D5D10D10C4×D5C4×D5C8×D5C8×D5D5×C16
kernelD5×C2×C16D5×C16C2×C52C16C2×C80D5×C2×C8C8×D5C2×C52C8C2×C4×D5C4×D5C2×Dic5C22×D5D10C2×C16C16C2×C8C8C2×C4C4C22C2
# reps1411142284432242448832

Matrix representation of D5×C2×C16 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] >;

D5×C2×C16 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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