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## G = C10×D16order 320 = 26·5

### Direct product of C10 and D16

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C10×D16
 Chief series C1 — C2 — C4 — C8 — C40 — C5×D8 — C5×D16 — C10×D16
 Lower central C1 — C2 — C4 — C8 — C10×D16
 Upper central C1 — C2×C10 — C2×C20 — C2×C40 — C10×D16

Generators and relations for C10×D16
G = < a,b,c | a10=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 274 in 98 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, D4, C23, C10, C10, C10, C16, C2×C8, D8, D8, C2×D4, C20, C2×C10, C2×C10, C2×C16, D16, C2×D8, C40, C2×C20, C5×D4, C22×C10, C2×D16, C80, C2×C40, C5×D8, C5×D8, D4×C10, C2×C80, C5×D16, C10×D8, C10×D16
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C2×C10, D16, C2×D8, C5×D4, C22×C10, C2×D16, C5×D8, D4×C10, C5×D16, C10×D8, C10×D16

Smallest permutation representation of C10×D16
On 160 points
Generators in S160
(1 139 147 98 76 126 59 32 35 82)(2 140 148 99 77 127 60 17 36 83)(3 141 149 100 78 128 61 18 37 84)(4 142 150 101 79 113 62 19 38 85)(5 143 151 102 80 114 63 20 39 86)(6 144 152 103 65 115 64 21 40 87)(7 129 153 104 66 116 49 22 41 88)(8 130 154 105 67 117 50 23 42 89)(9 131 155 106 68 118 51 24 43 90)(10 132 156 107 69 119 52 25 44 91)(11 133 157 108 70 120 53 26 45 92)(12 134 158 109 71 121 54 27 46 93)(13 135 159 110 72 122 55 28 47 94)(14 136 160 111 73 123 56 29 48 95)(15 137 145 112 74 124 57 30 33 96)(16 138 146 97 75 125 58 31 34 81)
(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)
(1 117)(2 116)(3 115)(4 114)(5 113)(6 128)(7 127)(8 126)(9 125)(10 124)(11 123)(12 122)(13 121)(14 120)(15 119)(16 118)(17 153)(18 152)(19 151)(20 150)(21 149)(22 148)(23 147)(24 146)(25 145)(26 160)(27 159)(28 158)(29 157)(30 156)(31 155)(32 154)(33 107)(34 106)(35 105)(36 104)(37 103)(38 102)(39 101)(40 100)(41 99)(42 98)(43 97)(44 112)(45 111)(46 110)(47 109)(48 108)(49 140)(50 139)(51 138)(52 137)(53 136)(54 135)(55 134)(56 133)(57 132)(58 131)(59 130)(60 129)(61 144)(62 143)(63 142)(64 141)(65 84)(66 83)(67 82)(68 81)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10AB 16A ··· 16H 20A ··· 20H 40A ··· 40P 80A ··· 80AF order 1 2 2 2 2 2 2 2 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 16 ··· 16 20 ··· 20 40 ··· 40 80 ··· 80 size 1 1 1 1 8 8 8 8 2 2 1 1 1 1 2 2 2 2 1 ··· 1 8 ··· 8 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 D4 D8 D8 D16 C5×D4 C5×D4 C5×D8 C5×D8 C5×D16 kernel C10×D16 C2×C80 C5×D16 C10×D8 C2×D16 C2×C16 D16 C2×D8 C40 C2×C20 C20 C2×C10 C10 C8 C2×C4 C4 C22 C2 # reps 1 1 4 2 4 4 16 8 1 1 2 2 8 4 4 8 8 32

Matrix representation of C10×D16 in GL3(𝔽241) generated by

 240 0 0 0 91 0 0 0 91
,
 1 0 0 0 156 27 0 214 156
,
 1 0 0 0 85 214 0 214 156
G:=sub<GL(3,GF(241))| [240,0,0,0,91,0,0,0,91],[1,0,0,0,156,214,0,27,156],[1,0,0,0,85,214,0,214,156] >;

C10×D16 in GAP, Magma, Sage, TeX

C_{10}\times D_{16}
% in TeX

G:=Group("C10xD16");
// GroupNames label

G:=SmallGroup(320,1006);
// by ID

G=gap.SmallGroup(320,1006);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,4204,2111,242,10085,5052,124]);
// Polycyclic

G:=Group<a,b,c|a^10=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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