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## G = C2×C16⋊D5order 320 = 26·5

### Direct product of C2 and C16⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C2×C16⋊D5
 Chief series C1 — C5 — C10 — C20 — C40 — D40 — C2×D40 — C2×C16⋊D5
 Lower central C5 — C10 — C20 — C40 — C2×C16⋊D5
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C2×C16⋊D5
G = < a,b,c,d | a2=b16=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b7, dcd=c-1 >

Subgroups: 574 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C16, C2×C8, D8, Q16, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C2×C16, SD32, C2×D8, C2×Q16, C40, Dic10, D20, C2×Dic5, C2×C20, C22×D5, C2×SD32, C80, D40, D40, Dic20, Dic20, C2×C40, C2×Dic10, C2×D20, C16⋊D5, C2×C80, C2×D40, C2×Dic20, C2×C16⋊D5
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, SD32, C2×D8, D20, C22×D5, C2×SD32, D40, C2×D20, C16⋊D5, C2×D40, C2×C16⋊D5

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

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

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

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

86 conjugacy classes

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

86 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 SD32 D20 D20 D40 D40 C16⋊D5 kernel C2×C16⋊D5 C16⋊D5 C2×C80 C2×D40 C2×Dic20 C40 C2×C20 C2×C16 C20 C2×C10 C16 C2×C8 C10 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 2 2 2 4 2 8 4 4 8 8 32

Matrix representation of C2×C16⋊D5 in GL3(𝔽241) generated by

 240 0 0 0 240 0 0 0 240
,
 240 0 0 0 220 130 0 234 227
,
 1 0 0 0 190 240 0 191 240
,
 1 0 0 0 0 189 0 190 0
G:=sub<GL(3,GF(241))| [240,0,0,0,240,0,0,0,240],[240,0,0,0,220,234,0,130,227],[1,0,0,0,190,191,0,240,240],[1,0,0,0,0,190,0,189,0] >;

C2×C16⋊D5 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC16:D5");
// GroupNames label

G:=SmallGroup(320,530);
// by ID

G=gap.SmallGroup(320,530);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,142,1571,80,1684,102,12550]);
// 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,d*b*d=b^7,d*c*d=c^-1>;
// generators/relations

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