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## G = C2×D5⋊C16order 320 = 26·5

### Direct product of C2 and D5⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×D5⋊C16
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C5⋊C16 — C2×C5⋊C16 — C2×D5⋊C16
 Lower central C5 — C2×D5⋊C16
 Upper central C1 — C2×C8

Generators and relations for C2×D5⋊C16
G = < a,b,c,d | a2=b5=c2=d16=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >

Subgroups: 250 in 98 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C16, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C2×C16, C22×C8, C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C16, C5⋊C16, C8×D5, C2×C52C8, C2×C40, C2×C4×D5, D5⋊C16, C2×C5⋊C16, D5×C2×C8, C2×D5⋊C16
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C16, C2×C8, C22×C4, F5, C2×C16, C22×C8, C2×F5, C22×C16, D5⋊C8, C22×F5, D5⋊C16, C2×D5⋊C8, C2×D5⋊C16

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

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C8 C8 C8 C16 F5 C2×F5 C2×F5 D5⋊C8 D5⋊C8 D5⋊C16 kernel C2×D5⋊C16 D5⋊C16 C2×C5⋊C16 D5×C2×C8 C8×D5 C2×C40 C2×C4×D5 C4×D5 C2×Dic5 C22×D5 D10 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 2 1 4 2 2 8 4 4 32 1 2 1 2 2 8

Matrix representation of C2×D5⋊C16 in GL5(𝔽241)

 240 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 240
,
 1 0 0 0 0 0 240 240 240 240 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 240 0 0 0 0 0 240 240 240 240 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0
,
 240 0 0 0 0 0 154 0 76 76 0 76 76 0 154 0 165 78 165 0 0 87 163 163 87

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240],[1,0,0,0,0,0,240,1,0,0,0,240,0,1,0,0,240,0,0,1,0,240,0,0,0],[240,0,0,0,0,0,240,0,0,0,0,240,0,0,1,0,240,0,1,0,0,240,1,0,0],[240,0,0,0,0,0,154,76,165,87,0,0,76,78,163,0,76,0,165,163,0,76,154,0,87] >;

C2×D5⋊C16 in GAP, Magma, Sage, TeX

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

G:=Group("C2xD5:C16");
// GroupNames label

G:=SmallGroup(320,1051);
// by ID

G=gap.SmallGroup(320,1051);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,100,80,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^2=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations

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