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## G = C2×C5⋊D16order 320 = 26·5

### Direct product of C2 and C5⋊D16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C2×C5⋊D16
 Chief series C1 — C5 — C10 — C20 — C40 — D40 — C2×D40 — C2×C5⋊D16
 Lower central C5 — C10 — C20 — C40 — C2×C5⋊D16
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×D8

Generators and relations for C2×C5⋊D16
G = < a,b,c,d | a2=b5=c16=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 542 in 98 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, D4, C23, D5, C10, C10, C10, C16, C2×C8, D8, D8, C2×D4, C20, D10, C2×C10, C2×C10, C2×C16, D16, C2×D8, C2×D8, C40, D20, C2×C20, C5×D4, C22×D5, C22×C10, C2×D16, C52C16, D40, D40, C2×C40, C5×D8, C5×D8, C2×D20, D4×C10, C2×C52C16, C5⋊D16, C2×D40, C10×D8, C2×C5⋊D16
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, D16, C2×D8, C5⋊D4, C22×D5, C2×D16, D4⋊D5, C2×C5⋊D4, C5⋊D16, C2×D4⋊D5, C2×C5⋊D16

Smallest permutation representation of C2×C5⋊D16
On 160 points
Generators in S160
(1 31)(2 32)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(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 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 65)(61 66)(62 67)(63 68)(64 69)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)(121 145)(122 146)(123 147)(124 148)(125 149)(126 150)(127 151)(128 152)
(1 119 137 91 54)(2 55 92 138 120)(3 121 139 93 56)(4 57 94 140 122)(5 123 141 95 58)(6 59 96 142 124)(7 125 143 81 60)(8 61 82 144 126)(9 127 129 83 62)(10 63 84 130 128)(11 113 131 85 64)(12 49 86 132 114)(13 115 133 87 50)(14 51 88 134 116)(15 117 135 89 52)(16 53 90 136 118)(17 145 99 38 77)(18 78 39 100 146)(19 147 101 40 79)(20 80 41 102 148)(21 149 103 42 65)(22 66 43 104 150)(23 151 105 44 67)(24 68 45 106 152)(25 153 107 46 69)(26 70 47 108 154)(27 155 109 48 71)(28 72 33 110 156)(29 157 111 34 73)(30 74 35 112 158)(31 159 97 36 75)(32 76 37 98 160)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(30 32)(33 100)(34 99)(35 98)(36 97)(37 112)(38 111)(39 110)(40 109)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 124)(50 123)(51 122)(52 121)(53 120)(54 119)(55 118)(56 117)(57 116)(58 115)(59 114)(60 113)(61 128)(62 127)(63 126)(64 125)(65 153)(66 152)(67 151)(68 150)(69 149)(70 148)(71 147)(72 146)(73 145)(74 160)(75 159)(76 158)(77 157)(78 156)(79 155)(80 154)(81 131)(82 130)(83 129)(84 144)(85 143)(86 142)(87 141)(88 140)(89 139)(90 138)(91 137)(92 136)(93 135)(94 134)(95 133)(96 132)

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 D16 C5⋊D4 C5⋊D4 D4⋊D5 D4⋊D5 C5⋊D16 kernel C2×C5⋊D16 C2×C5⋊2C16 C5⋊D16 C2×D40 C10×D8 C40 C2×C20 C2×D8 C20 C2×C10 C2×C8 D8 C10 C8 C2×C4 C4 C22 C2 # reps 1 1 4 1 1 1 1 2 2 2 2 4 8 4 4 2 2 8

Matrix representation of C2×C5⋊D16 in GL5(𝔽241)

 240 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 189 240 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 240 0 0 0 0 0 214 138 0 0 0 96 27 0 0 0 0 0 27 85 0 0 0 156 27
,
 240 0 0 0 0 0 1 0 0 0 0 189 240 0 0 0 0 0 1 0 0 0 0 0 240

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,189,1,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,1],[240,0,0,0,0,0,214,96,0,0,0,138,27,0,0,0,0,0,27,156,0,0,0,85,27],[240,0,0,0,0,0,1,189,0,0,0,0,240,0,0,0,0,0,1,0,0,0,0,0,240] >;

C2×C5⋊D16 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes D_{16}
% in TeX

G:=Group("C2xC5:D16");
// GroupNames label

G:=SmallGroup(320,773);
// by ID

G=gap.SmallGroup(320,773);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,675,185,192,1684,438,102,12550]);
// Polycyclic

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

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