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## G = C2×C5⋊C16order 160 = 25·5

### Direct product of C2 and C5⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C5⋊C16
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C5⋊C16 — C2×C5⋊C16
 Lower central C5 — C2×C5⋊C16
 Upper central C1 — C2×C4

Generators and relations for C2×C5⋊C16
G = < a,b,c | a2=b5=c16=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C2×C5⋊C16
Regular action on 160 points
Generators in S160
(1 160)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 46)(18 47)(19 48)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(49 140)(50 141)(51 142)(52 143)(53 144)(54 129)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 121)(96 122)
(1 83 62 21 99)(2 22 84 100 63)(3 101 23 64 85)(4 49 102 86 24)(5 87 50 25 103)(6 26 88 104 51)(7 105 27 52 89)(8 53 106 90 28)(9 91 54 29 107)(10 30 92 108 55)(11 109 31 56 93)(12 57 110 94 32)(13 95 58 17 111)(14 18 96 112 59)(15 97 19 60 81)(16 61 98 82 20)(33 159 136 74 124)(34 75 160 125 137)(35 126 76 138 145)(36 139 127 146 77)(37 147 140 78 128)(38 79 148 113 141)(39 114 80 142 149)(40 143 115 150 65)(41 151 144 66 116)(42 67 152 117 129)(43 118 68 130 153)(44 131 119 154 69)(45 155 132 70 120)(46 71 156 121 133)(47 122 72 134 157)(48 135 123 158 73)
(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)| (1,160)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,46)(18,47)(19,48)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(49,140)(50,141)(51,142)(52,143)(53,144)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122), (1,83,62,21,99)(2,22,84,100,63)(3,101,23,64,85)(4,49,102,86,24)(5,87,50,25,103)(6,26,88,104,51)(7,105,27,52,89)(8,53,106,90,28)(9,91,54,29,107)(10,30,92,108,55)(11,109,31,56,93)(12,57,110,94,32)(13,95,58,17,111)(14,18,96,112,59)(15,97,19,60,81)(16,61,98,82,20)(33,159,136,74,124)(34,75,160,125,137)(35,126,76,138,145)(36,139,127,146,77)(37,147,140,78,128)(38,79,148,113,141)(39,114,80,142,149)(40,143,115,150,65)(41,151,144,66,116)(42,67,152,117,129)(43,118,68,130,153)(44,131,119,154,69)(45,155,132,70,120)(46,71,156,121,133)(47,122,72,134,157)(48,135,123,158,73), (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( (1,160)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,46)(18,47)(19,48)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(49,140)(50,141)(51,142)(52,143)(53,144)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122), (1,83,62,21,99)(2,22,84,100,63)(3,101,23,64,85)(4,49,102,86,24)(5,87,50,25,103)(6,26,88,104,51)(7,105,27,52,89)(8,53,106,90,28)(9,91,54,29,107)(10,30,92,108,55)(11,109,31,56,93)(12,57,110,94,32)(13,95,58,17,111)(14,18,96,112,59)(15,97,19,60,81)(16,61,98,82,20)(33,159,136,74,124)(34,75,160,125,137)(35,126,76,138,145)(36,139,127,146,77)(37,147,140,78,128)(38,79,148,113,141)(39,114,80,142,149)(40,143,115,150,65)(41,151,144,66,116)(42,67,152,117,129)(43,118,68,130,153)(44,131,119,154,69)(45,155,132,70,120)(46,71,156,121,133)(47,122,72,134,157)(48,135,123,158,73), (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([[(1,160),(2,145),(3,146),(4,147),(5,148),(6,149),(7,150),(8,151),(9,152),(10,153),(11,154),(12,155),(13,156),(14,157),(15,158),(16,159),(17,46),(18,47),(19,48),(20,33),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(49,140),(50,141),(51,142),(52,143),(53,144),(54,129),(55,130),(56,131),(57,132),(58,133),(59,134),(60,135),(61,136),(62,137),(63,138),(64,139),(65,105),(66,106),(67,107),(68,108),(69,109),(70,110),(71,111),(72,112),(73,97),(74,98),(75,99),(76,100),(77,101),(78,102),(79,103),(80,104),(81,123),(82,124),(83,125),(84,126),(85,127),(86,128),(87,113),(88,114),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,121),(96,122)], [(1,83,62,21,99),(2,22,84,100,63),(3,101,23,64,85),(4,49,102,86,24),(5,87,50,25,103),(6,26,88,104,51),(7,105,27,52,89),(8,53,106,90,28),(9,91,54,29,107),(10,30,92,108,55),(11,109,31,56,93),(12,57,110,94,32),(13,95,58,17,111),(14,18,96,112,59),(15,97,19,60,81),(16,61,98,82,20),(33,159,136,74,124),(34,75,160,125,137),(35,126,76,138,145),(36,139,127,146,77),(37,147,140,78,128),(38,79,148,113,141),(39,114,80,142,149),(40,143,115,150,65),(41,151,144,66,116),(42,67,152,117,129),(43,118,68,130,153),(44,131,119,154,69),(45,155,132,70,120),(46,71,156,121,133),(47,122,72,134,157),(48,135,123,158,73)], [(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)]])

C2×C5⋊C16 is a maximal subgroup of
C20⋊C16  C42.4F5  Dic5⋊C16  C40.C8  D10⋊C16  C10.M5(2)  D20.C8  C10.6M5(2)  D4.(C5⋊C8)  Dic10.C8  C5⋊C16.C22
C2×C5⋊C16 is a maximal quotient of
C20⋊C16  C5⋊M6(2)  C10.6M5(2)

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5 8A ··· 8H 10A 10B 10C 16A ··· 16P 20A 20B 20C 20D order 1 2 2 2 4 4 4 4 5 8 ··· 8 10 10 10 16 ··· 16 20 20 20 20 size 1 1 1 1 1 1 1 1 4 5 ··· 5 4 4 4 5 ··· 5 4 4 4 4

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 4 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 C16 F5 C5⋊C8 C2×F5 C5⋊C8 C5⋊C16 kernel C2×C5⋊C16 C5⋊C16 C2×C5⋊2C8 C5⋊2C8 C2×C20 C20 C2×C10 C10 C2×C4 C4 C4 C22 C2 # reps 1 2 1 2 2 4 4 16 1 1 1 1 4

Matrix representation of C2×C5⋊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 0 0 0 240 0 1 0 0 240 0 0 1 0 240 0 0 0 1 240
,
 130 0 0 0 0 0 24 4 29 166 0 53 170 47 190 0 71 194 51 219 0 75 223 217 237

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,240,240,240,240],[130,0,0,0,0,0,24,53,71,75,0,4,170,194,223,0,29,47,51,217,0,166,190,219,237] >;

C2×C5⋊C16 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC5:C16");
// GroupNames label

G:=SmallGroup(160,72);
// by ID

G=gap.SmallGroup(160,72);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,50,69,2309,1169]);
// Polycyclic

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

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