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G = C2×C52C16order 160 = 25·5

Direct product of C2 and C52C16

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×C52C16, C102C16, C20.6C8, C40.8C4, C8.21D10, C8.4Dic5, C40.21C22, C54(C2×C16), (C2×C8).9D5, (C2×C40).9C2, (C2×C10).4C8, C4.3(C52C8), C10.17(C2×C8), C20.59(C2×C4), (C2×C20).22C4, (C2×C4).8Dic5, C4.10(C2×Dic5), C22.2(C52C8), C2.2(C2×C52C8), SmallGroup(160,18)

Series: Derived Chief Lower central Upper central

C1C5 — C2×C52C16
C1C5C10C20C40C52C16 — C2×C52C16
C5 — C2×C52C16
C1C2×C8

Generators and relations for C2×C52C16
 G = < a,b,c | a2=b5=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

5C16
5C16
5C2×C16

Smallest permutation representation of C2×C52C16
Regular action on 160 points
Generators in S160
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 97)(13 98)(14 99)(15 100)(16 101)(17 124)(18 125)(19 126)(20 127)(21 128)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 120)(30 121)(31 122)(32 123)(33 76)(34 77)(35 78)(36 79)(37 80)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 145)(57 146)(58 147)(59 148)(60 149)(61 150)(62 151)(63 152)(64 153)(81 144)(82 129)(83 130)(84 131)(85 132)(86 133)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 141)(95 142)(96 143)
(1 125 88 80 148)(2 149 65 89 126)(3 127 90 66 150)(4 151 67 91 128)(5 113 92 68 152)(6 153 69 93 114)(7 115 94 70 154)(8 155 71 95 116)(9 117 96 72 156)(10 157 73 81 118)(11 119 82 74 158)(12 159 75 83 120)(13 121 84 76 160)(14 145 77 85 122)(15 123 86 78 146)(16 147 79 87 124)(17 101 58 36 134)(18 135 37 59 102)(19 103 60 38 136)(20 137 39 61 104)(21 105 62 40 138)(22 139 41 63 106)(23 107 64 42 140)(24 141 43 49 108)(25 109 50 44 142)(26 143 45 51 110)(27 111 52 46 144)(28 129 47 53 112)(29 97 54 48 130)(30 131 33 55 98)(31 99 56 34 132)(32 133 35 57 100)
(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)| (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,97)(13,98)(14,99)(15,100)(16,101)(17,124)(18,125)(19,126)(20,127)(21,128)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,121)(31,122)(32,123)(33,76)(34,77)(35,78)(36,79)(37,80)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,145)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(81,144)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143), (1,125,88,80,148)(2,149,65,89,126)(3,127,90,66,150)(4,151,67,91,128)(5,113,92,68,152)(6,153,69,93,114)(7,115,94,70,154)(8,155,71,95,116)(9,117,96,72,156)(10,157,73,81,118)(11,119,82,74,158)(12,159,75,83,120)(13,121,84,76,160)(14,145,77,85,122)(15,123,86,78,146)(16,147,79,87,124)(17,101,58,36,134)(18,135,37,59,102)(19,103,60,38,136)(20,137,39,61,104)(21,105,62,40,138)(22,139,41,63,106)(23,107,64,42,140)(24,141,43,49,108)(25,109,50,44,142)(26,143,45,51,110)(27,111,52,46,144)(28,129,47,53,112)(29,97,54,48,130)(30,131,33,55,98)(31,99,56,34,132)(32,133,35,57,100), (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( (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,97)(13,98)(14,99)(15,100)(16,101)(17,124)(18,125)(19,126)(20,127)(21,128)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,121)(31,122)(32,123)(33,76)(34,77)(35,78)(36,79)(37,80)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,145)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(81,144)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143), (1,125,88,80,148)(2,149,65,89,126)(3,127,90,66,150)(4,151,67,91,128)(5,113,92,68,152)(6,153,69,93,114)(7,115,94,70,154)(8,155,71,95,116)(9,117,96,72,156)(10,157,73,81,118)(11,119,82,74,158)(12,159,75,83,120)(13,121,84,76,160)(14,145,77,85,122)(15,123,86,78,146)(16,147,79,87,124)(17,101,58,36,134)(18,135,37,59,102)(19,103,60,38,136)(20,137,39,61,104)(21,105,62,40,138)(22,139,41,63,106)(23,107,64,42,140)(24,141,43,49,108)(25,109,50,44,142)(26,143,45,51,110)(27,111,52,46,144)(28,129,47,53,112)(29,97,54,48,130)(30,131,33,55,98)(31,99,56,34,132)(32,133,35,57,100), (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([[(1,102),(2,103),(3,104),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,111),(11,112),(12,97),(13,98),(14,99),(15,100),(16,101),(17,124),(18,125),(19,126),(20,127),(21,128),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,120),(30,121),(31,122),(32,123),(33,76),(34,77),(35,78),(36,79),(37,80),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,154),(50,155),(51,156),(52,157),(53,158),(54,159),(55,160),(56,145),(57,146),(58,147),(59,148),(60,149),(61,150),(62,151),(63,152),(64,153),(81,144),(82,129),(83,130),(84,131),(85,132),(86,133),(87,134),(88,135),(89,136),(90,137),(91,138),(92,139),(93,140),(94,141),(95,142),(96,143)], [(1,125,88,80,148),(2,149,65,89,126),(3,127,90,66,150),(4,151,67,91,128),(5,113,92,68,152),(6,153,69,93,114),(7,115,94,70,154),(8,155,71,95,116),(9,117,96,72,156),(10,157,73,81,118),(11,119,82,74,158),(12,159,75,83,120),(13,121,84,76,160),(14,145,77,85,122),(15,123,86,78,146),(16,147,79,87,124),(17,101,58,36,134),(18,135,37,59,102),(19,103,60,38,136),(20,137,39,61,104),(21,105,62,40,138),(22,139,41,63,106),(23,107,64,42,140),(24,141,43,49,108),(25,109,50,44,142),(26,143,45,51,110),(27,111,52,46,144),(28,129,47,53,112),(29,97,54,48,130),(30,131,33,55,98),(31,99,56,34,132),(32,133,35,57,100)], [(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×C52C16 is a maximal subgroup of
C40.10C8  C203C16  C40.2Q8  C10.SD32  C40.5D4  C10.Q32  C40.7Q8  D40.5C4  C16×Dic5  C40.88D4  C8017C4  D101C16  D20.4C8  C40.91D4  C40.92D4  C10.D16  C40.15D4  C20.58D8  C5⋊M6(2)  D5×C2×C16  D20.5C8  C40.70C23  C40.30C23
C2×C52C16 is a maximal quotient of
C203C16  C80.9C4  C40.91D4

64 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B8A···8H10A···10F16A···16P20A···20H40A···40P
order12224444558···810···1016···1620···2040···40
size11111111221···12···25···52···22···2

64 irreducible representations

dim111111112222222
type++++-+-
imageC1C2C2C4C4C8C8C16D5Dic5D10Dic5C52C8C52C8C52C16
kernelC2×C52C16C52C16C2×C40C40C2×C20C20C2×C10C10C2×C8C8C8C2×C4C4C22C2
# reps12122441622224416

Matrix representation of C2×C52C16 in GL3(𝔽241) generated by

100
02400
00240
,
100
0189240
010
,
19700
0170114
019171
G:=sub<GL(3,GF(241))| [1,0,0,0,240,0,0,0,240],[1,0,0,0,189,1,0,240,0],[197,0,0,0,170,191,0,114,71] >;

C2×C52C16 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes_2C_{16}
% in TeX

G:=Group("C2xC5:2C16");
// GroupNames label

G:=SmallGroup(160,18);
// by ID

G=gap.SmallGroup(160,18);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,50,69,4613]);
// 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^-1>;
// generators/relations

Export

Subgroup lattice of C2×C52C16 in TeX

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