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G = C2×C5⋊2C16order 160 = 25·5

Direct product of C2 and C5⋊2C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C5⋊2C16
 Chief series C1 — C5 — C10 — C20 — C40 — C5⋊2C16 — C2×C5⋊2C16
 Lower central C5 — C2×C5⋊2C16
 Upper central C1 — C2×C8

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

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

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 C16 D5 Dic5 D10 Dic5 C5⋊2C8 C5⋊2C8 C5⋊2C16 kernel C2×C5⋊2C16 C5⋊2C16 C2×C40 C40 C2×C20 C20 C2×C10 C10 C2×C8 C8 C8 C2×C4 C4 C22 C2 # reps 1 2 1 2 2 4 4 16 2 2 2 2 4 4 16

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

 1 0 0 0 240 0 0 0 240
,
 1 0 0 0 189 240 0 1 0
,
 197 0 0 0 170 114 0 191 71
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

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