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G = Dic40order 160 = 25·5

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic40, C16.D5, C51Q32, C80.1C2, C10.3D8, C2.5D40, C4.3D20, C20.26D4, C8.15D10, C40.16C22, Dic20.1C2, SmallGroup(160,8)

Series: Derived Chief Lower central Upper central

C1C40 — Dic40
C1C5C10C20C40Dic20 — Dic40
C5C10C20C40 — Dic40
C1C2C4C8C16

Generators and relations for Dic40
 G = < a,b | a80=1, b2=a40, bab-1=a-1 >

20C4
20C4
10Q8
10Q8
4Dic5
4Dic5
5Q16
5Q16
2Dic10
2Dic10
5Q32

Smallest permutation representation of Dic40
Regular action on 160 points
Generators in S160
(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)
(1 103 41 143)(2 102 42 142)(3 101 43 141)(4 100 44 140)(5 99 45 139)(6 98 46 138)(7 97 47 137)(8 96 48 136)(9 95 49 135)(10 94 50 134)(11 93 51 133)(12 92 52 132)(13 91 53 131)(14 90 54 130)(15 89 55 129)(16 88 56 128)(17 87 57 127)(18 86 58 126)(19 85 59 125)(20 84 60 124)(21 83 61 123)(22 82 62 122)(23 81 63 121)(24 160 64 120)(25 159 65 119)(26 158 66 118)(27 157 67 117)(28 156 68 116)(29 155 69 115)(30 154 70 114)(31 153 71 113)(32 152 72 112)(33 151 73 111)(34 150 74 110)(35 149 75 109)(36 148 76 108)(37 147 77 107)(38 146 78 106)(39 145 79 105)(40 144 80 104)

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

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

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

Dic40 is a maximal subgroup of
C160⋊C2  Dic80  D16.D5  C5⋊Q64  D807C2  C16.D10  D163D5  SD32⋊D5  D5×Q32  C3⋊Dic40  Dic120
Dic40 is a maximal quotient of
C40.78D4  C8013C4  C3⋊Dic40  Dic120

43 conjugacy classes

class 1  2 4A4B4C5A5B8A8B10A10B16A16B16C16D20A20B20C20D40A···40H80A···80P
order1244455881010161616162020202040···4080···80
size1124040222222222222222···22···2

43 irreducible representations

dim11122222222
type+++++++-++-
imageC1C2C2D4D5D8D10Q32D20D40Dic40
kernelDic40C80Dic20C20C16C10C8C5C4C2C1
# reps112122244816

Matrix representation of Dic40 in GL2(𝔽241) generated by

23958
183115
,
14842
5893
G:=sub<GL(2,GF(241))| [239,183,58,115],[148,58,42,93] >;

Dic40 in GAP, Magma, Sage, TeX

{\rm Dic}_{40}
% in TeX

G:=Group("Dic40");
// GroupNames label

G:=SmallGroup(160,8);
// by ID

G=gap.SmallGroup(160,8);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,73,79,218,122,579,69,4613]);
// Polycyclic

G:=Group<a,b|a^80=1,b^2=a^40,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic40 in TeX

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