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## G = Dic40order 160 = 25·5

### Dicyclic group

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

 Derived series C1 — C40 — Dic40
 Chief series C1 — C5 — C10 — C20 — C40 — Dic20 — Dic40
 Lower central C5 — C10 — C20 — C40 — Dic40
 Upper central C1 — C2 — C4 — C8 — C16

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

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 4A 4B 4C 5A 5B 8A 8B 10A 10B 16A 16B 16C 16D 20A 20B 20C 20D 40A ··· 40H 80A ··· 80P order 1 2 4 4 4 5 5 8 8 10 10 16 16 16 16 20 20 20 20 40 ··· 40 80 ··· 80 size 1 1 2 40 40 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

43 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + - + + - image C1 C2 C2 D4 D5 D8 D10 Q32 D20 D40 Dic40 kernel Dic40 C80 Dic20 C20 C16 C10 C8 C5 C4 C2 C1 # reps 1 1 2 1 2 2 2 4 4 8 16

Matrix representation of Dic40 in GL2(𝔽241) generated by

 239 58 183 115
,
 148 42 58 93
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

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