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G = Dic41order 164 = 22·41

Dicyclic group

Aliases: Dic41, C412C4, C82.C2, C2.D41, SmallGroup(164,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C41 — Dic41
 Chief series C1 — C41 — C82 — Dic41
 Lower central C41 — Dic41
 Upper central C1 — C2

Generators and relations for Dic41
G = < a,b | a82=1, b2=a41, bab-1=a-1 >

Smallest permutation representation of Dic41
Regular action on 164 points
Generators in S164
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164)
(1 114 42 155)(2 113 43 154)(3 112 44 153)(4 111 45 152)(5 110 46 151)(6 109 47 150)(7 108 48 149)(8 107 49 148)(9 106 50 147)(10 105 51 146)(11 104 52 145)(12 103 53 144)(13 102 54 143)(14 101 55 142)(15 100 56 141)(16 99 57 140)(17 98 58 139)(18 97 59 138)(19 96 60 137)(20 95 61 136)(21 94 62 135)(22 93 63 134)(23 92 64 133)(24 91 65 132)(25 90 66 131)(26 89 67 130)(27 88 68 129)(28 87 69 128)(29 86 70 127)(30 85 71 126)(31 84 72 125)(32 83 73 124)(33 164 74 123)(34 163 75 122)(35 162 76 121)(36 161 77 120)(37 160 78 119)(38 159 79 118)(39 158 80 117)(40 157 81 116)(41 156 82 115)

G:=sub<Sym(164)| (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)(83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,114,42,155)(2,113,43,154)(3,112,44,153)(4,111,45,152)(5,110,46,151)(6,109,47,150)(7,108,48,149)(8,107,49,148)(9,106,50,147)(10,105,51,146)(11,104,52,145)(12,103,53,144)(13,102,54,143)(14,101,55,142)(15,100,56,141)(16,99,57,140)(17,98,58,139)(18,97,59,138)(19,96,60,137)(20,95,61,136)(21,94,62,135)(22,93,63,134)(23,92,64,133)(24,91,65,132)(25,90,66,131)(26,89,67,130)(27,88,68,129)(28,87,69,128)(29,86,70,127)(30,85,71,126)(31,84,72,125)(32,83,73,124)(33,164,74,123)(34,163,75,122)(35,162,76,121)(36,161,77,120)(37,160,78,119)(38,159,79,118)(39,158,80,117)(40,157,81,116)(41,156,82,115)>;

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)(83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,114,42,155)(2,113,43,154)(3,112,44,153)(4,111,45,152)(5,110,46,151)(6,109,47,150)(7,108,48,149)(8,107,49,148)(9,106,50,147)(10,105,51,146)(11,104,52,145)(12,103,53,144)(13,102,54,143)(14,101,55,142)(15,100,56,141)(16,99,57,140)(17,98,58,139)(18,97,59,138)(19,96,60,137)(20,95,61,136)(21,94,62,135)(22,93,63,134)(23,92,64,133)(24,91,65,132)(25,90,66,131)(26,89,67,130)(27,88,68,129)(28,87,69,128)(29,86,70,127)(30,85,71,126)(31,84,72,125)(32,83,73,124)(33,164,74,123)(34,163,75,122)(35,162,76,121)(36,161,77,120)(37,160,78,119)(38,159,79,118)(39,158,80,117)(40,157,81,116)(41,156,82,115) );

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),(83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164)], [(1,114,42,155),(2,113,43,154),(3,112,44,153),(4,111,45,152),(5,110,46,151),(6,109,47,150),(7,108,48,149),(8,107,49,148),(9,106,50,147),(10,105,51,146),(11,104,52,145),(12,103,53,144),(13,102,54,143),(14,101,55,142),(15,100,56,141),(16,99,57,140),(17,98,58,139),(18,97,59,138),(19,96,60,137),(20,95,61,136),(21,94,62,135),(22,93,63,134),(23,92,64,133),(24,91,65,132),(25,90,66,131),(26,89,67,130),(27,88,68,129),(28,87,69,128),(29,86,70,127),(30,85,71,126),(31,84,72,125),(32,83,73,124),(33,164,74,123),(34,163,75,122),(35,162,76,121),(36,161,77,120),(37,160,78,119),(38,159,79,118),(39,158,80,117),(40,157,81,116),(41,156,82,115)])

Dic41 is a maximal subgroup of   C412C8  Dic82  C4×D41  C41⋊D4  Dic123
Dic41 is a maximal quotient of   C413C8  Dic123

44 conjugacy classes

 class 1 2 4A 4B 41A ··· 41T 82A ··· 82T order 1 2 4 4 41 ··· 41 82 ··· 82 size 1 1 41 41 2 ··· 2 2 ··· 2

44 irreducible representations

 dim 1 1 1 2 2 type + + + - image C1 C2 C4 D41 Dic41 kernel Dic41 C82 C41 C2 C1 # reps 1 1 2 20 20

Matrix representation of Dic41 in GL3(𝔽821) generated by

 820 0 0 0 782 820 0 1 0
,
 526 0 0 0 725 390 0 29 96
G:=sub<GL(3,GF(821))| [820,0,0,0,782,1,0,820,0],[526,0,0,0,725,29,0,390,96] >;

Dic41 in GAP, Magma, Sage, TeX

{\rm Dic}_{41}
% in TeX

G:=Group("Dic41");
// GroupNames label

G:=SmallGroup(164,1);
// by ID

G=gap.SmallGroup(164,1);
# by ID

G:=PCGroup([3,-2,-2,-41,6,1442]);
// Polycyclic

G:=Group<a,b|a^82=1,b^2=a^41,b*a*b^-1=a^-1>;
// generators/relations

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