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G = Dic37order 148 = 22·37

Dicyclic group

Aliases: Dic37, C372C4, C74.C2, C2.D37, SmallGroup(148,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C37 — Dic37
 Chief series C1 — C37 — C74 — Dic37
 Lower central C37 — Dic37
 Upper central C1 — C2

Generators and relations for Dic37
G = < a,b | a74=1, b2=a37, bab-1=a-1 >

Smallest permutation representation of Dic37
Regular action on 148 points
Generators in S148
(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)
(1 120 38 83)(2 119 39 82)(3 118 40 81)(4 117 41 80)(5 116 42 79)(6 115 43 78)(7 114 44 77)(8 113 45 76)(9 112 46 75)(10 111 47 148)(11 110 48 147)(12 109 49 146)(13 108 50 145)(14 107 51 144)(15 106 52 143)(16 105 53 142)(17 104 54 141)(18 103 55 140)(19 102 56 139)(20 101 57 138)(21 100 58 137)(22 99 59 136)(23 98 60 135)(24 97 61 134)(25 96 62 133)(26 95 63 132)(27 94 64 131)(28 93 65 130)(29 92 66 129)(30 91 67 128)(31 90 68 127)(32 89 69 126)(33 88 70 125)(34 87 71 124)(35 86 72 123)(36 85 73 122)(37 84 74 121)

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

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

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

Dic37 is a maximal subgroup of   C37⋊C8  Dic74  C4×D37  C37⋊D4  C74.C6  Dic111
Dic37 is a maximal quotient of   C372C8  Dic111

40 conjugacy classes

 class 1 2 4A 4B 37A ··· 37R 74A ··· 74R order 1 2 4 4 37 ··· 37 74 ··· 74 size 1 1 37 37 2 ··· 2 2 ··· 2

40 irreducible representations

 dim 1 1 1 2 2 type + + + - image C1 C2 C4 D37 Dic37 kernel Dic37 C74 C37 C2 C1 # reps 1 1 2 18 18

Matrix representation of Dic37 in GL2(𝔽149) generated by

 18 1 148 0
,
 10 110 79 139
G:=sub<GL(2,GF(149))| [18,148,1,0],[10,79,110,139] >;

Dic37 in GAP, Magma, Sage, TeX

{\rm Dic}_{37}
% in TeX

G:=Group("Dic37");
// GroupNames label

G:=SmallGroup(148,1);
// by ID

G=gap.SmallGroup(148,1);
# by ID

G:=PCGroup([3,-2,-2,-37,6,1298]);
// Polycyclic

G:=Group<a,b|a^74=1,b^2=a^37,b*a*b^-1=a^-1>;
// generators/relations

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