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

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

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

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

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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