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G = Dic37order 148 = 22·37

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C37 — Dic37
C1C37C74 — Dic37
C37 — Dic37
C1C2

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

37C4

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 4A4B37A···37R74A···74R
order124437···3774···74
size1137372···22···2

40 irreducible representations

dim11122
type+++-
imageC1C2C4D37Dic37
kernelDic37C74C37C2C1
# reps1121818

Matrix representation of Dic37 in GL2(𝔽149) generated by

181
1480
,
10110
79139
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

Export

Subgroup lattice of Dic37 in TeX

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