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G = Dic38order 152 = 23·19

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic38, C19⋊Q8, C4.D19, C76.1C2, C2.3D38, Dic19.C2, C38.1C22, SmallGroup(152,3)

Series: Derived Chief Lower central Upper central

C1C38 — Dic38
C1C19C38Dic19 — Dic38
C19C38 — Dic38
C1C2C4

Generators and relations for Dic38
 G = < a,b | a76=1, b2=a38, bab-1=a-1 >

19C4
19C4
19Q8

Smallest permutation representation of Dic38
Regular action on 152 points
Generators in S152
(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)
(1 131 39 93)(2 130 40 92)(3 129 41 91)(4 128 42 90)(5 127 43 89)(6 126 44 88)(7 125 45 87)(8 124 46 86)(9 123 47 85)(10 122 48 84)(11 121 49 83)(12 120 50 82)(13 119 51 81)(14 118 52 80)(15 117 53 79)(16 116 54 78)(17 115 55 77)(18 114 56 152)(19 113 57 151)(20 112 58 150)(21 111 59 149)(22 110 60 148)(23 109 61 147)(24 108 62 146)(25 107 63 145)(26 106 64 144)(27 105 65 143)(28 104 66 142)(29 103 67 141)(30 102 68 140)(31 101 69 139)(32 100 70 138)(33 99 71 137)(34 98 72 136)(35 97 73 135)(36 96 74 134)(37 95 75 133)(38 94 76 132)

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

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

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

Dic38 is a maximal subgroup of
C152⋊C2  Dic76  D4.D19  C19⋊Q16  D765C2  D42D19  Q8×D19  Dic38⋊C3  C57⋊Q8  Dic114
Dic38 is a maximal quotient of
Dic19⋊C4  C76⋊C4  C57⋊Q8  Dic114

41 conjugacy classes

class 1  2 4A4B4C19A···19I38A···38I76A···76R
order1244419···1938···3876···76
size11238382···22···22···2

41 irreducible representations

dim1112222
type+++-++-
imageC1C2C2Q8D19D38Dic38
kernelDic38Dic19C76C19C4C2C1
# reps12119918

Matrix representation of Dic38 in GL2(𝔽229) generated by

41157
853
,
34156
88195
G:=sub<GL(2,GF(229))| [41,8,157,53],[34,88,156,195] >;

Dic38 in GAP, Magma, Sage, TeX

{\rm Dic}_{38}
% in TeX

G:=Group("Dic38");
// GroupNames label

G:=SmallGroup(152,3);
// by ID

G=gap.SmallGroup(152,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-19,16,49,21,2307]);
// Polycyclic

G:=Group<a,b|a^76=1,b^2=a^38,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic38 in TeX

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