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G = D152order 304 = 24·19

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D152, C191D8, C81D19, C1521C2, D761C2, C2.4D76, C4.9D38, C38.2D4, C76.9C22, sometimes denoted D304 or Dih152 or Dih304, SmallGroup(304,6)

Series: Derived Chief Lower central Upper central

C1C76 — D152
C1C19C38C76D76 — D152
C19C38C76 — D152
C1C2C4C8

Generators and relations for D152
 G = < a,b | a152=b2=1, bab=a-1 >

76C2
76C2
38C22
38C22
4D19
4D19
19D4
19D4
2D38
2D38
19D8

Smallest permutation representation of D152
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)
(1 152)(2 151)(3 150)(4 149)(5 148)(6 147)(7 146)(8 145)(9 144)(10 143)(11 142)(12 141)(13 140)(14 139)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(21 132)(22 131)(23 130)(24 129)(25 128)(26 127)(27 126)(28 125)(29 124)(30 123)(31 122)(32 121)(33 120)(34 119)(35 118)(36 117)(37 116)(38 115)(39 114)(40 113)(41 112)(42 111)(43 110)(44 109)(45 108)(46 107)(47 106)(48 105)(49 104)(50 103)(51 102)(52 101)(53 100)(54 99)(55 98)(56 97)(57 96)(58 95)(59 94)(60 93)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)

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

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

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

79 conjugacy classes

class 1 2A2B2C 4 8A8B19A···19I38A···38I76A···76R152A···152AJ
order122248819···1938···3876···76152···152
size1176762222···22···22···22···2

79 irreducible representations

dim111222222
type+++++++++
imageC1C2C2D4D8D19D38D76D152
kernelD152C152D76C38C19C8C4C2C1
# reps11212991836

Matrix representation of D152 in GL2(𝔽457) generated by

94270
18785
,
94270
145363
G:=sub<GL(2,GF(457))| [94,187,270,85],[94,145,270,363] >;

D152 in GAP, Magma, Sage, TeX

D_{152}
% in TeX

G:=Group("D152");
// GroupNames label

G:=SmallGroup(304,6);
// by ID

G=gap.SmallGroup(304,6);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-19,61,66,182,42,7204]);
// Polycyclic

G:=Group<a,b|a^152=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D152 in TeX

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