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G = D148order 296 = 23·37

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D148, C4⋊D37, C371D4, C1481C2, D741C2, C2.4D74, C74.3C22, sometimes denoted D296 or Dih148 or Dih296, SmallGroup(296,6)

Series: Derived Chief Lower central Upper central

C1C74 — D148
C1C37C74D74 — D148
C37C74 — D148
C1C2C4

Generators and relations for D148
 G = < a,b | a148=b2=1, bab=a-1 >

74C2
74C2
37C22
37C22
2D37
2D37
37D4

Smallest permutation representation of D148
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)
(1 148)(2 147)(3 146)(4 145)(5 144)(6 143)(7 142)(8 141)(9 140)(10 139)(11 138)(12 137)(13 136)(14 135)(15 134)(16 133)(17 132)(18 131)(19 130)(20 129)(21 128)(22 127)(23 126)(24 125)(25 124)(26 123)(27 122)(28 121)(29 120)(30 119)(31 118)(32 117)(33 116)(34 115)(35 114)(36 113)(37 112)(38 111)(39 110)(40 109)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 100)(50 99)(51 98)(52 97)(53 96)(54 95)(55 94)(56 93)(57 92)(58 91)(59 90)(60 89)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)

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

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

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

77 conjugacy classes

class 1 2A2B2C 4 37A···37R74A···74R148A···148AJ
order1222437···3774···74148···148
size11747422···22···22···2

77 irreducible representations

dim1112222
type+++++++
imageC1C2C2D4D37D74D148
kernelD148C148D74C37C4C2C1
# reps1121181836

Matrix representation of D148 in GL2(𝔽149) generated by

108123
2631
,
108123
9941
G:=sub<GL(2,GF(149))| [108,26,123,31],[108,99,123,41] >;

D148 in GAP, Magma, Sage, TeX

D_{148}
% in TeX

G:=Group("D148");
// GroupNames label

G:=SmallGroup(296,6);
// by ID

G=gap.SmallGroup(296,6);
# by ID

G:=PCGroup([4,-2,-2,-2,-37,49,21,4611]);
// Polycyclic

G:=Group<a,b|a^148=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D148 in TeX

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