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G = D154order 308 = 22·7·11

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D154, C2×D77, C22⋊D7, C14⋊D11, C72D22, C112D14, C1541C2, C772C22, sometimes denoted D308 or Dih154 or Dih308, SmallGroup(308,8)

Series: Derived Chief Lower central Upper central

C1C77 — D154
C1C11C77D77 — D154
C77 — D154
C1C2

Generators and relations for D154
 G = < a,b | a154=b2=1, bab=a-1 >

77C2
77C2
77C22
11D7
11D7
7D11
7D11
11D14
7D22

Smallest permutation representation of D154
On 154 points
Generators in S154
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154)
(1 154)(2 153)(3 152)(4 151)(5 150)(6 149)(7 148)(8 147)(9 146)(10 145)(11 144)(12 143)(13 142)(14 141)(15 140)(16 139)(17 138)(18 137)(19 136)(20 135)(21 134)(22 133)(23 132)(24 131)(25 130)(26 129)(27 128)(28 127)(29 126)(30 125)(31 124)(32 123)(33 122)(34 121)(35 120)(36 119)(37 118)(38 117)(39 116)(40 115)(41 114)(42 113)(43 112)(44 111)(45 110)(46 109)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 100)(56 99)(57 98)(58 97)(59 96)(60 95)(61 94)(62 93)(63 92)(64 91)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)

G:=sub<Sym(154)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154), (1,154)(2,153)(3,152)(4,151)(5,150)(6,149)(7,148)(8,147)(9,146)(10,145)(11,144)(12,143)(13,142)(14,141)(15,140)(16,139)(17,138)(18,137)(19,136)(20,135)(21,134)(22,133)(23,132)(24,131)(25,130)(26,129)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154), (1,154)(2,153)(3,152)(4,151)(5,150)(6,149)(7,148)(8,147)(9,146)(10,145)(11,144)(12,143)(13,142)(14,141)(15,140)(16,139)(17,138)(18,137)(19,136)(20,135)(21,134)(22,133)(23,132)(24,131)(25,130)(26,129)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154)], [(1,154),(2,153),(3,152),(4,151),(5,150),(6,149),(7,148),(8,147),(9,146),(10,145),(11,144),(12,143),(13,142),(14,141),(15,140),(16,139),(17,138),(18,137),(19,136),(20,135),(21,134),(22,133),(23,132),(24,131),(25,130),(26,129),(27,128),(28,127),(29,126),(30,125),(31,124),(32,123),(33,122),(34,121),(35,120),(36,119),(37,118),(38,117),(39,116),(40,115),(41,114),(42,113),(43,112),(44,111),(45,110),(46,109),(47,108),(48,107),(49,106),(50,105),(51,104),(52,103),(53,102),(54,101),(55,100),(56,99),(57,98),(58,97),(59,96),(60,95),(61,94),(62,93),(63,92),(64,91),(65,90),(66,89),(67,88),(68,87),(69,86),(70,85),(71,84),(72,83),(73,82),(74,81),(75,80),(76,79),(77,78)])

80 conjugacy classes

class 1 2A2B2C7A7B7C11A···11E14A14B14C22A···22E77A···77AD154A···154AD
order122277711···1114141422···2277···77154···154
size1177772222···22222···22···22···2

80 irreducible representations

dim111222222
type+++++++++
imageC1C2C2D7D11D14D22D77D154
kernelD154D77C154C22C14C11C7C2C1
# reps12135353030

Matrix representation of D154 in GL3(𝔽463) generated by

46200
0231236
0107424
,
100
0273385
0445190
G:=sub<GL(3,GF(463))| [462,0,0,0,231,107,0,236,424],[1,0,0,0,273,445,0,385,190] >;

D154 in GAP, Magma, Sage, TeX

D_{154}
% in TeX

G:=Group("D154");
// GroupNames label

G:=SmallGroup(308,8);
// by ID

G=gap.SmallGroup(308,8);
# by ID

G:=PCGroup([4,-2,-2,-7,-11,290,4483]);
// Polycyclic

G:=Group<a,b|a^154=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D154 in TeX

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