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G = D113order 226 = 2·113

Dihedral group

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

Aliases: D113, C113⋊C2, sometimes denoted D226 or Dih113 or Dih226, SmallGroup(226,1)

Series: Derived Chief Lower central Upper central

C1C113 — D113
C1C113 — D113
C113 — D113
C1

Generators and relations for D113
 G = < a,b | a113=b2=1, bab=a-1 >

113C2

Smallest permutation representation of D113
On 113 points: primitive
Generators in S113
(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)
(1 113)(2 112)(3 111)(4 110)(5 109)(6 108)(7 107)(8 106)(9 105)(10 104)(11 103)(12 102)(13 101)(14 100)(15 99)(16 98)(17 97)(18 96)(19 95)(20 94)(21 93)(22 92)(23 91)(24 90)(25 89)(26 88)(27 87)(28 86)(29 85)(30 84)(31 83)(32 82)(33 81)(34 80)(35 79)(36 78)(37 77)(38 76)(39 75)(40 74)(41 73)(42 72)(43 71)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)

G:=sub<Sym(113)| (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), (1,113)(2,112)(3,111)(4,110)(5,109)(6,108)(7,107)(8,106)(9,105)(10,104)(11,103)(12,102)(13,101)(14,100)(15,99)(16,98)(17,97)(18,96)(19,95)(20,94)(21,93)(22,92)(23,91)(24,90)(25,89)(26,88)(27,87)(28,86)(29,85)(30,84)(31,83)(32,82)(33,81)(34,80)(35,79)(36,78)(37,77)(38,76)(39,75)(40,74)(41,73)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)>;

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), (1,113)(2,112)(3,111)(4,110)(5,109)(6,108)(7,107)(8,106)(9,105)(10,104)(11,103)(12,102)(13,101)(14,100)(15,99)(16,98)(17,97)(18,96)(19,95)(20,94)(21,93)(22,92)(23,91)(24,90)(25,89)(26,88)(27,87)(28,86)(29,85)(30,84)(31,83)(32,82)(33,81)(34,80)(35,79)(36,78)(37,77)(38,76)(39,75)(40,74)(41,73)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58) );

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)], [(1,113),(2,112),(3,111),(4,110),(5,109),(6,108),(7,107),(8,106),(9,105),(10,104),(11,103),(12,102),(13,101),(14,100),(15,99),(16,98),(17,97),(18,96),(19,95),(20,94),(21,93),(22,92),(23,91),(24,90),(25,89),(26,88),(27,87),(28,86),(29,85),(30,84),(31,83),(32,82),(33,81),(34,80),(35,79),(36,78),(37,77),(38,76),(39,75),(40,74),(41,73),(42,72),(43,71),(44,70),(45,69),(46,68),(47,67),(48,66),(49,65),(50,64),(51,63),(52,62),(53,61),(54,60),(55,59),(56,58)])

D113 is a maximal subgroup of   C113⋊C4
D113 is a maximal quotient of   Dic113

58 conjugacy classes

class 1  2 113A···113BD
order12113···113
size11132···2

58 irreducible representations

dim112
type+++
imageC1C2D113
kernelD113C113C1
# reps1156

Matrix representation of D113 in GL2(𝔽227) generated by

156226
136117
,
63155
156164
G:=sub<GL(2,GF(227))| [156,136,226,117],[63,156,155,164] >;

D113 in GAP, Magma, Sage, TeX

D_{113}
% in TeX

G:=Group("D113");
// GroupNames label

G:=SmallGroup(226,1);
// by ID

G=gap.SmallGroup(226,1);
# by ID

G:=PCGroup([2,-2,-113,897]);
// Polycyclic

G:=Group<a,b|a^113=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D113 in TeX

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