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G = D103order 206 = 2·103

Dihedral group

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

Aliases: D103, C103⋊C2, sometimes denoted D206 or Dih103 or Dih206, SmallGroup(206,1)

Series: Derived Chief Lower central Upper central

C1C103 — D103
C1C103 — D103
C103 — D103
C1

Generators and relations for D103
 G = < a,b | a103=b2=1, bab=a-1 >

103C2

Smallest permutation representation of D103
On 103 points: primitive
Generators in S103
(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)
(1 103)(2 102)(3 101)(4 100)(5 99)(6 98)(7 97)(8 96)(9 95)(10 94)(11 93)(12 92)(13 91)(14 90)(15 89)(16 88)(17 87)(18 86)(19 85)(20 84)(21 83)(22 82)(23 81)(24 80)(25 79)(26 78)(27 77)(28 76)(29 75)(30 74)(31 73)(32 72)(33 71)(34 70)(35 69)(36 68)(37 67)(38 66)(39 65)(40 64)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)

G:=sub<Sym(103)| (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), (1,103)(2,102)(3,101)(4,100)(5,99)(6,98)(7,97)(8,96)(9,95)(10,94)(11,93)(12,92)(13,91)(14,90)(15,89)(16,88)(17,87)(18,86)(19,85)(20,84)(21,83)(22,82)(23,81)(24,80)(25,79)(26,78)(27,77)(28,76)(29,75)(30,74)(31,73)(32,72)(33,71)(34,70)(35,69)(36,68)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)>;

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), (1,103)(2,102)(3,101)(4,100)(5,99)(6,98)(7,97)(8,96)(9,95)(10,94)(11,93)(12,92)(13,91)(14,90)(15,89)(16,88)(17,87)(18,86)(19,85)(20,84)(21,83)(22,82)(23,81)(24,80)(25,79)(26,78)(27,77)(28,76)(29,75)(30,74)(31,73)(32,72)(33,71)(34,70)(35,69)(36,68)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53) );

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)], [(1,103),(2,102),(3,101),(4,100),(5,99),(6,98),(7,97),(8,96),(9,95),(10,94),(11,93),(12,92),(13,91),(14,90),(15,89),(16,88),(17,87),(18,86),(19,85),(20,84),(21,83),(22,82),(23,81),(24,80),(25,79),(26,78),(27,77),(28,76),(29,75),(30,74),(31,73),(32,72),(33,71),(34,70),(35,69),(36,68),(37,67),(38,66),(39,65),(40,64),(41,63),(42,62),(43,61),(44,60),(45,59),(46,58),(47,57),(48,56),(49,55),(50,54),(51,53)])

D103 is a maximal quotient of   Dic103

53 conjugacy classes

class 1  2 103A···103AY
order12103···103
size11032···2

53 irreducible representations

dim112
type+++
imageC1C2D103
kernelD103C103C1
# reps1151

Matrix representation of D103 in GL2(𝔽619) generated by

459618
60306
,
144350
321475
G:=sub<GL(2,GF(619))| [459,60,618,306],[144,321,350,475] >;

D103 in GAP, Magma, Sage, TeX

D_{103}
% in TeX

G:=Group("D103");
// GroupNames label

G:=SmallGroup(206,1);
// by ID

G=gap.SmallGroup(206,1);
# by ID

G:=PCGroup([2,-2,-103,817]);
// Polycyclic

G:=Group<a,b|a^103=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D103 in TeX

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