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G = D53order 106 = 2·53

Dihedral group

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

Aliases: D53, C53⋊C2, sometimes denoted D106 or Dih53 or Dih106, SmallGroup(106,1)

Series: Derived Chief Lower central Upper central

C1C53 — D53
C1C53 — D53
C53 — D53
C1

Generators and relations for D53
 G = < a,b | a53=b2=1, bab=a-1 >

53C2

Character table of D53

 class 1253A53B53C53D53E53F53G53H53I53J53K53L53M53N53O53P53Q53R53S53T53U53V53W53X53Y53Z
 size 15322222222222222222222222222
ρ11111111111111111111111111111    trivial
ρ21-111111111111111111111111111    linear of order 2
ρ320ζ5346537ζ53375316ζ53395314ζ5344539ζ53325321ζ5351532ζ53285325ζ5348535ζ53355318ζ53415312ζ53425311ζ53345319ζ5349534ζ53275326ζ5350533ζ53335320ζ53435310ζ53405313ζ53365317ζ5347536ζ53295324ζ535253ζ53315322ζ5345538ζ53385315ζ53305323    orthogonal faithful
ρ420ζ53425311ζ53435310ζ53315322ζ535253ζ53335320ζ53415312ζ5344539ζ53305323ζ5351532ζ53345319ζ53405313ζ5345538ζ53295324ζ5350533ζ53355318ζ53395314ζ5346537ζ53285325ζ5349534ζ53365317ζ53385315ζ5347536ζ53275326ζ5348535ζ53375316ζ53325321    orthogonal faithful
ρ520ζ5344539ζ53405313ζ53355318ζ5349534ζ53275326ζ5348535ζ53365317ζ53395314ζ5345538ζ53305323ζ535253ζ53325321ζ53435310ζ53415312ζ53345319ζ5350533ζ53285325ζ5347536ζ53375316ζ53385315ζ5346537ζ53295324ζ5351532ζ53335320ζ53425311ζ53315322    orthogonal faithful
ρ620ζ5348535ζ53345319ζ53435310ζ53395314ζ53385315ζ5344539ζ53335320ζ5349534ζ53285325ζ535253ζ53305323ζ5347536ζ53355318ζ53425311ζ53405313ζ53375316ζ5345538ζ53325321ζ5350533ζ53275326ζ5351532ζ53315322ζ5346537ζ53365317ζ53415312ζ53295324    orthogonal faithful
ρ720ζ53405313ζ5346537ζ53275326ζ5347536ζ53395314ζ53345319ζ535253ζ53325321ζ53415312ζ5345538ζ53285325ζ5348535ζ53385315ζ53355318ζ5351532ζ53315322ζ53425311ζ5344539ζ53295324ζ5349534ζ53375316ζ53365317ζ5350533ζ53305323ζ53435310ζ53335320    orthogonal faithful
ρ820ζ53285325ζ53425311ζ5350533ζ53365317ζ53315322ζ5345538ζ5347536ζ53335320ζ53345319ζ5348535ζ5344539ζ53305323ζ53375316ζ5351532ζ53415312ζ53275326ζ53405313ζ535253ζ53385315ζ53295324ζ53435310ζ5349534ζ53355318ζ53325321ζ5346537ζ53395314    orthogonal faithful
ρ920ζ53375316ζ53295324ζ53325321ζ53405313ζ5348535ζ5350533ζ53425311ζ53345319ζ53275326ζ53355318ζ53435310ζ5351532ζ5347536ζ53395314ζ53315322ζ53305323ζ53385315ζ5346537ζ535253ζ5344539ζ53365317ζ53285325ζ53335320ζ53415312ζ5349534ζ5345538    orthogonal faithful
ρ1020ζ53395314ζ53325321ζ53285325ζ53355318ζ53425311ζ5349534ζ5350533ζ53435310ζ53365317ζ53295324ζ53315322ζ53385315ζ5345538ζ535253ζ5347536ζ53405313ζ53335320ζ53275326ζ53345319ζ53415312ζ5348535ζ5351532ζ5344539ζ53375316ζ53305323ζ5346537    orthogonal faithful
ρ1120ζ53275326ζ53395314ζ535253ζ53415312ζ53285325ζ53385315ζ5351532ζ53425311ζ53295324ζ53375316ζ5350533ζ53435310ζ53305323ζ53365317ζ5349534ζ5344539ζ53315322ζ53355318ζ5348535ζ5345538ζ53325321ζ53345319ζ5347536ζ5346537ζ53335320ζ53405313    orthogonal faithful
ρ1220ζ53415312ζ53355318ζ53295324ζ53305323ζ53365317ζ53425311ζ5348535ζ535253ζ5346537ζ53405313ζ53345319ζ53285325ζ53315322ζ53375316ζ53435310ζ5349534ζ5351532ζ5345538ζ53395314ζ53335320ζ53275326ζ53325321ζ53385315ζ5344539ζ5350533ζ5347536    orthogonal faithful
ρ1320ζ53365317ζ535253ζ53345319ζ53375316ζ5351532ζ53335320ζ53385315ζ5350533ζ53325321ζ53395314ζ5349534ζ53315322ζ53405313ζ5348535ζ53305323ζ53415312ζ5347536ζ53295324ζ53425311ζ5346537ζ53285325ζ53435310ζ5345538ζ53275326ζ5344539ζ53355318    orthogonal faithful
ρ1420ζ5350533ζ53315322ζ5347536ζ53345319ζ5344539ζ53375316ζ53415312ζ53405313ζ53385315ζ53435310ζ53355318ζ5346537ζ53325321ζ5349534ζ53295324ζ535253ζ53275326ζ5351532ζ53305323ζ5348535ζ53335320ζ5345538ζ53365317ζ53425311ζ53395314ζ53285325    orthogonal faithful
ρ1520ζ535253ζ53285325ζ5351532ζ53295324ζ5350533ζ53305323ζ5349534ζ53315322ζ5348535ζ53325321ζ5347536ζ53335320ζ5346537ζ53345319ζ5345538ζ53355318ζ5344539ζ53365317ζ53435310ζ53375316ζ53425311ζ53385315ζ53415312ζ53395314ζ53405313ζ53275326    orthogonal faithful
ρ1620ζ53305323ζ5345538ζ5346537ζ53315322ζ53375316ζ535253ζ53395314ζ53295324ζ5344539ζ5347536ζ53325321ζ53365317ζ5351532ζ53405313ζ53285325ζ53435310ζ5348535ζ53335320ζ53355318ζ5350533ζ53415312ζ53275326ζ53425311ζ5349534ζ53345319ζ53385315    orthogonal faithful
ρ1720ζ53435310ζ53385315ζ53335320ζ53285325ζ53305323ζ53355318ζ53405313ζ5345538ζ5350533ζ5351532ζ5346537ζ53415312ζ53365317ζ53315322ζ53275326ζ53325321ζ53375316ζ53425311ζ5347536ζ535253ζ5349534ζ5344539ζ53395314ζ53345319ζ53295324ζ5348535    orthogonal faithful
ρ1820ζ53325321ζ5348535ζ53425311ζ53275326ζ53435310ζ5347536ζ53315322ζ53385315ζ535253ζ53365317ζ53335320ζ5349534ζ53415312ζ53285325ζ5344539ζ5346537ζ53305323ζ53395314ζ5351532ζ53355318ζ53345319ζ5350533ζ53405313ζ53295324ζ5345538ζ53375316    orthogonal faithful
ρ1920ζ5347536ζ5344539ζ53415312ζ53385315ζ53355318ζ53325321ζ53295324ζ53275326ζ53305323ζ53335320ζ53365317ζ53395314ζ53425311ζ5345538ζ5348535ζ5351532ζ535253ζ5349534ζ5346537ζ53435310ζ53405313ζ53375316ζ53345319ζ53315322ζ53285325ζ5350533    orthogonal faithful
ρ2020ζ53345319ζ5351532ζ53385315ζ53325321ζ5349534ζ53405313ζ53305323ζ5347536ζ53425311ζ53285325ζ5345538ζ5344539ζ53275326ζ53435310ζ5346537ζ53295324ζ53415312ζ5348535ζ53315322ζ53395314ζ5350533ζ53335320ζ53375316ζ535253ζ53355318ζ53365317    orthogonal faithful
ρ2120ζ53335320ζ53305323ζ53405313ζ5350533ζ5346537ζ53365317ζ53275326ζ53375316ζ5347536ζ5349534ζ53395314ζ53295324ζ53345319ζ5344539ζ535253ζ53425311ζ53325321ζ53315322ζ53415312ζ5351532ζ5345538ζ53355318ζ53285325ζ53385315ζ5348535ζ53435310    orthogonal faithful
ρ2220ζ5349534ζ5347536ζ5345538ζ53435310ζ53415312ζ53395314ζ53375316ζ53355318ζ53335320ζ53315322ζ53295324ζ53275326ζ53285325ζ53305323ζ53325321ζ53345319ζ53365317ζ53385315ζ53405313ζ53425311ζ5344539ζ5346537ζ5348535ζ5350533ζ535253ζ5351532    orthogonal faithful
ρ2320ζ53295324ζ53365317ζ5348535ζ5346537ζ53345319ζ53315322ζ53435310ζ5351532ζ53395314ζ53275326ζ53385315ζ5350533ζ5344539ζ53325321ζ53335320ζ5345538ζ5349534ζ53375316ζ53285325ζ53405313ζ535253ζ53425311ζ53305323ζ53355318ζ5347536ζ53415312    orthogonal faithful
ρ2420ζ5345538ζ53415312ζ53375316ζ53335320ζ53295324ζ53285325ζ53325321ζ53365317ζ53405313ζ5344539ζ5348535ζ535253ζ5350533ζ5346537ζ53425311ζ53385315ζ53345319ζ53305323ζ53275326ζ53315322ζ53355318ζ53395314ζ53435310ζ5347536ζ5351532ζ5349534    orthogonal faithful
ρ2520ζ5351532ζ5350533ζ5349534ζ5348535ζ5347536ζ5346537ζ5345538ζ5344539ζ53435310ζ53425311ζ53415312ζ53405313ζ53395314ζ53385315ζ53375316ζ53365317ζ53355318ζ53345319ζ53335320ζ53325321ζ53315322ζ53305323ζ53295324ζ53285325ζ53275326ζ535253    orthogonal faithful
ρ2620ζ53385315ζ5349534ζ53305323ζ53425311ζ5345538ζ53275326ζ5346537ζ53415312ζ53315322ζ5350533ζ53375316ζ53355318ζ535253ζ53335320ζ53395314ζ5348535ζ53295324ζ53435310ζ5344539ζ53285325ζ5347536ζ53405313ζ53325321ζ5351532ζ53365317ζ53345319    orthogonal faithful
ρ2720ζ53315322ζ53335320ζ5344539ζ5351532ζ53405313ζ53295324ζ53355318ζ5346537ζ5349534ζ53385315ζ53275326ζ53375316ζ5348535ζ5347536ζ53365317ζ53285325ζ53395314ζ5350533ζ5345538ζ53345319ζ53305323ζ53415312ζ535253ζ53435310ζ53325321ζ53425311    orthogonal faithful
ρ2820ζ53355318ζ53275326ζ53365317ζ5345538ζ535253ζ53435310ζ53345319ζ53285325ζ53375316ζ5346537ζ5351532ζ53425311ζ53335320ζ53295324ζ53385315ζ5347536ζ5350533ζ53415312ζ53325321ζ53305323ζ53395314ζ5348535ζ5349534ζ53405313ζ53315322ζ5344539    orthogonal faithful

Smallest permutation representation of D53
On 53 points: primitive
Generators in S53
(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)
(1 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)

G:=sub<Sym(53)| (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)>;

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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28) );

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)], [(1,53),(2,52),(3,51),(4,50),(5,49),(6,48),(7,47),(8,46),(9,45),(10,44),(11,43),(12,42),(13,41),(14,40),(15,39),(16,38),(17,37),(18,36),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28)])

D53 is a maximal subgroup of   C53⋊C4  D159
D53 is a maximal quotient of   Dic53  D159

Matrix representation of D53 in GL2(𝔽107) generated by

14106
10
,
14106
8893
G:=sub<GL(2,GF(107))| [14,1,106,0],[14,88,106,93] >;

D53 in GAP, Magma, Sage, TeX

D_{53}
% in TeX

G:=Group("D53");
// GroupNames label

G:=SmallGroup(106,1);
// by ID

G=gap.SmallGroup(106,1);
# by ID

G:=PCGroup([2,-2,-53,417]);
// Polycyclic

G:=Group<a,b|a^53=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D53 in TeX
Character table of D53 in TeX

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