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G = D31  order 62 = 2·31

Dihedral group

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

Aliases: D31, C31⋊C2, sometimes denoted D62 or Dih31 or Dih62, SmallGroup(62,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C31 — D31
Chief series
C1 — C31 — D31
Lower central
C31 — D31
Upper central
C1

Generators and relations for D31
 G = < a,b | a31=b2=1, bab=a-1 >

C1
31C2
C31
D31

Character table of D31

 class 1231A31B31C31D31E31F31G31H31I31J31K31L31M31N31O
 size 131222222222222222
ρ111111111111111111    trivial
ρ21-1111111111111111    linear of order 2
ρ320ζ3127+ζ314ζ3125+ζ316ζ3123+ζ318ζ3121+ζ3110ζ3119+ζ3112ζ3117+ζ3114ζ3116+ζ3115ζ3118+ζ3113ζ3120+ζ3111ζ3122+ζ319ζ3124+ζ317ζ3126+ζ315ζ3128+ζ313ζ3130+ζ31ζ3129+ζ312    orthogonal faithful
ρ420ζ3124+ζ317ζ3126+ζ315ζ3117+ζ3114ζ3129+ζ312ζ3121+ζ3110ζ3122+ζ319ζ3128+ζ313ζ3116+ζ3115ζ3127+ζ314ζ3123+ζ318ζ3120+ζ3111ζ3130+ζ31ζ3118+ζ3113ζ3125+ζ316ζ3119+ζ3112    orthogonal faithful
ρ520ζ3130+ζ31ζ3117+ζ3114ζ3129+ζ312ζ3118+ζ3113ζ3128+ζ313ζ3119+ζ3112ζ3127+ζ314ζ3120+ζ3111ζ3126+ζ315ζ3121+ζ3110ζ3125+ζ316ζ3122+ζ319ζ3124+ζ317ζ3123+ζ318ζ3116+ζ3115    orthogonal faithful
ρ620ζ3117+ζ3114ζ3121+ζ3110ζ3128+ζ313ζ3127+ζ314ζ3120+ζ3111ζ3118+ζ3113ζ3125+ζ316ζ3130+ζ31ζ3123+ζ318ζ3116+ζ3115ζ3122+ζ319ζ3129+ζ312ζ3126+ζ315ζ3119+ζ3112ζ3124+ζ317    orthogonal faithful
ρ720ζ3126+ζ315ζ3123+ζ318ζ3121+ζ3110ζ3128+ζ313ζ3116+ζ3115ζ3129+ζ312ζ3120+ζ3111ζ3124+ζ317ζ3125+ζ316ζ3119+ζ3112ζ3130+ζ31ζ3117+ζ3114ζ3127+ζ314ζ3122+ζ319ζ3118+ζ3113    orthogonal faithful
ρ820ζ3116+ζ3115ζ3124+ζ317ζ3130+ζ31ζ3122+ζ319ζ3117+ζ3114ζ3125+ζ316ζ3129+ζ312ζ3121+ζ3110ζ3118+ζ3113ζ3126+ζ315ζ3128+ζ313ζ3120+ζ3111ζ3119+ζ3112ζ3127+ζ314ζ3123+ζ318    orthogonal faithful
ρ920ζ3125+ζ316ζ3122+ζ319ζ3119+ζ3112ζ3116+ζ3115ζ3118+ζ3113ζ3121+ζ3110ζ3124+ζ317ζ3127+ζ314ζ3130+ζ31ζ3129+ζ312ζ3126+ζ315ζ3123+ζ318ζ3120+ζ3111ζ3117+ζ3114ζ3128+ζ313    orthogonal faithful
ρ1020ζ3129+ζ312ζ3128+ζ313ζ3127+ζ314ζ3126+ζ315ζ3125+ζ316ζ3124+ζ317ζ3123+ζ318ζ3122+ζ319ζ3121+ζ3110ζ3120+ζ3111ζ3119+ζ3112ζ3118+ζ3113ζ3117+ζ3114ζ3116+ζ3115ζ3130+ζ31    orthogonal faithful
ρ1120ζ3128+ζ313ζ3120+ζ3111ζ3125+ζ316ζ3123+ζ318ζ3122+ζ319ζ3126+ζ315ζ3119+ζ3112ζ3129+ζ312ζ3116+ζ3115ζ3130+ζ31ζ3118+ζ3113ζ3127+ζ314ζ3121+ζ3110ζ3124+ζ317ζ3117+ζ3114    orthogonal faithful
ρ1220ζ3121+ζ3110ζ3116+ζ3115ζ3120+ζ3111ζ3125+ζ316ζ3130+ζ31ζ3127+ζ314ζ3122+ζ319ζ3117+ζ3114ζ3119+ζ3112ζ3124+ζ317ζ3129+ζ312ζ3128+ζ313ζ3123+ζ318ζ3118+ζ3113ζ3126+ζ315    orthogonal faithful
ρ1320ζ3122+ζ319ζ3129+ζ312ζ3118+ζ3113ζ3124+ζ317ζ3127+ζ314ζ3116+ζ3115ζ3126+ζ315ζ3125+ζ316ζ3117+ζ3114ζ3128+ζ313ζ3123+ζ318ζ3119+ζ3112ζ3130+ζ31ζ3121+ζ3110ζ3120+ζ3111    orthogonal faithful
ρ1420ζ3123+ζ318ζ3119+ζ3112ζ3116+ζ3115ζ3120+ζ3111ζ3124+ζ317ζ3128+ζ313ζ3130+ζ31ζ3126+ζ315ζ3122+ζ319ζ3118+ζ3113ζ3117+ζ3114ζ3121+ζ3110ζ3125+ζ316ζ3129+ζ312ζ3127+ζ314    orthogonal faithful
ρ1520ζ3118+ζ3113ζ3127+ζ314ζ3126+ζ315ζ3117+ζ3114ζ3123+ζ318ζ3130+ζ31ζ3121+ζ3110ζ3119+ζ3112ζ3128+ζ313ζ3125+ζ316ζ3116+ζ3115ζ3124+ζ317ζ3129+ζ312ζ3120+ζ3111ζ3122+ζ319    orthogonal faithful
ρ1620ζ3119+ζ3112ζ3118+ζ3113ζ3124+ζ317ζ3130+ζ31ζ3126+ζ315ζ3120+ζ3111ζ3117+ζ3114ζ3123+ζ318ζ3129+ζ312ζ3127+ζ314ζ3121+ζ3110ζ3116+ζ3115ζ3122+ζ319ζ3128+ζ313ζ3125+ζ316    orthogonal faithful
ρ1720ζ3120+ζ3111ζ3130+ζ31ζ3122+ζ319ζ3119+ζ3112ζ3129+ζ312ζ3123+ζ318ζ3118+ζ3113ζ3128+ζ313ζ3124+ζ317ζ3117+ζ3114ζ3127+ζ314ζ3125+ζ316ζ3116+ζ3115ζ3126+ζ315ζ3121+ζ3110    orthogonal faithful

Permutation representations of D31
►On 31 points: primitive - transitive group 31T2
Generators in S31
(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)

(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)
 
G:=sub<Sym(31)| (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)>;
 
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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17) );
 
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)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17)]])
 
G:=TransitiveGroup(31,2);
 

D31 is a maximal subgroup of   C31⋊C6  D93  C31⋊C10  D155  D217
D31 is a maximal quotient of   Dic31  D93  D155  D217

Matrix representation of D31 ►in GL2(𝔽311) generated by

200310
201310
,
9199
134302
G:=sub<GL(2,GF(311))| [200,201,310,310],[9,134,199,302] >;
 

D31 in GAP, Magma, Sage, TeX 

D_{31}
 
% in TeX
 
G:=Group("D31");
 
// GroupNames label
 
G:=SmallGroup(62,1);
 
// by ID
 
G=gap.SmallGroup(62,1);
 
# by ID
 
G:=PCGroup([2,-2,-31,241]);
 
// Polycyclic
 
G:=Group<a,b|a^31=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D31 in TeX
Character table of D31 in TeX

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