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G = D31order 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

C1C31 — D31
C1C31 — D31
C31 — D31
C1

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

31C2

Character table of D31

 class 1231A31B31C31D31E31F31G31H31I31J31K31L31M31N31O
 size 131222222222222222
ρ111111111111111111    trivial
ρ21-1111111111111111    linear of order 2
ρ320ζ3127314ζ3125316ζ3123318ζ31213110ζ31193112ζ31173114ζ31163115ζ31183113ζ31203111ζ3122319ζ3124317ζ3126315ζ3128313ζ313031ζ3129312    orthogonal faithful
ρ420ζ3124317ζ3126315ζ31173114ζ3129312ζ31213110ζ3122319ζ3128313ζ31163115ζ3127314ζ3123318ζ31203111ζ313031ζ31183113ζ3125316ζ31193112    orthogonal faithful
ρ520ζ313031ζ31173114ζ3129312ζ31183113ζ3128313ζ31193112ζ3127314ζ31203111ζ3126315ζ31213110ζ3125316ζ3122319ζ3124317ζ3123318ζ31163115    orthogonal faithful
ρ620ζ31173114ζ31213110ζ3128313ζ3127314ζ31203111ζ31183113ζ3125316ζ313031ζ3123318ζ31163115ζ3122319ζ3129312ζ3126315ζ31193112ζ3124317    orthogonal faithful
ρ720ζ3126315ζ3123318ζ31213110ζ3128313ζ31163115ζ3129312ζ31203111ζ3124317ζ3125316ζ31193112ζ313031ζ31173114ζ3127314ζ3122319ζ31183113    orthogonal faithful
ρ820ζ31163115ζ3124317ζ313031ζ3122319ζ31173114ζ3125316ζ3129312ζ31213110ζ31183113ζ3126315ζ3128313ζ31203111ζ31193112ζ3127314ζ3123318    orthogonal faithful
ρ920ζ3125316ζ3122319ζ31193112ζ31163115ζ31183113ζ31213110ζ3124317ζ3127314ζ313031ζ3129312ζ3126315ζ3123318ζ31203111ζ31173114ζ3128313    orthogonal faithful
ρ1020ζ3129312ζ3128313ζ3127314ζ3126315ζ3125316ζ3124317ζ3123318ζ3122319ζ31213110ζ31203111ζ31193112ζ31183113ζ31173114ζ31163115ζ313031    orthogonal faithful
ρ1120ζ3128313ζ31203111ζ3125316ζ3123318ζ3122319ζ3126315ζ31193112ζ3129312ζ31163115ζ313031ζ31183113ζ3127314ζ31213110ζ3124317ζ31173114    orthogonal faithful
ρ1220ζ31213110ζ31163115ζ31203111ζ3125316ζ313031ζ3127314ζ3122319ζ31173114ζ31193112ζ3124317ζ3129312ζ3128313ζ3123318ζ31183113ζ3126315    orthogonal faithful
ρ1320ζ3122319ζ3129312ζ31183113ζ3124317ζ3127314ζ31163115ζ3126315ζ3125316ζ31173114ζ3128313ζ3123318ζ31193112ζ313031ζ31213110ζ31203111    orthogonal faithful
ρ1420ζ3123318ζ31193112ζ31163115ζ31203111ζ3124317ζ3128313ζ313031ζ3126315ζ3122319ζ31183113ζ31173114ζ31213110ζ3125316ζ3129312ζ3127314    orthogonal faithful
ρ1520ζ31183113ζ3127314ζ3126315ζ31173114ζ3123318ζ313031ζ31213110ζ31193112ζ3128313ζ3125316ζ31163115ζ3124317ζ3129312ζ31203111ζ3122319    orthogonal faithful
ρ1620ζ31193112ζ31183113ζ3124317ζ313031ζ3126315ζ31203111ζ31173114ζ3123318ζ3129312ζ3127314ζ31213110ζ31163115ζ3122319ζ3128313ζ3125316    orthogonal faithful
ρ1720ζ31203111ζ313031ζ3122319ζ31193112ζ3129312ζ3123318ζ31183113ζ3128313ζ3124317ζ31173114ζ3127314ζ3125316ζ31163115ζ3126315ζ31213110    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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