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G = D47  order 94 = 2·47

Dihedral group

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

Aliases: D47, C47⋊C2, sometimes denoted D94 or Dih47 or Dih94, SmallGroup(94,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C47 — D47
Chief series
C1 — C47 — D47
Lower central
C47 — D47
Upper central
C1

Generators and relations for D47
 G = < a,b | a47=b2=1, bab=a-1 >

C1
47C2
C47
D47

Character table of D47

 class 1247A47B47C47D47E47F47G47H47I47J47K47L47M47N47O47P47Q47R47S47T47U47V47W
 size 14722222222222222222222222
ρ11111111111111111111111111    trivial
ρ21-111111111111111111111111    linear of order 2
ρ320ζ4725+ζ4722ζ4733+ζ4714ζ4744+ζ473ζ4739+ζ478ζ4728+ζ4719ζ4730+ζ4717ζ4741+ζ476ζ4742+ζ475ζ4731+ζ4716ζ4727+ζ4720ζ4738+ζ479ζ4745+ζ472ζ4734+ζ4713ζ4724+ζ4723ζ4735+ζ4712ζ4746+ζ47ζ4737+ζ4710ζ4726+ζ4721ζ4732+ζ4715ζ4743+ζ474ζ4740+ζ477ζ4729+ζ4718ζ4736+ζ4711    orthogonal faithful
ρ420ζ4732+ζ4715ζ4746+ζ47ζ4730+ζ4717ζ4733+ζ4714ζ4745+ζ472ζ4729+ζ4718ζ4734+ζ4713ζ4744+ζ473ζ4728+ζ4719ζ4735+ζ4712ζ4743+ζ474ζ4727+ζ4720ζ4736+ζ4711ζ4742+ζ475ζ4726+ζ4721ζ4737+ζ4710ζ4741+ζ476ζ4725+ζ4722ζ4738+ζ479ζ4740+ζ477ζ4724+ζ4723ζ4739+ζ478ζ4731+ζ4716    orthogonal faithful
ρ520ζ4734+ζ4713ζ4743+ζ474ζ4726+ζ4721ζ4738+ζ479ζ4739+ζ478ζ4725+ζ4722ζ4742+ζ475ζ4735+ζ4712ζ4729+ζ4718ζ4746+ζ47ζ4731+ζ4716ζ4733+ζ4714ζ4744+ζ473ζ4727+ζ4720ζ4737+ζ4710ζ4740+ζ477ζ4724+ζ4723ζ4741+ζ476ζ4736+ζ4711ζ4728+ζ4719ζ4745+ζ472ζ4732+ζ4715ζ4730+ζ4717    orthogonal faithful
ρ620ζ4745+ζ472ζ4744+ζ473ζ4743+ζ474ζ4742+ζ475ζ4741+ζ476ζ4740+ζ477ζ4739+ζ478ζ4738+ζ479ζ4737+ζ4710ζ4736+ζ4711ζ4735+ζ4712ζ4734+ζ4713ζ4733+ζ4714ζ4732+ζ4715ζ4731+ζ4716ζ4730+ζ4717ζ4729+ζ4718ζ4728+ζ4719ζ4727+ζ4720ζ4726+ζ4721ζ4725+ζ4722ζ4724+ζ4723ζ4746+ζ47    orthogonal faithful
ρ720ζ4735+ζ4712ζ4729+ζ4718ζ4724+ζ4723ζ4730+ζ4717ζ4736+ζ4711ζ4742+ζ475ζ4746+ζ47ζ4740+ζ477ζ4734+ζ4713ζ4728+ζ4719ζ4725+ζ4722ζ4731+ζ4716ζ4737+ζ4710ζ4743+ζ474ζ4745+ζ472ζ4739+ζ478ζ4733+ζ4714ζ4727+ζ4720ζ4726+ζ4721ζ4732+ζ4715ζ4738+ζ479ζ4744+ζ473ζ4741+ζ476    orthogonal faithful
ρ820ζ4741+ζ476ζ4738+ζ479ζ4735+ζ4712ζ4732+ζ4715ζ4729+ζ4718ζ4726+ζ4721ζ4724+ζ4723ζ4727+ζ4720ζ4730+ζ4717ζ4733+ζ4714ζ4736+ζ4711ζ4739+ζ478ζ4742+ζ475ζ4745+ζ472ζ4746+ζ47ζ4743+ζ474ζ4740+ζ477ζ4737+ζ4710ζ4734+ζ4713ζ4731+ζ4716ζ4728+ζ4719ζ4725+ζ4722ζ4744+ζ473    orthogonal faithful
ρ920ζ4739+ζ478ζ4735+ζ4712ζ4731+ζ4716ζ4727+ζ4720ζ4724+ζ4723ζ4728+ζ4719ζ4732+ζ4715ζ4736+ζ4711ζ4740+ζ477ζ4744+ζ473ζ4746+ζ47ζ4742+ζ475ζ4738+ζ479ζ4734+ζ4713ζ4730+ζ4717ζ4726+ζ4721ζ4725+ζ4722ζ4729+ζ4718ζ4733+ζ4714ζ4737+ζ4710ζ4741+ζ476ζ4745+ζ472ζ4743+ζ474    orthogonal faithful
ρ1020ζ4746+ζ47ζ4725+ζ4722ζ4745+ζ472ζ4726+ζ4721ζ4744+ζ473ζ4727+ζ4720ζ4743+ζ474ζ4728+ζ4719ζ4742+ζ475ζ4729+ζ4718ζ4741+ζ476ζ4730+ζ4717ζ4740+ζ477ζ4731+ζ4716ζ4739+ζ478ζ4732+ζ4715ζ4738+ζ479ζ4733+ζ4714ζ4737+ζ4710ζ4734+ζ4713ζ4736+ζ4711ζ4735+ζ4712ζ4724+ζ4723    orthogonal faithful
ρ1120ζ4736+ζ4711ζ4740+ζ477ζ4725+ζ4722ζ4743+ζ474ζ4733+ζ4714ζ4732+ζ4715ζ4744+ζ473ζ4726+ζ4721ζ4739+ζ478ζ4737+ζ4710ζ4728+ζ4719ζ4746+ζ47ζ4730+ζ4717ζ4735+ζ4712ζ4741+ζ476ζ4724+ζ4723ζ4742+ζ475ζ4734+ζ4713ζ4731+ζ4716ζ4745+ζ472ζ4727+ζ4720ζ4738+ζ479ζ4729+ζ4718    orthogonal faithful
ρ1220ζ4729+ζ4718ζ4727+ζ4720ζ4736+ζ4711ζ4745+ζ472ζ4740+ζ477ζ4731+ζ4716ζ4725+ζ4722ζ4734+ζ4713ζ4743+ζ474ζ4742+ζ475ζ4733+ζ4714ζ4724+ζ4723ζ4732+ζ4715ζ4741+ζ476ζ4744+ζ473ζ4735+ζ4712ζ4726+ζ4721ζ4730+ζ4717ζ4739+ζ478ζ4746+ζ47ζ4737+ζ4710ζ4728+ζ4719ζ4738+ζ479    orthogonal faithful
ρ1320ζ4724+ζ4723ζ4736+ζ4711ζ4746+ζ47ζ4734+ζ4713ζ4725+ζ4722ζ4737+ζ4710ζ4745+ζ472ζ4733+ζ4714ζ4726+ζ4721ζ4738+ζ479ζ4744+ζ473ζ4732+ζ4715ζ4727+ζ4720ζ4739+ζ478ζ4743+ζ474ζ4731+ζ4716ζ4728+ζ4719ζ4740+ζ477ζ4742+ζ475ζ4730+ζ4717ζ4729+ζ4718ζ4741+ζ476ζ4735+ζ4712    orthogonal faithful
ρ1420ζ4737+ζ4710ζ4732+ζ4715ζ4727+ζ4720ζ4725+ζ4722ζ4730+ζ4717ζ4735+ζ4712ζ4740+ζ477ζ4745+ζ472ζ4744+ζ473ζ4739+ζ478ζ4734+ζ4713ζ4729+ζ4718ζ4724+ζ4723ζ4728+ζ4719ζ4733+ζ4714ζ4738+ζ479ζ4743+ζ474ζ4746+ζ47ζ4741+ζ476ζ4736+ζ4711ζ4731+ζ4716ζ4726+ζ4721ζ4742+ζ475    orthogonal faithful
ρ1520ζ4742+ζ475ζ4731+ζ4716ζ4737+ζ4710ζ4736+ζ4711ζ4732+ζ4715ζ4741+ζ476ζ4727+ζ4720ζ4746+ζ47ζ4725+ζ4722ζ4743+ζ474ζ4730+ζ4717ζ4738+ζ479ζ4735+ζ4712ζ4733+ζ4714ζ4740+ζ477ζ4728+ζ4719ζ4745+ζ472ζ4724+ζ4723ζ4744+ζ473ζ4729+ζ4718ζ4739+ζ478ζ4734+ζ4713ζ4726+ζ4721    orthogonal faithful
ρ1620ζ4727+ζ4720ζ4730+ζ4717ζ4740+ζ477ζ4744+ζ473ζ4734+ζ4713ζ4724+ζ4723ζ4733+ζ4714ζ4743+ζ474ζ4741+ζ476ζ4731+ζ4716ζ4726+ζ4721ζ4736+ζ4711ζ4746+ζ47ζ4738+ζ479ζ4728+ζ4719ζ4729+ζ4718ζ4739+ζ478ζ4745+ζ472ζ4735+ζ4712ζ4725+ζ4722ζ4732+ζ4715ζ4742+ζ475ζ4737+ζ4710    orthogonal faithful
ρ1720ζ4726+ζ4721ζ4739+ζ478ζ4742+ζ475ζ4729+ζ4718ζ4731+ζ4716ζ4744+ζ473ζ4737+ζ4710ζ4724+ζ4723ζ4736+ζ4711ζ4745+ζ472ζ4732+ζ4715ζ4728+ζ4719ζ4741+ζ476ζ4740+ζ477ζ4727+ζ4720ζ4733+ζ4714ζ4746+ζ47ζ4735+ζ4712ζ4725+ζ4722ζ4738+ζ479ζ4743+ζ474ζ4730+ζ4717ζ4734+ζ4713    orthogonal faithful
ρ1820ζ4730+ζ4717ζ4745+ζ472ζ4734+ζ4713ζ4728+ζ4719ζ4743+ζ474ζ4736+ζ4711ζ4726+ζ4721ζ4741+ζ476ζ4738+ζ479ζ4724+ζ4723ζ4739+ζ478ζ4740+ζ477ζ4725+ζ4722ζ4737+ζ4710ζ4742+ζ475ζ4727+ζ4720ζ4735+ζ4712ζ4744+ζ473ζ4729+ζ4718ζ4733+ζ4714ζ4746+ζ47ζ4731+ζ4716ζ4732+ζ4715    orthogonal faithful
ρ1920ζ4738+ζ479ζ4737+ζ4710ζ4729+ζ4718ζ4746+ζ47ζ4727+ζ4720ζ4739+ζ478ζ4736+ζ4711ζ4730+ζ4717ζ4745+ζ472ζ4726+ζ4721ζ4740+ζ477ζ4735+ζ4712ζ4731+ζ4716ζ4744+ζ473ζ4725+ζ4722ζ4741+ζ476ζ4734+ζ4713ζ4732+ζ4715ζ4743+ζ474ζ4724+ζ4723ζ4742+ζ475ζ4733+ζ4714ζ4728+ζ4719    orthogonal faithful
ρ2020ζ4728+ζ4719ζ4742+ζ475ζ4738+ζ479ζ4724+ζ4723ζ4737+ζ4710ζ4743+ζ474ζ4729+ζ4718ζ4732+ζ4715ζ4746+ζ47ζ4734+ζ4713ζ4727+ζ4720ζ4741+ζ476ζ4739+ζ478ζ4725+ζ4722ζ4736+ζ4711ζ4744+ζ473ζ4730+ζ4717ζ4731+ζ4716ζ4745+ζ472ζ4735+ζ4712ζ4726+ζ4721ζ4740+ζ477ζ4733+ζ4714    orthogonal faithful
ρ2120ζ4744+ζ473ζ4728+ζ4719ζ4741+ζ476ζ4731+ζ4716ζ4738+ζ479ζ4734+ζ4713ζ4735+ζ4712ζ4737+ζ4710ζ4732+ζ4715ζ4740+ζ477ζ4729+ζ4718ζ4743+ζ474ζ4726+ζ4721ζ4746+ζ47ζ4724+ζ4723ζ4745+ζ472ζ4727+ζ4720ζ4742+ζ475ζ4730+ζ4717ζ4739+ζ478ζ4733+ζ4714ζ4736+ζ4711ζ4725+ζ4722    orthogonal faithful
ρ2220ζ4743+ζ474ζ4741+ζ476ζ4739+ζ478ζ4737+ζ4710ζ4735+ζ4712ζ4733+ζ4714ζ4731+ζ4716ζ4729+ζ4718ζ4727+ζ4720ζ4725+ζ4722ζ4724+ζ4723ζ4726+ζ4721ζ4728+ζ4719ζ4730+ζ4717ζ4732+ζ4715ζ4734+ζ4713ζ4736+ζ4711ζ4738+ζ479ζ4740+ζ477ζ4742+ζ475ζ4744+ζ473ζ4746+ζ47ζ4745+ζ472    orthogonal faithful
ρ2320ζ4731+ζ4716ζ4724+ζ4723ζ4732+ζ4715ζ4740+ζ477ζ4746+ζ47ζ4738+ζ479ζ4730+ζ4717ζ4725+ζ4722ζ4733+ζ4714ζ4741+ζ476ζ4745+ζ472ζ4737+ζ4710ζ4729+ζ4718ζ4726+ζ4721ζ4734+ζ4713ζ4742+ζ475ζ4744+ζ473ζ4736+ζ4711ζ4728+ζ4719ζ4727+ζ4720ζ4735+ζ4712ζ4743+ζ474ζ4739+ζ478    orthogonal faithful
ρ2420ζ4733+ζ4714ζ4726+ζ4721ζ4728+ζ4719ζ4735+ζ4712ζ4742+ζ475ζ4745+ζ472ζ4738+ζ479ζ4731+ζ4716ζ4724+ζ4723ζ4730+ζ4717ζ4737+ζ4710ζ4744+ζ473ζ4743+ζ474ζ4736+ζ4711ζ4729+ζ4718ζ4725+ζ4722ζ4732+ζ4715ζ4739+ζ478ζ4746+ζ47ζ4741+ζ476ζ4734+ζ4713ζ4727+ζ4720ζ4740+ζ477    orthogonal faithful
ρ2520ζ4740+ζ477ζ4734+ζ4713ζ4733+ζ4714ζ4741+ζ476ζ4726+ζ4721ζ4746+ζ47ζ4728+ζ4719ζ4739+ζ478ζ4735+ζ4712ζ4732+ζ4715ζ4742+ζ475ζ4725+ζ4722ζ4745+ζ472ζ4729+ζ4718ζ4738+ζ479ζ4736+ζ4711ζ4731+ζ4716ζ4743+ζ474ζ4724+ζ4723ζ4744+ζ473ζ4730+ζ4717ζ4737+ζ4710ζ4727+ζ4720    orthogonal faithful

Smallest permutation representation of D47
►On 47 points: primitive
Generators in S47
(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)

(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)
 
G:=sub<Sym(47)| (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)>;
 
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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25) );
 
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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25)]])
 

D47 is a maximal subgroup of   D141  D235
D47 is a maximal quotient of   Dic47  D141  D235

Matrix representation of D47 ►in GL2(𝔽283) generated by

27282
10
,
27282
162256
G:=sub<GL(2,GF(283))| [27,1,282,0],[27,162,282,256] >;
 

D47 in GAP, Magma, Sage, TeX 

D_{47}
 
% in TeX
 
G:=Group("D47");
 
// GroupNames label
 
G:=SmallGroup(94,1);
 
// by ID
 
G=gap.SmallGroup(94,1);
 
# by ID
 
G:=PCGroup([2,-2,-47,369]);
 
// Polycyclic
 
G:=Group<a,b|a^47=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D47 in TeX
Character table of D47 in TeX

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