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G = D35  order 70 = 2·5·7

Dihedral group

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

Aliases: D35, C7⋊D5, C5⋊D7, C35⋊1C2, sometimes denoted D70 or Dih35 or Dih70, SmallGroup(70,3)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C35 — D35
Chief series
C1 — C7 — C35 — D35
Lower central
C35 — D35
Upper central
C1

Generators and relations for D35
 G = < a,b | a35=b2=1, bab=a-1 >

C1
35C2
C5
C7
7D5
5D7
C35
D35

Character table of D35

 class 125A5B7A7B7C35A35B35C35D35E35F35G35H35I35J35K35L
 size 13522222222222222222
ρ11111111111111111111    trivial
ρ21-111111111111111111    linear of order 2
ρ320-1-√5/2-1+√5/2222-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ420-1+√5/2-1-√5/2222-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ52022ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ7ζ75+ζ72    orthogonal lifted from D7
ρ62022ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ72ζ74+ζ73    orthogonal lifted from D7
ρ72022ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ73ζ76+ζ7    orthogonal lifted from D7
ρ820-1-√5/2-1+√5/2ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ54ζ7+ζ5ζ76ζ54ζ76+ζ5ζ7ζ54ζ74+ζ5ζ73ζ54ζ72+ζ5ζ75ζ54ζ75+ζ5ζ72ζ53ζ75+ζ52ζ72ζ53ζ73+ζ52ζ74ζ53ζ7+ζ52ζ76ζ53ζ76+ζ52ζ7ζ53ζ74+ζ52ζ73ζ53ζ72+ζ52ζ75ζ54ζ73+ζ5ζ74    orthogonal faithful
ρ920-1+√5/2-1-√5/2ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ53ζ76+ζ52ζ7ζ53ζ7+ζ52ζ76ζ53ζ73+ζ52ζ74ζ53ζ75+ζ52ζ72ζ53ζ72+ζ52ζ75ζ54ζ75+ζ5ζ72ζ54ζ73+ζ5ζ74ζ54ζ7+ζ5ζ76ζ54ζ76+ζ5ζ7ζ54ζ74+ζ5ζ73ζ54ζ72+ζ5ζ75ζ53ζ74+ζ52ζ73    orthogonal faithful
ρ1020-1-√5/2-1+√5/2ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ54ζ76+ζ5ζ7ζ54ζ7+ζ5ζ76ζ54ζ73+ζ5ζ74ζ54ζ75+ζ5ζ72ζ54ζ72+ζ5ζ75ζ53ζ72+ζ52ζ75ζ53ζ74+ζ52ζ73ζ53ζ76+ζ52ζ7ζ53ζ7+ζ52ζ76ζ53ζ73+ζ52ζ74ζ53ζ75+ζ52ζ72ζ54ζ74+ζ5ζ73    orthogonal faithful
ρ1120-1+√5/2-1-√5/2ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ53ζ74+ζ52ζ73ζ53ζ73+ζ52ζ74ζ53ζ72+ζ52ζ75ζ53ζ7+ζ52ζ76ζ53ζ76+ζ52ζ7ζ54ζ7+ζ5ζ76ζ54ζ72+ζ5ζ75ζ54ζ73+ζ5ζ74ζ54ζ74+ζ5ζ73ζ54ζ75+ζ5ζ72ζ54ζ76+ζ5ζ7ζ53ζ75+ζ52ζ72    orthogonal faithful
ρ1220-1-√5/2-1+√5/2ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ54ζ73+ζ5ζ74ζ54ζ74+ζ5ζ73ζ54ζ75+ζ5ζ72ζ54ζ76+ζ5ζ7ζ54ζ7+ζ5ζ76ζ53ζ7+ζ52ζ76ζ53ζ72+ζ52ζ75ζ53ζ73+ζ52ζ74ζ53ζ74+ζ52ζ73ζ53ζ75+ζ52ζ72ζ53ζ76+ζ52ζ7ζ54ζ72+ζ5ζ75    orthogonal faithful
ρ1320-1-√5/2-1+√5/2ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ54ζ75+ζ5ζ72ζ54ζ72+ζ5ζ75ζ54ζ76+ζ5ζ7ζ54ζ73+ζ5ζ74ζ54ζ74+ζ5ζ73ζ53ζ74+ζ52ζ73ζ53ζ7+ζ52ζ76ζ53ζ75+ζ52ζ72ζ53ζ72+ζ52ζ75ζ53ζ76+ζ52ζ7ζ53ζ73+ζ52ζ74ζ54ζ7+ζ5ζ76    orthogonal faithful
ρ1420-1+√5/2-1-√5/2ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ53ζ75+ζ52ζ72ζ53ζ72+ζ52ζ75ζ53ζ76+ζ52ζ7ζ53ζ73+ζ52ζ74ζ53ζ74+ζ52ζ73ζ54ζ73+ζ5ζ74ζ54ζ76+ζ5ζ7ζ54ζ72+ζ5ζ75ζ54ζ75+ζ5ζ72ζ54ζ7+ζ5ζ76ζ54ζ74+ζ5ζ73ζ53ζ7+ζ52ζ76    orthogonal faithful
ρ1520-1-√5/2-1+√5/2ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ54ζ74+ζ5ζ73ζ54ζ73+ζ5ζ74ζ54ζ72+ζ5ζ75ζ54ζ7+ζ5ζ76ζ54ζ76+ζ5ζ7ζ53ζ76+ζ52ζ7ζ53ζ75+ζ52ζ72ζ53ζ74+ζ52ζ73ζ53ζ73+ζ52ζ74ζ53ζ72+ζ52ζ75ζ53ζ7+ζ52ζ76ζ54ζ75+ζ5ζ72    orthogonal faithful
ρ1620-1+√5/2-1-√5/2ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ53ζ72+ζ52ζ75ζ53ζ75+ζ52ζ72ζ53ζ7+ζ52ζ76ζ53ζ74+ζ52ζ73ζ53ζ73+ζ52ζ74ζ54ζ74+ζ5ζ73ζ54ζ7+ζ5ζ76ζ54ζ75+ζ5ζ72ζ54ζ72+ζ5ζ75ζ54ζ76+ζ5ζ7ζ54ζ73+ζ5ζ74ζ53ζ76+ζ52ζ7    orthogonal faithful
ρ1720-1+√5/2-1-√5/2ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ53ζ7+ζ52ζ76ζ53ζ76+ζ52ζ7ζ53ζ74+ζ52ζ73ζ53ζ72+ζ52ζ75ζ53ζ75+ζ52ζ72ζ54ζ72+ζ5ζ75ζ54ζ74+ζ5ζ73ζ54ζ76+ζ5ζ7ζ54ζ7+ζ5ζ76ζ54ζ73+ζ5ζ74ζ54ζ75+ζ5ζ72ζ53ζ73+ζ52ζ74    orthogonal faithful
ρ1820-1-√5/2-1+√5/2ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ54ζ72+ζ5ζ75ζ54ζ75+ζ5ζ72ζ54ζ7+ζ5ζ76ζ54ζ74+ζ5ζ73ζ54ζ73+ζ5ζ74ζ53ζ73+ζ52ζ74ζ53ζ76+ζ52ζ7ζ53ζ72+ζ52ζ75ζ53ζ75+ζ52ζ72ζ53ζ7+ζ52ζ76ζ53ζ74+ζ52ζ73ζ54ζ76+ζ5ζ7    orthogonal faithful
ρ1920-1+√5/2-1-√5/2ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ53ζ73+ζ52ζ74ζ53ζ74+ζ52ζ73ζ53ζ75+ζ52ζ72ζ53ζ76+ζ52ζ7ζ53ζ7+ζ52ζ76ζ54ζ76+ζ5ζ7ζ54ζ75+ζ5ζ72ζ54ζ74+ζ5ζ73ζ54ζ73+ζ5ζ74ζ54ζ72+ζ5ζ75ζ54ζ7+ζ5ζ76ζ53ζ72+ζ52ζ75    orthogonal faithful

Smallest permutation representation of D35
►On 35 points
Generators in S35
(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)

(1 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)
 
G:=sub<Sym(35)| (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)>;
 
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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19) );
 
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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19)]])
 

D35 is a maximal subgroup of   D5×D7  C5⋊F7  D105  D175  C5⋊D35  D245  C7⋊D35
D35 is a maximal quotient of   Dic35  D105  D175  C5⋊D35  D245  C7⋊D35

Matrix representation of D35 ►in GL2(𝔽71) generated by

441
306
,
441
3667
G:=sub<GL(2,GF(71))| [4,30,41,6],[4,36,41,67] >;
 

D35 in GAP, Magma, Sage, TeX 

D_{35}
 
% in TeX
 
G:=Group("D35");
 
// GroupNames label
 
G:=SmallGroup(70,3);
 
// by ID
 
G=gap.SmallGroup(70,3);
 
# by ID
 
G:=PCGroup([3,-2,-5,-7,49,542]);
 
// Polycyclic
 
G:=Group<a,b|a^35=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D35 in TeX
Character table of D35 in TeX

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