D8oD8 - GroupNames

G = D₈oD₈ order 128 = 2⁷

Central product of D₈ and D₈

p-group, metabelian, nilpotent (class 3), monomial

Aliases: D₈ o D₈, Q₁₆ o Q₁₆, D₈:₁₄D₄, Q₁₆:₁₄D₄, SD₁₆:₄D₄, C₄².₄₅₇C₂³, M₄(2).₁₇C₂³, 2₊¹⁺⁴:₄C₂², C₂².₅2₊¹⁺⁴, C₂.₇₆D₄², D₄oD₈:₄C₂, C₈oD₈:₁₀C₂, C₈.₁₃(C₂xD₄), D₄:₄D₄:₇C₂, C₈:₄D₄:₂₂C₂, (C₄xC₈):₃₅C₂², D₄.₃₅(C₂xD₄), C₈oD₄:₆C₂², C₄wrC₂:₁₄C₂², Q₈.₃₅(C₂xD₄), D₄.₄D₄:₇C₂, (C₂xD₈):₃₁C₂², C₈:C₂²:₃C₂², (C₂xC₄).₂₅C₂⁴, C₄:₁D₄:₁₄C₂², (C₂xC₈).₂₈₉C₂³, C₄oD₄.₁₄C₂³, C₄oD₈.₂₉C₂², (C₂xD₄).₁₁C₂³, C₄.D₄:₅C₂², C₄.₁₀₆(C₂²xD₄), C₈.C₄:₂₁C₂², 2-Sylow(Omega+(4,7)), SmallGroup(128,2024)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂xC₄ — D₈oD₈

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₄oD₄ — 2₊¹⁺⁴ — D₄oD₈ — D₈oD₈

Lower central

C₁ — C₂ — C₂xC₄ — D₈oD₈

Upper central

C₁ — C₂ — C₂xC₄ — D₈oD₈

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — D₈oD₈

Generators and relations for D₈oD₈
G = < a,b,c,d | a⁸=b²=d²=1, c⁴=a⁴, bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a⁴c³ >

Subgroups: 572 in 238 conjugacy classes, 92 normal (8 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, C₂³, C₄², C₂xC₈, C₂xC₈, M₄(2), M₄(2), D₈, D₈, SD₁₆, SD₁₆, Q₁₆, C₂xD₄, C₂xD₄, C₄oD₄, C₄oD₄, C₄xC₈, C₄.D₄, C₄wrC₂, C₈.C₄, C₄:₁D₄, C₈oD₄, C₂xD₈, C₂xD₈, C₄oD₈, C₄oD₈, C₈:C₂², C₈:C₂², 2₊¹⁺⁴, C₈oD₈, D₄:₄D₄, D₄.₄D₄, C₈:₄D₄, D₄oD₈, D₈oD₈
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, C₂⁴, C₂²xD₄, 2₊¹⁺⁴, D₄², D₈oD₈

Character table of D₈oD₈

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	2	4	4	4	4	8	8	8	8	2	2	4	4	4	4	4	4	2	2	2	2	4	4	8	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	1	1	-1	-1	1	1	1	1	-1	1	1	-1	1	-1	-1	-1	-1	-1	-1	1	1	-1	-1	linear of order 2
ρ₃	1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₄	1	1	1	-1	1	1	-1	1	-1	-1	1	1	1	1	1	-1	-1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	1	1	linear of order 2
ρ₅	1	1	1	1	1	1	-1	-1	1	-1	1	1	1	-1	1	1	-1	1	-1	1	1	-1	-1	-1	1	1	-1	-1	-1	linear of order 2
ρ₆	1	1	1	1	-1	-1	-1	-1	-1	1	1	1	1	-1	-1	1	-1	-1	-1	-1	-1	1	1	1	-1	1	-1	1	1	linear of order 2
ρ₇	1	1	1	-1	-1	-1	1	-1	1	-1	1	1	1	-1	-1	-1	1	-1	-1	1	1	-1	-1	-1	1	-1	1	1	1	linear of order 2
ρ₈	1	1	1	-1	1	1	1	-1	-1	1	1	1	1	-1	1	-1	1	1	-1	-1	-1	1	1	1	-1	-1	1	-1	-1	linear of order 2
ρ₉	1	1	1	1	1	-1	1	1	-1	1	-1	1	1	-1	-1	1	1	1	-1	1	1	-1	-1	-1	1	-1	-1	1	-1	linear of order 2
ρ₁₀	1	1	1	1	-1	1	1	1	1	-1	-1	1	1	-1	1	1	1	-1	-1	-1	-1	1	1	1	-1	-1	-1	-1	1	linear of order 2
ρ₁₁	1	1	1	-1	-1	1	-1	1	-1	1	-1	1	1	-1	1	-1	-1	-1	-1	1	1	-1	-1	-1	1	1	1	-1	1	linear of order 2
ρ₁₂	1	1	1	-1	1	-1	-1	1	1	-1	-1	1	1	-1	-1	-1	-1	1	-1	-1	-1	1	1	1	-1	1	1	1	-1	linear of order 2
ρ₁₃	1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	1	1	-1	1	-1	1	1	1	1	1	1	1	1	-1	1	-1	1	linear of order 2
ρ₁₄	1	1	1	1	-1	1	-1	-1	1	1	-1	1	1	1	1	1	-1	-1	1	-1	-1	-1	-1	-1	-1	-1	1	1	-1	linear of order 2
ρ₁₅	1	1	1	-1	-1	1	1	-1	-1	-1	-1	1	1	1	1	-1	1	-1	1	1	1	1	1	1	1	1	-1	1	-1	linear of order 2
ρ₁₆	1	1	1	-1	1	-1	1	-1	1	1	-1	1	1	1	-1	-1	1	1	1	-1	-1	-1	-1	-1	-1	1	-1	-1	1	linear of order 2
ρ₁₇	2	2	-2	-2	2	0	0	0	0	0	0	-2	2	0	0	2	0	-2	0	-2	-2	0	0	0	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	2	-2	0	0	-2	2	0	0	0	0	2	-2	0	2	0	-2	0	0	0	0	-2	-2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	2	-2	0	0	-2	-2	0	0	0	0	2	-2	0	2	0	2	0	0	0	0	2	2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	2	-2	2	-2	0	0	0	0	0	0	-2	2	0	0	-2	0	2	0	-2	-2	0	0	0	2	0	0	0	0	orthogonal lifted from D₄
ρ₂₁	2	2	-2	0	0	2	2	0	0	0	0	2	-2	0	-2	0	-2	0	0	0	0	2	2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₂	2	2	-2	2	2	0	0	0	0	0	0	-2	2	0	0	-2	0	-2	0	2	2	0	0	0	-2	0	0	0	0	orthogonal lifted from D₄
ρ₂₃	2	2	-2	-2	-2	0	0	0	0	0	0	-2	2	0	0	2	0	2	0	2	2	0	0	0	-2	0	0	0	0	orthogonal lifted from D₄
ρ₂₄	2	2	-2	0	0	2	-2	0	0	0	0	2	-2	0	-2	0	2	0	0	0	0	-2	-2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₅	4	4	4	0	0	0	0	0	0	0	0	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from 2₊¹⁺⁴
ρ₂₆	4	-4	0	0	0	0	0	0	0	0	0	0	0	-2	0	0	0	0	2	-2√2	2√2	-2√2	2√2	0	0	0	0	0	0	orthogonal faithful
ρ₂₇	4	-4	0	0	0	0	0	0	0	0	0	0	0	2	0	0	0	0	-2	2√2	-2√2	-2√2	2√2	0	0	0	0	0	0	orthogonal faithful
ρ₂₈	4	-4	0	0	0	0	0	0	0	0	0	0	0	2	0	0	0	0	-2	-2√2	2√2	2√2	-2√2	0	0	0	0	0	0	orthogonal faithful
ρ₂₉	4	-4	0	0	0	0	0	0	0	0	0	0	0	-2	0	0	0	0	2	2√2	-2√2	2√2	-2√2	0	0	0	0	0	0	orthogonal faithful

Permutation representations of D₈oD₈
►On 16 points - transitive group 16T296

Generators in S₁₆

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)

(1 2 3 4 5 6 7 8)(9 16 15 14 13 12 11 10)

(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9), (1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9), (1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)], [(1,2,3,4,5,6,7,8),(9,16,15,14,13,12,11,10)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9)]])

G:=TransitiveGroup(16,296);

Matrix representation of D₈oD₈ ►in GL₄(F₇) generated by

2	0	2	6
6	2	5	6
6	1	5	2
2	2	6	6

5	0	6	3
0	2	4	5
5	4	0	1
3	6	4	0

2	2	4	5
1	2	4	2
1	1	4	5
5	2	1	0

4	6	2	1
3	4	1	3
6	3	2	1
4	2	3	4

G:=sub<GL(4,GF(7))| [2,6,6,2,0,2,1,2,2,5,5,6,6,6,2,6],[5,0,5,3,0,2,4,6,6,4,0,4,3,5,1,0],[2,1,1,5,2,2,1,2,4,4,4,1,5,2,5,0],[4,3,6,4,6,4,3,2,2,1,2,3,1,3,1,4] >;

D₈oD₈ in GAP, Magma, Sage, TeX

D_8\circ D_8

% in TeX

G:=Group("D8oD8");

// GroupNames label

G:=SmallGroup(128,2024);

// by ID

G=gap.SmallGroup(128,2024);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,723,346,2804,1411,375,172,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^4=a^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^4*c^3>;

// generators/relations

Export

Character table of D₈oD₈ in TeX

G = D8oD8 order 128 = 27

Central product of D8 and D8

G = D₈oD₈ order 128 = 2⁷

Central product of D₈ and D₈