C8oD8 - GroupNames

G = C₈oD₈ order 64 = 2⁶

Central product of C₈ and D₈

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

Aliases: C₈ o D₈, C₈ o Q₁₆, D₈:₅C₄, C₈ o SD₁₆, Q₁₆:₅C₄, C₈.₃₀D₄, SD₁₆:₃C₄, C₄².₇₄C₂², M₄(2).₁₁C₂², C₈ o C₄wrC₂, C₄wrC₂:₇C₂, (C₄xC₈):₁₀C₂, C₈ o(C₄oD₈), C₈oD₄:₆C₂, C₈.₁₁(C₂xC₄), C₄oD₈.₅C₂, D₄.₃(C₂xC₄), C₂.₁₈(C₄xD₄), C₄.₇₉(C₂xD₄), Q₈.₃(C₂xC₄), C₈ o(C₈.C₄), C₈.C₄:₈C₂, C₄.₁₅(C₂²xC₄), (C₂xC₄).₇₉C₂³, C₄oD₄.₇C₂², (C₂xC₈).₁₀₂C₂², C₂².₁(C₄oD₄), 2-Sylow(CU(2,3)), SmallGroup(64,124)

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

Derived series

C₁ — C₄ — C₈oD₈

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂xC₈ — C₈oD₄ — C₈oD₈

Lower central

C₁ — C₂ — C₄ — C₈oD₈

Upper central

C₁ — C₈ — C₂xC₈ — C₈oD₈

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — C₈oD₈

Generators and relations for C₈oD₈
G = < a,b,c | a⁸=c²=1, b⁴=a⁴, ab=ba, ac=ca, cbc=a⁴b³ >

Subgroups: 77 in 53 conjugacy classes, 33 normal (19 characteristic)
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, C₂²xC₄, C₂xD₄, C₄oD₄, C₄xD₄, C₈oD₈

C₁

C₂

₂C₂

₄C₂

C₄

C₂²

C₄

₂C₄

₂C₂²

Q₈

D₄

C₂xC₄

C₈

Q₈

C₈

D₄

₂C₂xC₄

₂D₄

₂C₂xC₄

₂D₄

₂C₈

₂C₂xC₄

SD₁₆

C₄²

D₈

Q₁₆

SD₁₆

C₂xC₈

M₄(2)

C₄oD₄

₂M₄(2)

₂C₂xC₈

C₄wrC₂

C₈.C₄

C₈oD₄

C₄xC₈

C₈oD₄

C₄oD₈

C₈oD₈

Character table of C₈oD₈

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	8M	8N
size	1	1	2	4	4	1	1	2	2	2	2	2	4	4	1	1	1	1	2	2	2	2	2	2	4	4	4	4
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	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	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	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	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	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	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	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	linear of order 2
ρ₉	1	1	-1	-1	-1	-1	-1	-i	i	1	-i	i	1	1	i	-i	-i	i	-1	-1	-i	1	i	1	i	i	-i	-i	linear of order 4
ρ₁₀	1	1	-1	-1	1	-1	-1	-i	i	1	-i	i	1	-1	-i	i	i	-i	1	1	i	-1	-i	-1	-i	i	i	-i	linear of order 4
ρ₁₁	1	1	-1	1	-1	-1	-1	-i	i	1	-i	i	-1	1	-i	i	i	-i	1	1	i	-1	-i	-1	i	-i	-i	i	linear of order 4
ρ₁₂	1	1	-1	1	1	-1	-1	-i	i	1	-i	i	-1	-1	i	-i	-i	i	-1	-1	-i	1	i	1	-i	-i	i	i	linear of order 4
ρ₁₃	1	1	-1	1	1	-1	-1	i	-i	1	i	-i	-1	-1	-i	i	i	-i	-1	-1	i	1	-i	1	i	i	-i	-i	linear of order 4
ρ₁₄	1	1	-1	1	-1	-1	-1	i	-i	1	i	-i	-1	1	i	-i	-i	i	1	1	-i	-1	i	-1	-i	i	i	-i	linear of order 4
ρ₁₅	1	1	-1	-1	1	-1	-1	i	-i	1	i	-i	1	-1	i	-i	-i	i	1	1	-i	-1	i	-1	i	-i	-i	i	linear of order 4
ρ₁₆	1	1	-1	-1	-1	-1	-1	i	-i	1	i	-i	1	1	-i	i	i	-i	-1	-1	i	1	-i	1	-i	-i	i	i	linear of order 4
ρ₁₇	2	2	-2	0	0	2	2	0	0	-2	0	0	0	0	2	2	2	2	0	0	-2	0	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	2	-2	0	0	2	2	0	0	-2	0	0	0	0	-2	-2	-2	-2	0	0	2	0	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	2	2	0	0	-2	-2	0	0	-2	0	0	0	0	2i	-2i	-2i	2i	0	0	2i	0	-2i	0	0	0	0	0	complex lifted from C₄oD₄
ρ₂₀	2	2	2	0	0	-2	-2	0	0	-2	0	0	0	0	-2i	2i	2i	-2i	0	0	-2i	0	2i	0	0	0	0	0	complex lifted from C₄oD₄
ρ₂₁	2	-2	0	0	0	-2i	2i	-1-i	1-i	0	1+i	-1+i	0	0	2ζ₈³	2ζ₈⁵	2ζ₈	2ζ₈⁷	√-2	-√-2	0	-√2	0	√2	0	0	0	0	complex faithful
ρ₂₂	2	-2	0	0	0	-2i	2i	1+i	-1+i	0	-1-i	1-i	0	0	2ζ₈⁷	2ζ₈	2ζ₈⁵	2ζ₈³	√-2	-√-2	0	-√2	0	√2	0	0	0	0	complex faithful
ρ₂₃	2	-2	0	0	0	2i	-2i	-1+i	1+i	0	1-i	-1-i	0	0	2ζ₈⁵	2ζ₈³	2ζ₈⁷	2ζ₈	-√-2	√-2	0	-√2	0	√2	0	0	0	0	complex faithful
ρ₂₄	2	-2	0	0	0	2i	-2i	1-i	-1-i	0	-1+i	1+i	0	0	2ζ₈⁵	2ζ₈³	2ζ₈⁷	2ζ₈	√-2	-√-2	0	√2	0	-√2	0	0	0	0	complex faithful
ρ₂₅	2	-2	0	0	0	2i	-2i	1-i	-1-i	0	-1+i	1+i	0	0	2ζ₈	2ζ₈⁷	2ζ₈³	2ζ₈⁵	-√-2	√-2	0	-√2	0	√2	0	0	0	0	complex faithful
ρ₂₆	2	-2	0	0	0	2i	-2i	-1+i	1+i	0	1-i	-1-i	0	0	2ζ₈	2ζ₈⁷	2ζ₈³	2ζ₈⁵	√-2	-√-2	0	√2	0	-√2	0	0	0	0	complex faithful
ρ₂₇	2	-2	0	0	0	-2i	2i	1+i	-1+i	0	-1-i	1-i	0	0	2ζ₈³	2ζ₈⁵	2ζ₈	2ζ₈⁷	-√-2	√-2	0	√2	0	-√2	0	0	0	0	complex faithful
ρ₂₈	2	-2	0	0	0	-2i	2i	-1-i	1-i	0	1+i	-1+i	0	0	2ζ₈⁷	2ζ₈	2ζ₈⁵	2ζ₈³	-√-2	√-2	0	√2	0	-√2	0	0	0	0	complex faithful

Permutation representations of C₈oD₈
►On 16 points - transitive group 16T114

Generators in S₁₆

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

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

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

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

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

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

G:=TransitiveGroup(16,114);

C₈oD₈ is a maximal subgroup of
C₄².₂₈₃C₂³ M₄(2).₅₁D₄ D₈oSD₁₆ D₈:₆D₄ D₈oD₈ D₈oQ₁₆ CU₂(F₃) SD₁₆:₃F₅
D_8p:C₄: C₈oD₁₆ D₁₆:₅C₄ D₂₄:₁₁C₄ D₂₄:₇C₄ D₄₀:₁₇C₄ D₄₀:₁₃C₄ Q₁₆:₅F₅ D₅₆:₁₁C₄ ...
C₄².D_2p: C₈wrC₂ C₈.₃₂D₈ C₈.₃D₈ D₈:₃D₄ C₈.₅D₈ D₈:₃Q₈ D₈.₂Q₈ C₄².₁₉₆D₆ ...
C_8p.D₄: C₁₆oD₈ D₈.C₈ C₂₄.₁₀₀D₄ C₄₀.₉₃D₄ C₅₆.₉₃D₄ ...
(C_pxD₈):C₄: M₄(2)oD₈ D₈:₅Dic₃ D₈:₅Dic₅ D₈:₅F₅ D₈:₅Dic₇ ...
C₈oD₈ is a maximal quotient of
C₈xD₈ C₈xSD₁₆ C₈xQ₁₆ D₄.M₄(2) D₄:₂M₄(2) Q₈.M₄(2) Q₈:₂M₄(2) C₈:₆D₈ C₈:₉SD₁₆ C₈:₆Q₁₆ M₄(2).₃Q₈ C₄².₆₂Q₈
C₄².D_2p: C₄².₄₂₈D₄ C₄².₃₂₆D₄ D₂₄:₁₁C₄ C₄².₁₉₆D₆ D₄₀:₁₇C₄ C₄².₁₉₆D₁₀ D₅₆:₁₁C₄ C₄².₁₉₆D₁₄ ...
M₄(2).D_2p: M₄(2).₄₂D₄ M₄(2).₄₃D₄ M₄(2).₂₄D₄ D₂₄:₇C₄ C₂₄.₁₀₀D₄ D₄₀:₁₃C₄ C₄₀.₉₃D₄ D₅₆:₇C₄ ...
C_2p.(C₄xD₄): Q₈.C₄² C₈.₁₄C₄² D₈:₅Dic₃ D₈:₅Dic₅ D₈:₅F₅ SD₁₆:₃F₅ Q₁₆:₅F₅ D₈:₅Dic₇ ...

Matrix representation of C₈oD₈ ►in GL₂(F₁₇) generated by

2	0
0	2

2	0
0	9

0	1
1	0

G:=sub<GL(2,GF(17))| [2,0,0,2],[2,0,0,9],[0,1,1,0] >;

C₈oD₈ in GAP, Magma, Sage, TeX

C_8\circ D_8

% in TeX

G:=Group("C8oD8");

// GroupNames label

G:=SmallGroup(64,124);

// by ID

G=gap.SmallGroup(64,124);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,86,963,489,117,88]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₈oD₈ in TeX
Character table of C₈oD₈ in TeX

G = C8oD8 order 64 = 26

Central product of C8 and D8

G = C₈oD₈ order 64 = 2⁶

Central product of C₈ and D₈