C2xC8 - GroupNames

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G = C₂×C₈ order 16 = 2⁴

Abelian group of type [2,8]

direct product, p-group, abelian, monomial

Aliases: C₂×C₈, SmallGroup(16,5)

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

C₁ — C₂×C₈

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₈

C₁ — C₂×C₈

C₁ — C₂×C₈

C₁ — C₂ — C₂ — C₄ — C₂×C₈

Generators and relations for C₂×C₈
G = < a,b | a²=b⁸=1, ab=ba >

C₁

C₂

C₂

C₂

C₄

C₂²

C₄

C₈

C₈

Character table of C₂×C₈

class	1	2A	2B	2C	4A	4B	4C	4D	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
ρ₁	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	linear of order 2
ρ₃	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	linear of order 2
ρ₅	1	-1	1	-1	i	i	-i	-i	ζ₈⁷	ζ₈	ζ₈⁵	ζ₈⁵	ζ₈³	ζ₈³	ζ₈⁷	ζ₈	linear of order 8
ρ₆	1	-1	-1	1	i	-i	-i	i	ζ₈³	ζ₈⁵	ζ₈⁵	ζ₈	ζ₈³	ζ₈⁷	ζ₈⁷	ζ₈	linear of order 8
ρ₇	1	-1	1	-1	-i	-i	i	i	ζ₈⁵	ζ₈³	ζ₈⁷	ζ₈⁷	ζ₈	ζ₈	ζ₈⁵	ζ₈³	linear of order 8
ρ₈	1	-1	-1	1	-i	i	i	-i	ζ₈	ζ₈⁷	ζ₈⁷	ζ₈³	ζ₈	ζ₈⁵	ζ₈⁵	ζ₈³	linear of order 8
ρ₉	1	1	1	1	-1	-1	-1	-1	-i	i	i	i	-i	-i	-i	i	linear of order 4
ρ₁₀	1	1	-1	-1	-1	1	-1	1	i	-i	i	-i	-i	i	-i	i	linear of order 4
ρ₁₁	1	-1	1	-1	i	i	-i	-i	ζ₈³	ζ₈⁵	ζ₈	ζ₈	ζ₈⁷	ζ₈⁷	ζ₈³	ζ₈⁵	linear of order 8
ρ₁₂	1	-1	-1	1	i	-i	-i	i	ζ₈⁷	ζ₈	ζ₈	ζ₈⁵	ζ₈⁷	ζ₈³	ζ₈³	ζ₈⁵	linear of order 8
ρ₁₃	1	1	1	1	-1	-1	-1	-1	i	-i	-i	-i	i	i	i	-i	linear of order 4
ρ₁₄	1	1	-1	-1	-1	1	-1	1	-i	i	-i	i	i	-i	i	-i	linear of order 4
ρ₁₅	1	-1	1	-1	-i	-i	i	i	ζ₈	ζ₈⁷	ζ₈³	ζ₈³	ζ₈⁵	ζ₈⁵	ζ₈	ζ₈⁷	linear of order 8
ρ₁₆	1	-1	-1	1	-i	i	i	-i	ζ₈⁵	ζ₈³	ζ₈³	ζ₈⁷	ζ₈⁵	ζ₈	ζ₈	ζ₈⁷	linear of order 8

Permutation representations of C₂×C₈
►Regular action on 16 points - transitive group 16T5

Generators in S₁₆

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

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

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

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

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

G:=TransitiveGroup(16,5);

C₂×C₈ is a maximal subgroup of
C₈⋊C₄ C₂²⋊C₈ D₄⋊C₄ Q₈⋊C₄ C₄⋊C₈ C₄.Q₈ C₂.D₈ C₈.C₄ M₅(2) C₈○D₄ C₄○D₈ C₃⋊S₃⋊₃C₈
D_p⋊C₈, p=1 mod 4: D₅⋊C₈ D₁₃⋊C₈ C₆₈.C₄ D₂₉⋊C₈ ...
C₂×C₈ is a maximal quotient of
C₂²⋊C₈ C₄⋊C₈ M₅(2) C₃⋊S₃⋊₃C₈
D_p⋊C₈, p=1 mod 4: D₅⋊C₈ D₁₃⋊C₈ C₆₈.C₄ D₂₉⋊C₈ ...

Polynomial with Galois group C₂×C₈ over ℚ

action	f(x)	Disc(f)
16T5	x¹⁶+1	2⁶⁴

Matrix representation of C₂×C₈ ►in GL₂(𝔽₁₇) generated by

16	0
0	16

,

1	0
0	9

G:=sub<GL(2,GF(17))| [16,0,0,16],[1,0,0,9] >;

C₂×C₈ in GAP, Magma, Sage, TeX

C_2\times C_8

% in TeX

G:=Group("C2xC8");

// GroupNames label

G:=SmallGroup(16,5);

// by ID

G=gap.SmallGroup(16,5);

# by ID

G:=PCGroup([4,-2,2,-2,-2,16,34]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂×C₈ in TeX
Character table of C₂×C₈ in TeX

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