C4^2 - GroupNames

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G = C₄² order 16 = 2⁴

Abelian group of type [4,4]

direct product, p-group, abelian, monomial

Aliases: C₄², SmallGroup(16,2)

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

C₁ — C₄²

C₁ — C₂ — C₂² — C₂×C₄ — C₄²

C₁ — C₄²

C₁ — C₄²

C₁ — C₂² — C₄²

Generators and relations for C₄²
G = < a,b | a⁴=b⁴=1, ab=ba >

C₁

C₂

C₂

C₂

C₄

C₄

C₂²

C₄

C₄

C₄

C₄

C₄²

Character table of C₄²

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L
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	1	-1	-1	i	-i	-i	i	-i	1	linear of order 4
ρ₆	1	1	-1	-1	i	-i	-i	1	-1	-1	-i	i	i	-i	i	1	linear of order 4
ρ₇	1	-1	-1	1	-1	i	-1	-i	-i	i	-i	-i	i	1	1	i	linear of order 4
ρ₈	1	-1	-1	1	1	-i	1	-i	-i	i	i	i	-i	-1	-1	i	linear of order 4
ρ₉	1	-1	1	-1	-i	1	i	-i	i	-i	-1	1	-1	-i	i	i	linear of order 4
ρ₁₀	1	1	-1	-1	i	i	-i	-1	1	1	i	-i	-i	-i	i	-1	linear of order 4
ρ₁₁	1	-1	1	-1	i	-1	-i	-i	i	-i	1	-1	1	i	-i	i	linear of order 4
ρ₁₂	1	1	-1	-1	-i	-i	i	-1	1	1	-i	i	i	i	-i	-1	linear of order 4
ρ₁₃	1	-1	1	-1	i	1	-i	i	-i	i	-1	1	-1	i	-i	-i	linear of order 4
ρ₁₄	1	-1	-1	1	1	i	1	i	i	-i	-i	-i	i	-1	-1	-i	linear of order 4
ρ₁₅	1	-1	1	-1	-i	-1	i	i	-i	i	1	-1	1	-i	i	-i	linear of order 4
ρ₁₆	1	-1	-1	1	-1	-i	-1	i	i	-i	i	i	-i	1	1	-i	linear of order 4

Permutation representations of C₄²
►Regular action on 16 points - transitive group 16T4

Generators in S₁₆

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

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

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

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

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

G:=TransitiveGroup(16,4);

C₄² is a maximal subgroup of
C₈⋊C₄ C₄≀C₂ C₄⋊C₈ C₄²⋊C₂ C₄.₄D₄ C₄².C₂ C₄²⋊₂C₂ C₄⋊₁D₄ C₄⋊Q₈ C₄²⋊C₃
C₄² is a maximal quotient of
C₂.C₄² C₈⋊C₄

Matrix representation of C₄² ►in GL₂(𝔽₅) generated by

4	0
0	3

,

3	0
0	2

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

C₄² in GAP, Magma, Sage, TeX

C_4^2

% in TeX

G:=Group("C4^2");

// GroupNames label

G:=SmallGroup(16,2);

// by ID

G=gap.SmallGroup(16,2);

# by ID

G:=PCGroup([4,-2,2,-2,2,16,37]);

// Polycyclic

G:=Group<a,b|a^4=b^4=1,a*b=b*a>;

// generators/relations

Export

Subgroup lattice of C₄² in TeX
Character table of C₄² in TeX

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