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G = C₂³  order 8 = 2³

Elementary abelian group of type [2,2,2]

direct product, p-group, elementary abelian, monomial, rational

Aliases: C₂³, SmallGroup(8,5)

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

Generators and relations for C₂³
 G = < a,b,c | a²=b²=c²=1, ab=ba, ac=ca, bc=cb >

Character table of C₂³

(1 2)(3 4)(5 6)(7 8)

(1 3)(2 4)(5 7)(6 8)

(1 7)(2 8)(3 5)(4 6)

G:=sub<Sym(8)| (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8), (1,7)(2,8)(3,5)(4,6)>;

G:=Group( (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8), (1,7)(2,8)(3,5)(4,6) );

G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8)], [(1,3),(2,4),(5,7),(6,8)], [(1,7),(2,8),(3,5),(4,6)]])

G:=TransitiveGroup(8,3);

C₂³ is a maximal subgroup of   C₂²⋊C₄  F₈
C₂³ is a maximal quotient of   C₄○D₄

Matrix representation of C₂³ ►in GL₃(ℤ) generated by

G:=sub<GL(3,Integers())| [1,0,0,0,1,0,0,0,-1],[1,0,0,0,-1,0,0,0,-1],[-1,0,0,0,-1,0,0,0,1] >;

C₂³ in GAP, Magma, Sage, TeX

C_2^3

% in TeX

G:=Group("C2^3");

// GroupNames label

G:=SmallGroup(8,5);

// by ID

G=gap.SmallGroup(8,5);

# by ID

G:=PCGroup([3,-2,2,2]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂³ in TeX
Character table of C₂³ in TeX