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G = C23order 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

C1 — C23
C1C2C22 — C23
C1 — C23
C1 — C23
C1 — C23

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


Character table of C23

 class 12A2B2C2D2E2F2G
 size 11111111
ρ111111111    trivial
ρ21-111-1-1-11    linear of order 2
ρ31-1-1-111-11    linear of order 2
ρ411-1-1-1-111    linear of order 2
ρ51-11-11-11-1    linear of order 2
ρ6111-1-11-1-1    linear of order 2
ρ711-111-1-1-1    linear of order 2
ρ81-1-11-111-1    linear of order 2

Permutation representations of C23
Regular action on 8 points - transitive group 8T3
Generators in S8
(1 2)(3 4)(5 6)(7 8)
(1 3)(2 4)(5 8)(6 7)
(1 7)(2 8)(3 6)(4 5)

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

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

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

G:=TransitiveGroup(8,3);

Polynomial with Galois group C23 over ℚ
actionf(x)Disc(f)
8T3x8-x4+1216·34

Matrix representation of C23 in GL3(ℤ) generated by

100
010
00-1
,
100
0-10
00-1
,
-100
0-10
001
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

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