Copied to
clipboard

G = C2×C4order 8 = 23

Abelian group of type [2,4]

direct product, p-group, abelian, monomial

Aliases: C2×C4, SmallGroup(8,2)

Series: Derived Chief Lower central Upper central Jennings

C1 — C2×C4
C1C2C22 — C2×C4
C1 — C2×C4
C1 — C2×C4
C1C2 — C2×C4

Generators and relations for C2×C4
 G = < a,b | a2=b4=1, ab=ba >


Character table of C2×C4

 class 12A2B2C4A4B4C4D
 size 11111111
ρ111111111    trivial
ρ211-1-11-11-1    linear of order 2
ρ31111-1-1-1-1    linear of order 2
ρ411-1-1-11-11    linear of order 2
ρ51-11-1ii-i-i    linear of order 4
ρ61-1-11i-i-ii    linear of order 4
ρ71-11-1-i-iii    linear of order 4
ρ81-1-11-iii-i    linear of order 4

Permutation representations of C2×C4
Regular action on 8 points - transitive group 8T2
Generators in S8
(1 5)(2 6)(3 7)(4 8)
(1 2 3 4)(5 6 7 8)

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

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

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

G:=TransitiveGroup(8,2);

Polynomial with Galois group C2×C4 over ℚ
actionf(x)Disc(f)
8T2x8+1224

Matrix representation of C2×C4 in GL2(𝔽5) generated by

40
01
,
20
03
G:=sub<GL(2,GF(5))| [4,0,0,1],[2,0,0,3] >;

C2×C4 in GAP, Magma, Sage, TeX

C_2\times C_4
% in TeX

G:=Group("C2xC4");
// GroupNames label

G:=SmallGroup(8,2);
// by ID

G=gap.SmallGroup(8,2);
# by ID

G:=PCGroup([3,-2,2,-2,12]);
// Polycyclic

G:=Group<a,b|a^2=b^4=1,a*b=b*a>;
// generators/relations

׿
×
𝔽