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## G = C2×C4order 8 = 23

### Abelian group of type [2,4]

Aliases: C2×C4, SmallGroup(8,2)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4
 Chief series C1 — C2 — C22 — C2×C4
 Lower central C1 — C2×C4
 Upper central C1 — C2×C4
 Jennings C1 — C2 — C2×C4

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

Character table of C2×C4

 class 1 2A 2B 2C 4A 4B 4C 4D size 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ3 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 -1 1 -1 1 linear of order 2 ρ5 1 -1 1 -1 i i -i -i linear of order 4 ρ6 1 -1 -1 1 i -i -i i linear of order 4 ρ7 1 -1 1 -1 -i -i i i linear of order 4 ρ8 1 -1 -1 1 -i i i -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);

C2×C4 is a maximal subgroup of   C22⋊C4  C4⋊C4  M4(2)  C4○D4
C2×C4 is a maximal quotient of   C22⋊C4  C4⋊C4  M4(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

 4 0 0 1
,
 2 0 0 3
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

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