direct product, p-group, abelian, monomial
Aliases: C2×C4, SmallGroup(8,2)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C2×C4 |
| C1 — C2×C4 |
| C1 — 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 |
►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)
| action | f(x) | Disc(f) |
|---|---|---|
| 8T2 | x8+1 | 224 |
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
Export
Subgroup lattice of C2×C4 in TeX
Character table of C2×C4 in TeX