direct product, p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C2×D4, C4⋊C22, C23⋊C2, C22⋊C22, C2.1C23, (C2×C4)⋊2C2, 2-Sylow(S6), SmallGroup(16,11)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×D4
G = < a,b,c | a2=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C2×D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
►On 8 points - transitive group 8T9
Generators in S8
(1 7)(2 8)(3 5)(4 6)
(1 2 3 4)(5 6 7 8)
(1 4)(2 3)(5 8)(6 7)
G:=sub<Sym(8)| (1,7)(2,8)(3,5)(4,6), (1,2,3,4)(5,6,7,8), (1,4)(2,3)(5,8)(6,7)>;
G:=Group( (1,7)(2,8)(3,5)(4,6), (1,2,3,4)(5,6,7,8), (1,4)(2,3)(5,8)(6,7) );
G=PermutationGroup([[(1,7),(2,8),(3,5),(4,6)], [(1,2,3,4),(5,6,7,8)], [(1,4),(2,3),(5,8),(6,7)]])
G:=TransitiveGroup(8,9);
►Regular action on 16 points - transitive group 16T9
Generators in S16
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 10)(2 9)(3 12)(4 11)(5 13)(6 16)(7 15)(8 14)
G:=sub<Sym(16)| (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10)(2,9)(3,12)(4,11)(5,13)(6,16)(7,15)(8,14)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10)(2,9)(3,12)(4,11)(5,13)(6,16)(7,15)(8,14) );
G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,10),(2,9),(3,12),(4,11),(5,13),(6,16),(7,15),(8,14)]])
G:=TransitiveGroup(16,9);
C2×D4 is a maximal subgroup of
C23⋊C4 C4.D4 D4⋊C4 C22≀C2 C4⋊D4 C22.D4 C4.4D4 C4⋊1D4 C8⋊C22 2+ 1+4
C2×D4 is a maximal quotient of
C22≀C2 C4⋊D4 C22⋊Q8 C22.D4 C4.4D4 C4⋊1D4 C4⋊Q8 C4○D8 C8⋊C22 C8.C22
| action | f(x) | Disc(f) |
|---|---|---|
| 8T9 | x8-10x6-4x5+21x4+8x3-12x2-4x+1 | 216·34·172·192 |
Matrix representation of C2×D4 ►in GL3(ℤ) generated by
| -1 | 0 | 0 |
| 0 | -1 | 0 |
| 0 | 0 | -1 |
| 1 | 0 | 0 |
| 0 | 0 | -1 |
| 0 | 1 | 0 |
| -1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | -1 |
G:=sub<GL(3,Integers())| [-1,0,0,0,-1,0,0,0,-1],[1,0,0,0,0,1,0,-1,0],[-1,0,0,0,1,0,0,0,-1] >;
C2×D4 in GAP, Magma, Sage, TeX
C_2\times D_4
% in TeX
G:=Group("C2xD4"); // GroupNames label
G:=SmallGroup(16,11);
// by ID
G=gap.SmallGroup(16,11);
# by ID
G:=PCGroup([4,-2,2,2,-2,81]);
// Polycyclic
G:=Group<a,b,c|a^2=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C2×D4 in TeX
Character table of C2×D4 in TeX