direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×D4, C42⋊4C2, C22.7C23, C23.8C22, C4⋊C4⋊7C2, C4⋊1(C2×C4), C4○2(C4⋊C4), C2.3(C2×D4), C22⋊C4⋊6C2, (C22×C4)⋊2C2, C22⋊1(C2×C4), (C2×D4).7C2, C4○2(C22⋊C4), C2.2(C4○D4), C2.4(C22×C4), (C2×C4).11C22, (C2×C4)○(C2×D4), (C2×C4)○(C4⋊C4), (C2×C4)○(C22⋊C4), 2-Sylow(CO(3,5)), SmallGroup(32,25)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×D4
G = < a,b,c | a4=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C4×D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 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 | 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 | 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 | -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 | -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 | -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 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | -i | i | -i | i | -i | -i | i | 1 | i | -1 | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | -i | -i | i | i | i | -1 | -i | 1 | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | -i | -i | -i | -i | i | i | -1 | i | 1 | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | i | -i | i | -i | -i | i | i | -i | i | 1 | -i | -1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | i | i | i | i | -i | -i | -1 | -i | 1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | i | -i | i | i | -i | -i | i | -i | 1 | i | -1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | i | -i | i | -i | i | i | -i | 1 | -i | -1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | i | i | -i | -i | -i | -1 | i | 1 | linear of order 4 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 16 points - transitive group 16T19
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16 11 7)(2 13 12 8)(3 14 9 5)(4 15 10 6)
(5 14)(6 15)(7 16)(8 13)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,11,7)(2,13,12,8)(3,14,9,5)(4,15,10,6), (5,14)(6,15)(7,16)(8,13)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,11,7)(2,13,12,8)(3,14,9,5)(4,15,10,6), (5,14)(6,15)(7,16)(8,13) );
G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,16,11,7),(2,13,12,8),(3,14,9,5),(4,15,10,6)], [(5,14),(6,15),(7,16),(8,13)])
G:=TransitiveGroup(16,19);
Matrix representation of C4×D4 ►in GL3(𝔽5) generated by
| 2 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 0 | 4 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 4 |
G:=sub<GL(3,GF(5))| [2,0,0,0,4,0,0,0,4],[1,0,0,0,0,1,0,4,0],[1,0,0,0,1,0,0,0,4] >;
C4×D4 in GAP, Magma, Sage, TeX
C_4\times D_4
% in TeX
G:=Group("C4xD4"); // GroupNames label
G:=SmallGroup(32,25);
// by ID
G=gap.SmallGroup(32,25);
# by ID
G:=PCGroup([5,-2,2,2,-2,2,80,101,72]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations