Copied to
clipboard

## G = C4×D4order 32 = 25

### Direct product of C4 and D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C4×D4, C424C2, C22.7C23, C23.8C22, C4⋊C47C2, C41(C2×C4), C42(C4⋊C4), C2.3(C2×D4), C22⋊C46C2, (C22×C4)⋊2C2, C221(C2×C4), (C2×D4).7C2, C42(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

 Derived series C1 — C2 — C4×D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C4×D4
 Lower central C1 — C2 — C4×D4
 Upper central C1 — C2×C4 — C4×D4
 Jennings C1 — C22 — C4×D4

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

Permutation representations of 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 14 11 7)(2 15 12 8)(3 16 9 5)(4 13 10 6)
(5 16)(6 13)(7 14)(8 15)

G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,11,7)(2,15,12,8)(3,16,9,5)(4,13,10,6), (5,16)(6,13)(7,14)(8,15)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,11,7)(2,15,12,8)(3,16,9,5)(4,13,10,6), (5,16)(6,13)(7,14)(8,15) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,14,11,7),(2,15,12,8),(3,16,9,5),(4,13,10,6)], [(5,16),(6,13),(7,14),(8,15)])

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

Export

׿
×
𝔽