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## G = C2×D4order 16 = 24

### Direct product of C2 and D4

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

 Derived series C1 — C2 — C2×D4
 Chief series C1 — C2 — C22 — C23 — C2×D4
 Lower central C1 — C2 — C2×D4
 Upper central C1 — C22 — C2×D4
 Jennings C1 — C2 — C2×D4

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

Permutation representations of C2×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 10)(6 11)(7 12)(8 9)
(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 15)(6 14)(7 13)(8 16)

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

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

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

G:=TransitiveGroup(16,9);

Polynomial with Galois group C2×D4 over ℚ
actionf(x)Disc(f)
8T9x8-10x6-4x5+21x4+8x3-12x2-4x+1216·34·172·192

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

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