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## G = C2×Q8order 16 = 24

### Direct product of C2 and Q8

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

Aliases: C2×Q8, C2.2C23, C4.4C22, C22.4C22, (C2×C4).3C2, SmallGroup(16,12)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×Q8
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8
 Lower central C1 — C2 — C2×Q8
 Upper central C1 — C22 — C2×Q8
 Jennings C1 — C2 — C2×Q8

Generators and relations for C2×Q8
G = < a,b,c | a2=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C2×Q8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 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 symplectic lifted from Q8, Schur index 2 ρ10 2 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2

Permutation representations of C2×Q8
Regular action on 16 points - transitive group 16T7
Generators in S16
(1 12)(2 9)(3 10)(4 11)(5 16)(6 13)(7 14)(8 15)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16 3 14)(2 15 4 13)(5 10 7 12)(6 9 8 11)

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

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

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

G:=TransitiveGroup(16,7);

Matrix representation of C2×Q8 in GL3(𝔽5) generated by

 4 0 0 0 4 0 0 0 4
,
 1 0 0 0 3 0 0 0 2
,
 1 0 0 0 0 1 0 4 0
G:=sub<GL(3,GF(5))| [4,0,0,0,4,0,0,0,4],[1,0,0,0,3,0,0,0,2],[1,0,0,0,0,4,0,1,0] >;

C2×Q8 in GAP, Magma, Sage, TeX

C_2\times Q_8
% in TeX

G:=Group("C2xQ8");
// GroupNames label

G:=SmallGroup(16,12);
// by ID

G=gap.SmallGroup(16,12);
# by ID

G:=PCGroup([4,-2,2,2,-2,32,81,37]);
// Polycyclic

G:=Group<a,b,c|a^2=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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