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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

C1C2 — C2×Q8
C1C2C22C2×C4 — C2×Q8
C1C2 — C2×Q8
C1C22 — C2×Q8
C1C2 — 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 12A2B2C4A4B4C4D4E4F
 size 1111222222
ρ11111111111    trivial
ρ21-11-1-11-111-1    linear of order 2
ρ31-11-11-11-11-1    linear of order 2
ρ41111-1-1-1-111    linear of order 2
ρ51-11-11-1-11-11    linear of order 2
ρ61-11-1-111-1-11    linear of order 2
ρ71111-1-111-1-1    linear of order 2
ρ8111111-1-1-1-1    linear of order 2
ρ92-2-22000000    symplectic lifted from Q8, Schur index 2
ρ1022-2-2000000    symplectic lifted from Q8, Schur index 2

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

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

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

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

G:=TransitiveGroup(16,7);

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

400
040
004
,
100
030
002
,
100
001
040
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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