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
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 |
►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);
C2×Q8 is a maximal subgroup of
C4.10D4 Q8⋊C4 C22⋊Q8 C4.4D4 C4⋊Q8 C8.C22 2- 1+4
C2×Q8 is a maximal quotient of C22⋊Q8 C42.C2 C4⋊Q8
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
Export
Subgroup lattice of C2×Q8 in TeX
Character table of C2×Q8 in TeX