direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×D8, C4.6D4, C8⋊2C22, D4⋊1C22, C4.1C23, C22.14D4, (C2×C8)⋊3C2, (C2×D4)⋊4C2, C2.11(C2×D4), (C2×C4).26C22, 2-Sylow(SO-(4,7)), SmallGroup(32,39)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×D8
G = < a,b,c | a2=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C2×D8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
►On 16 points - transitive group 16T29
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)
G:=sub<Sym(16)| (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)>;
G:=Group( (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15) );
G=PermutationGroup([[(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15)]])
G:=TransitiveGroup(16,29);
C2×D8 is a maximal subgroup of
C2.D16 M5(2)⋊C2 D8⋊C4 C22⋊D8 D4⋊D4 C4⋊D8 D4.2D4 C8⋊7D4 C8⋊2D4 D4.4D4 C8⋊4D4 C8.12D4 C8⋊3D4 C16⋊C22 D4○D8 C3⋊S3⋊D8
C2×D8 is a maximal quotient of
C22⋊D8 C4⋊D8 C8⋊7D4 D4⋊Q8 C22.D8 C4.4D8 C8⋊4D4 C8⋊2Q8 C4○D16 C16⋊C22 Q32⋊C2 C3⋊S3⋊D8
Matrix representation of C2×D8 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 1 | 0 | 0 |
| 0 | 0 | 6 |
| 0 | 14 | 6 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 1 | 16 |
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,0,14,0,6,6],[16,0,0,0,1,1,0,0,16] >;
C2×D8 in GAP, Magma, Sage, TeX
C_2\times D_8
% in TeX
G:=Group("C2xD8"); // GroupNames label
G:=SmallGroup(32,39);
// by ID
G=gap.SmallGroup(32,39);
# by ID
G:=PCGroup([5,-2,2,2,-2,-2,101,483,248,58]);
// Polycyclic
G:=Group<a,b,c|a^2=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C2×D8 in TeX
Character table of C2×D8 in TeX