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## G = C2×D8order 32 = 25

### Direct product of C2 and D8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×D8, C4.6D4, C82C22, D41C22, 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

 Derived series C1 — C4 — C2×D8
 Chief series C1 — C2 — C4 — C2×C4 — C2×D4 — C2×D8
 Lower central C1 — C2 — C4 — C2×D8
 Upper central C1 — C22 — C2×C4 — C2×D8
 Jennings C1 — C2 — C2 — C4 — C2×D8

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

Permutation representations of C2×D8
On 16 points - transitive group 16T29
Generators in S16
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(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 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(16,29);

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

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