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## G = D4×Q8order 64 = 26

### Direct product of D4 and Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — D4×Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×Q8 — D4×Q8
 Lower central C1 — C22 — D4×Q8
 Upper central C1 — C22 — D4×Q8
 Jennings C1 — C22 — D4×Q8

Generators and relations for D4×Q8
G = < a,b,c,d | a4=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 189 in 140 conjugacy classes, 91 normal (10 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×9], C22, C22 [×4], C22 [×4], C2×C4, C2×C4 [×12], C2×C4 [×12], D4 [×4], Q8 [×4], Q8 [×12], C23 [×2], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×12], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×8], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×3], C22×Q8 [×2], D4×Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C22×D4, C22×Q8, 2- 1+4, D4×Q8

Character table of D4×Q8

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q size 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 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 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 -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 -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 -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 -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 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 1 -1 -1 1 -1 1 1 1 -1 1 -1 linear of order 2 ρ9 1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 1 -1 -1 -1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ10 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 linear of order 2 ρ11 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ12 1 1 1 1 1 -1 -1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 linear of order 2 ρ13 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 linear of order 2 ρ14 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ15 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 linear of order 2 ρ16 1 1 1 1 -1 1 1 -1 -1 1 1 -1 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 linear of order 2 ρ17 2 2 -2 -2 0 0 0 0 0 2 -2 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 0 0 0 0 0 -2 2 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 0 0 0 0 0 -2 2 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 -2 0 0 0 0 0 2 -2 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 -2 2 -2 2 2 -2 -2 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ22 2 -2 2 -2 -2 2 -2 2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ23 2 -2 2 -2 2 -2 2 -2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ24 2 -2 2 -2 -2 -2 2 2 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of D4×Q8
On 32 points
Generators in S32
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 13)(2 16)(3 15)(4 14)(5 19)(6 18)(7 17)(8 20)(9 25)(10 28)(11 27)(12 26)(21 30)(22 29)(23 32)(24 31)
(1 20 13 8)(2 17 14 5)(3 18 15 6)(4 19 16 7)(9 21 27 32)(10 22 28 29)(11 23 25 30)(12 24 26 31)
(1 29 13 22)(2 30 14 23)(3 31 15 24)(4 32 16 21)(5 11 17 25)(6 12 18 26)(7 9 19 27)(8 10 20 28)

G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,13)(2,16)(3,15)(4,14)(5,19)(6,18)(7,17)(8,20)(9,25)(10,28)(11,27)(12,26)(21,30)(22,29)(23,32)(24,31), (1,20,13,8)(2,17,14,5)(3,18,15,6)(4,19,16,7)(9,21,27,32)(10,22,28,29)(11,23,25,30)(12,24,26,31), (1,29,13,22)(2,30,14,23)(3,31,15,24)(4,32,16,21)(5,11,17,25)(6,12,18,26)(7,9,19,27)(8,10,20,28)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,13)(2,16)(3,15)(4,14)(5,19)(6,18)(7,17)(8,20)(9,25)(10,28)(11,27)(12,26)(21,30)(22,29)(23,32)(24,31), (1,20,13,8)(2,17,14,5)(3,18,15,6)(4,19,16,7)(9,21,27,32)(10,22,28,29)(11,23,25,30)(12,24,26,31), (1,29,13,22)(2,30,14,23)(3,31,15,24)(4,32,16,21)(5,11,17,25)(6,12,18,26)(7,9,19,27)(8,10,20,28) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,13),(2,16),(3,15),(4,14),(5,19),(6,18),(7,17),(8,20),(9,25),(10,28),(11,27),(12,26),(21,30),(22,29),(23,32),(24,31)], [(1,20,13,8),(2,17,14,5),(3,18,15,6),(4,19,16,7),(9,21,27,32),(10,22,28,29),(11,23,25,30),(12,24,26,31)], [(1,29,13,22),(2,30,14,23),(3,31,15,24),(4,32,16,21),(5,11,17,25),(6,12,18,26),(7,9,19,27),(8,10,20,28)])

Matrix representation of D4×Q8 in GL4(𝔽5) generated by

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

D4×Q8 in GAP, Magma, Sage, TeX

D_4\times Q_8
% in TeX

G:=Group("D4xQ8");
// GroupNames label

G:=SmallGroup(64,230);
// by ID

G=gap.SmallGroup(64,230);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,103,650,297,69]);
// Polycyclic

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

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