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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

Aliases: D4×Q8, C22.43C24, C42.45C22, C23.46C23, C2.112- 1+4, C42(C2×Q8), C4⋊Q814C2, (C4×Q8)⋊12C2, C4.39(C2×D4), C222(C2×Q8), (C4×D4).10C2, C22⋊Q814C2, (C22×Q8)⋊7C2, C2.9(C22×Q8), C4⋊C4.34C22, C2.21(C22×D4), (C2×C4).133C23, (C2×D4).80C22, (C2×Q8).63C22, C22⋊C4.20C22, (C22×C4).70C22, (C2×D4)(C2×Q8), SmallGroup(64,230)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — D4×Q8
C1C2C22C2×C4C22×C4C22×Q8 — D4×Q8
C1C22 — D4×Q8
C1C22 — D4×Q8
C1C22 — 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 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q
 size 1111222222222222444444444
ρ11111111111111111111111111    trivial
ρ211111-1-11-111-11111-11-1-1-111-1-1    linear of order 2
ρ3111111111-1-11-11-111-1-1-1-11-11-1    linear of order 2
ρ411111-1-11-1-1-1-1-11-11-1-11111-1-11    linear of order 2
ρ51111-1-1-1-11111-1-1-1-11-1-11111-1-1    linear of order 2
ρ61111-111-1-111-1-1-1-1-1-1-11-1-11111    linear of order 2
ρ71111-1-1-1-11-1-111-11-1111-1-11-1-11    linear of order 2
ρ81111-111-1-1-1-1-11-11-1-11-1111-11-1    linear of order 2
ρ9111111111-1-111-11-1-1-1-1-11-11-11    linear of order 2
ρ1011111-1-11-1-1-1-11-11-11-111-1-111-1    linear of order 2
ρ11111111111111-1-1-1-1-1111-1-1-1-1-1    linear of order 2
ρ1211111-1-11-111-1-1-1-1-111-1-11-1-111    linear of order 2
ρ131111-1-1-1-11-1-11-11-11-111-11-111-1    linear of order 2
ρ141111-111-1-1-1-1-1-11-1111-11-1-11-11    linear of order 2
ρ151111-1-1-1-111111111-1-1-11-1-1-111    linear of order 2
ρ161111-111-1-111-111111-11-11-1-1-1-1    linear of order 2
ρ1722-2-2000002-20-2-222000000000    orthogonal lifted from D4
ρ1822-2-200000-2202-2-22000000000    orthogonal lifted from D4
ρ1922-2-200000-220-222-2000000000    orthogonal lifted from D4
ρ2022-2-2000002-2022-2-2000000000    orthogonal lifted from D4
ρ212-22-222-2-2-20020000000000000    symplectic lifted from Q8, Schur index 2
ρ222-22-2-22-22200-20000000000000    symplectic lifted from Q8, Schur index 2
ρ232-22-22-22-2200-20000000000000    symplectic lifted from Q8, Schur index 2
ρ242-22-2-2-222-20020000000000000    symplectic lifted from Q8, Schur index 2
ρ254-4-44000000000000000000000    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)])

D4×Q8 is a maximal subgroup of
Q83D8  D4.3Q16  D44Q16  Q84SD16  SD166D4  Q169D4  SD163D4  Q165D4  D48SD16  D45Q16  C42.47C23  C42.55C23  C42.477C23  C42.478C23  C22.75C25  C22.78C25  C4⋊2- 1+4  C22.88C25  C22.90C25  C22.103C25  C22.105C25  C23.144C24  C22.127C25  C22.130C25  C22.150C25
 D4p⋊Q8: D84Q8  D128Q8  D208Q8  D288Q8 ...
 C2p.2- 1+4: Q84D8  Q87SD16  C42.508C23  C42.514C23  SD162Q8  C22.71C25  C22.92C25  C22.107C25 ...
D4×Q8 is a maximal quotient of
C24.558C23  C23.247C24  C23.309C24  C24.252C23  C23.323C24  C23.329C24  C23.334C24  C24.568C23  C23.346C24  C23.349C24  C23.351C24  C23.352C24  C23.362C24  C24.285C23  C24.572C23  C23.392C24  C23.406C24  C42.166D4  C42.167D4  C42.169D4  C427Q8  C23.456C24  C24.385C23  C23.583C24  C23.592C24  C24.408C23  C23.613C24  C23.620C24  C23.626C24  C24.421C23  C23.632C24  C23.634C24
 D4p⋊Q8: D86Q8  D84Q8  D85Q8  D128Q8  D208Q8  D288Q8 ...
 C2p.2- 1+4: SD164Q8  Q166Q8  SD16⋊Q8  SD162Q8  Q164Q8  SD163Q8  Q165Q8  Dic621D4 ...

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

1200
4400
0010
0001
,
1000
4400
0040
0004
,
1000
0100
0001
0040
,
4000
0400
0002
0020
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

Export

Character table of D4×Q8 in TeX

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