direct product, p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: D4×Q8, C22.43C24, C42.45C22, C23.46C23, C2.112- 1+4, C4⋊2(C2×Q8), C4⋊Q8⋊14C2, (C4×Q8)⋊12C2, C4.39(C2×D4), C22⋊2(C2×Q8), (C4×D4).10C2, C22⋊Q8⋊14C2, (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
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, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, D4×Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, 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 |
►On 32 points
(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
Q8⋊3D8 D4.3Q16 D4⋊4Q16 Q8⋊4SD16 SD16⋊6D4 Q16⋊9D4 SD16⋊3D4 Q16⋊5D4 D4⋊8SD16 D4⋊5Q16 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: D8⋊4Q8 D12⋊8Q8 D20⋊8Q8 D28⋊8Q8 ...
C2p.2- 1+4: Q8⋊4D8 Q8⋊7SD16 C42.508C23 C42.514C23 SD16⋊2Q8 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 C42⋊7Q8 C23.456C24 C24.385C23 C23.583C24 C23.592C24 C24.408C23 C23.613C24 C23.620C24 C23.626C24 C24.421C23 C23.632C24 C23.634C24
D4p⋊Q8: D8⋊6Q8 D8⋊4Q8 D8⋊5Q8 D12⋊8Q8 D20⋊8Q8 D28⋊8Q8 ...
C2p.2- 1+4: SD16⋊4Q8 Q16⋊6Q8 SD16⋊Q8 SD16⋊2Q8 Q16⋊4Q8 SD16⋊3Q8 Q16⋊5Q8 Dic6⋊21D4 ...
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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