p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8○D8, D4○Q16, D4.13D4, C8.5C23, Q8.13D4, C4.10C24, D8.4C22, D4.7C23, SD16.C22, Q8.7C23, Q16.4C22, 2- 1+4⋊3C2, M4(2).6C22, C8○D4⋊5C2, C4○D8⋊6C2, C4.43(C2×D4), (C2×Q16)⋊12C2, C8.C22⋊5C2, C22.7(C2×D4), (C2×C8).26C22, (C2×C4).45C23, C2.32(C22×D4), C4○D4.13C22, (C2×Q8).37C22, SmallGroup(64,259)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8○D8
G = < a,b,c,d | a4=d2=1, b2=c4=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c3 >
Subgroups: 173 in 124 conjugacy classes, 79 normal (9 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, Q8○D8
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, Q8○D8
Character table of Q8○D8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 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 | trivial |
| ρ2 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 32 points
(1 26 5 30)(2 27 6 31)(3 28 7 32)(4 29 8 25)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)
(1 13 5 9)(2 14 6 10)(3 15 7 11)(4 16 8 12)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)
(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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 24)(18 23)(19 22)(20 21)(25 26)(27 32)(28 31)(29 30)
G:=sub<Sym(32)| (1,26,5,30)(2,27,6,31)(3,28,7,32)(4,29,8,25)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24), (1,13,5,9)(2,14,6,10)(3,15,7,11)(4,16,8,12)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,24)(18,23)(19,22)(20,21)(25,26)(27,32)(28,31)(29,30)>;
G:=Group( (1,26,5,30)(2,27,6,31)(3,28,7,32)(4,29,8,25)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24), (1,13,5,9)(2,14,6,10)(3,15,7,11)(4,16,8,12)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,24)(18,23)(19,22)(20,21)(25,26)(27,32)(28,31)(29,30) );
G=PermutationGroup([[(1,26,5,30),(2,27,6,31),(3,28,7,32),(4,29,8,25),(9,17,13,21),(10,18,14,22),(11,19,15,23),(12,20,16,24)], [(1,13,5,9),(2,14,6,10),(3,15,7,11),(4,16,8,12),(17,30,21,26),(18,31,22,27),(19,32,23,28),(20,25,24,29)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,24),(18,23),(19,22),(20,21),(25,26),(27,32),(28,31),(29,30)]])
Q8○D8 is a maximal subgroup of
D8.A4 D4.5S4
D4p.D4: Q16.D4 D8.12D4 D8.13D4 D8○SD16 D8○Q16 D12.30D4 D8.10D6 SD16.D6 ...
D4p.C23: D4○SD32 Q8○D16 C8.C24 C4.C25 D4.13D12 D12.35C23 D4.13D20 D20.35C23 ...
Q8○D8 is a maximal quotient of
C42.17C23 C42.21C23 C42.22C23 C42.367C23 M4(2)⋊4Q8 C42.409C23 C42.411C23 C42.424C23 C42.425C23 C42.25C23 C42.28C23 C42.29C23 Q8⋊7SD16 C42.508C23 C42.515C23 Q8×D8 Q16⋊6Q8 SD16⋊2Q8 Q16⋊5Q8 Q8⋊6Q16 C42.73C23
(Cp×D4).D4: 2- 1+4⋊4C4 C4○D4.8Q8 Q8.(C2×D4) (C2×Q8)⋊17D4 C8.D4⋊C2 (C2×C8)⋊14D4 M4(2)⋊17D4 (C2×D4).302D4 ...
M4(2).D2p: M4(2).20D4 SD16.D6 D20.44D4 D28.44D4 ...
(C2p×Q16)⋊C2: C42.276C23 C42.279C23 C42.280C23 C42.354C23 C42.355C23 C42.358C23 C42.361C23 C42.387C23 ...
C4○D4.D2p: C42.19C23 D8.10D6 D20.47D4 D8.10D14 ...
Matrix representation of Q8○D8 ►in GL4(𝔽7) generated by
| 1 | 6 | 4 | 2 |
| 3 | 1 | 2 | 6 |
| 3 | 4 | 5 | 1 |
| 5 | 0 | 3 | 0 |
| 3 | 5 | 5 | 6 |
| 6 | 3 | 6 | 4 |
| 0 | 3 | 2 | 2 |
| 5 | 3 | 6 | 6 |
| 5 | 0 | 5 | 1 |
| 1 | 5 | 2 | 1 |
| 1 | 6 | 2 | 5 |
| 5 | 5 | 1 | 1 |
| 0 | 6 | 2 | 6 |
| 2 | 0 | 5 | 6 |
| 6 | 1 | 4 | 1 |
| 2 | 2 | 3 | 3 |
G:=sub<GL(4,GF(7))| [1,3,3,5,6,1,4,0,4,2,5,3,2,6,1,0],[3,6,0,5,5,3,3,3,5,6,2,6,6,4,2,6],[5,1,1,5,0,5,6,5,5,2,2,1,1,1,5,1],[0,2,6,2,6,0,1,2,2,5,4,3,6,6,1,3] >;
Q8○D8 in GAP, Magma, Sage, TeX
Q_8\circ D_8
% in TeX
G:=Group("Q8oD8"); // GroupNames label
G:=SmallGroup(64,259);
// by ID
G=gap.SmallGroup(64,259);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,-2,192,217,199,255,1444,730,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=d^2=1,b^2=c^4=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^3>;
// generators/relations
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