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G = Q8○D8order 64 = 26

Central product of Q8 and D8

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

Aliases: Q8D8, D4Q16, D4.13D4, C8.5C23, Q8.13D4, C4.10C24, D8.4C22, D4.7C23, SD16.C22, Q8.7C23, Q16.4C22, 2- 1+43C2, M4(2).6C22, C8○D45C2, C4○D86C2, C4.43(C2×D4), (C2×Q16)⋊12C2, C8.C225C2, 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

C1C4 — Q8○D8
C1C2C4C2×C4C4○D42- 1+4 — Q8○D8
C1C2C4 — Q8○D8
C1C2C4○D4 — Q8○D8
C1C2C2C4 — Q8○D8

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 [×5], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C8, C8 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×5], D4 [×6], Q8, Q8 [×6], Q8 [×6], C2×C8 [×3], M4(2) [×3], D8, SD16 [×6], Q16 [×9], C2×Q8 [×6], C2×Q8 [×2], C4○D4, C4○D4 [×6], C4○D4 [×6], C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22 [×6], 2- 1+4 [×2], Q8○D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, Q8○D8

Character table of Q8○D8

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E
 size 1122244222244444422444
ρ11111111111111111111111    trivial
ρ211-11-1-1-1-111-1111-11-1111-1-1    linear of order 2
ρ3111111-11111-1111-1-1-1-1-1-1-1    linear of order 2
ρ411-11-1-11-111-1-111-1-11-1-1-111    linear of order 2
ρ5111-1-111-11-11-1-11-11-111-11-1    linear of order 2
ρ611-1-11-1-111-1-1-1-1111111-1-11    linear of order 2
ρ7111-1-11-1-11-111-11-1-11-1-11-11    linear of order 2
ρ811-1-11-1111-1-11-111-1-1-1-111-1    linear of order 2
ρ911111-1111111-1-1-111-1-1-1-1-1    linear of order 2
ρ1011-11-11-1-111-11-1-111-1-1-1-111    linear of order 2
ρ1111111-1-11111-1-1-1-1-1-111111    linear of order 2
ρ1211-11-111-111-1-1-1-11-11111-1-1    linear of order 2
ρ13111-1-1-11-11-11-11-111-1-1-11-11    linear of order 2
ρ1411-1-111-111-1-1-11-1-111-1-111-1    linear of order 2
ρ15111-1-1-1-1-11-1111-11-1111-11-1    linear of order 2
ρ1611-1-111111-1-111-1-1-1-111-1-11    linear of order 2
ρ1722-22200-2-2-2200000000000    orthogonal lifted from D4
ρ182222-2002-2-2-200000000000    orthogonal lifted from D4
ρ1922-2-2-2002-22200000000000    orthogonal lifted from D4
ρ20222-2200-2-22-200000000000    orthogonal lifted from D4
ρ214-400000000000000022-22000    symplectic faithful, Schur index 2
ρ224-4000000000000000-2222000    symplectic faithful, Schur index 2

Smallest permutation representation of Q8○D8
On 32 points
Generators in S32
(1 28 5 32)(2 29 6 25)(3 30 7 26)(4 31 8 27)(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 32 21 28)(18 25 22 29)(19 26 23 30)(20 27 24 31)
(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 30)(26 29)(27 28)(31 32)

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

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

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

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  Q87SD16  C42.508C23  C42.515C23  Q8×D8  Q166Q8  SD162Q8  Q165Q8  Q86Q16  C42.73C23
 (Cp×D4).D4: 2- 1+44C4  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

1642
3126
3451
5030
,
3556
6364
0322
5366
,
5051
1521
1625
5511
,
0626
2056
6141
2233
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

Export

Character table of Q8○D8 in TeX

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