Copied to
clipboard

G = D8○Q16order 128 = 27

Central product of D8 and Q16

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

Aliases: D8Q16, D8.14D4, Q16.14D4, SD16.1D4, C42.458C23, M4(2).18C23, C22.62+ 1+4, 2- 1+4.5C22, C2.77D42, Q8○D84C2, C8○D811C2, C4≀C2.C22, C8.14(C2×D4), D4.36(C2×D4), Q8.36(C2×D4), C4⋊Q1623C2, D4.5D47C2, (C2×C4).26C24, C8○D4.6C22, D4.10D47C2, (C2×Q8).9C23, C8.C22.C22, (C2×C8).290C23, (C4×C8).192C22, C4○D8.30C22, C4○D4.15C23, C4.107(C22×D4), C4⋊Q8.140C22, (C2×Q16).82C22, C8.C4.23C22, C4.10D4.4C22, SmallGroup(128,2025)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D8○Q16
C1C2C4C2×C4C4○D42- 1+4Q8○D8 — D8○Q16
C1C2C2×C4 — D8○Q16
C1C2C2×C4 — D8○Q16
C1C2C2C2×C4 — D8○Q16

Generators and relations for D8○Q16
 G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a4c3 >

Subgroups: 412 in 222 conjugacy classes, 92 normal (8 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×10], C22, C22 [×4], C8 [×4], C8 [×4], C2×C4, C2×C4 [×17], D4 [×4], D4 [×8], Q8 [×4], Q8 [×12], C42, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×4], M4(2) [×4], M4(2) [×4], D8 [×2], SD16 [×4], SD16 [×8], Q16 [×2], Q16 [×16], C2×Q8 [×4], C2×Q8 [×8], C4○D4 [×4], C4○D4 [×12], C4×C8, C4.10D4 [×4], C4≀C2 [×4], C8.C4 [×2], C4⋊Q8 [×2], C8○D4 [×4], C2×Q16 [×4], C2×Q16 [×4], C4○D8 [×2], C4○D8 [×4], C8.C22 [×8], C8.C22 [×8], 2- 1+4 [×4], C8○D8 [×2], D4.10D4 [×4], D4.5D4 [×4], C4⋊Q16, Q8○D8 [×4], D8○Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ 1+4, D42, D8○Q16

Character table of D8○Q16

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H8I8J
 size 11244442244444488882222448888
ρ111111111111111111111111111111    trivial
ρ21111-1-11111-11-111-111-1-1-1-1-1-1-1-1-111    linear of order 2
ρ311111-1111-1-1111-1-1-111-1-111-11-11-1-1    linear of order 2
ρ41111-11111-111-11-11-11-111-1-11-11-1-1-1    linear of order 2
ρ511111-1-1111-111-11-1-1-1-11111111-11-1    linear of order 2
ρ61111-11-111111-1-111-1-11-1-1-1-1-1-1-111-1    linear of order 2
ρ7111111-111-1111-1-111-1-1-1-111-11-1-1-11    linear of order 2
ρ81111-1-1-111-1-11-1-1-1-11-1111-1-11-111-11    linear of order 2
ρ9111-111-11111-11-11-111-1-1-1-1-1-1-111-1-1    linear of order 2
ρ10111-1-1-1-1111-1-1-1-111111111111-1-1-1-1    linear of order 2
ρ11111-11-1-111-1-1-11-1-11-11-111-1-11-1-1111    linear of order 2
ρ12111-1-11-111-11-1-1-1-1-1-111-1-111-111-111    linear of order 2
ρ13111-11-11111-1-11111-1-11-1-1-1-1-1-11-1-11    linear of order 2
ρ14111-1-1111111-1-111-1-1-1-1111111-11-11    linear of order 2
ρ15111-111111-11-111-1-11-1111-1-11-1-1-11-1    linear of order 2
ρ16111-1-1-1111-1-1-1-11-111-1-1-1-111-11111-1    linear of order 2
ρ1722-2002-2-220-200200000-2-200200000    orthogonal lifted from D4
ρ1822-200-22-220200-200000-2-200200000    orthogonal lifted from D4
ρ1922-2-22002-2002-200000000-2-2020000    orthogonal lifted from D4
ρ2022-22-2002-200-2200000000-2-2020000    orthogonal lifted from D4
ρ2122-2-2-2002-2002200000000220-20000    orthogonal lifted from D4
ρ2222-222002-200-2-200000000220-20000    orthogonal lifted from D4
ρ2322-20022-220-200-2000002200-200000    orthogonal lifted from D4
ρ2422-200-2-2-2202002000002200-200000    orthogonal lifted from D4
ρ254440000-4-400000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4000000020000-2000022-22-2222000000    symplectic faithful, Schur index 2
ρ274-40000000-2000020000-2222-2222000000    symplectic faithful, Schur index 2
ρ284-4000000020000-20000-222222-22000000    symplectic faithful, Schur index 2
ρ294-40000000-200002000022-2222-22000000    symplectic faithful, Schur index 2

Smallest permutation representation of D8○Q16
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 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)
(1 2 3 4 5 6 7 8)(9 16 15 14 13 12 11 10)(17 24 23 22 21 20 19 18)(25 26 27 28 29 30 31 32)
(1 20 5 24)(2 21 6 17)(3 22 7 18)(4 23 8 19)(9 32 13 28)(10 25 14 29)(11 26 15 30)(12 27 16 31)

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

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

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

Matrix representation of D8○Q16 in GL4(𝔽7) generated by

5026
6556
6112
2262
,
4253
3336
6463
4501
,
5245
1542
1105
5213
,
4221
1413
0455
2411
G:=sub<GL(4,GF(7))| [5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[4,3,6,4,2,3,4,5,5,3,6,0,3,6,3,1],[5,1,1,5,2,5,1,2,4,4,0,1,5,2,5,3],[4,1,0,2,2,4,4,4,2,1,5,1,1,3,5,1] >;

D8○Q16 in GAP, Magma, Sage, TeX

D_8\circ Q_{16}
% in TeX

G:=Group("D8oQ16");
// GroupNames label

G:=SmallGroup(128,2025);
// by ID

G=gap.SmallGroup(128,2025);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,723,352,346,2804,1411,375,172,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=d^2=a^4,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=a^4*c^3>;
// generators/relations

Export

Character table of D8○Q16 in TeX

׿
×
𝔽