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G = D8○Q16  order 128 = 27

Central product of D8 and Q16

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

Aliases: D8○Q16, D8.14D4, Q16.14D4, SD16.1D4, C42.458C23, M4(2).18C23, C22.62+ 1+4, 2- 1+4.5C22, C2.77D42, Q8○D8⋊4C2, C8○D8⋊11C2, C4≀C2.C22, C8.14(C2×D4), D4.36(C2×D4), Q8.36(C2×D4), C4⋊Q16⋊23C2, D4.5D4⋊7C2, (C2×C4).26C24, C8○D4.6C22, D4.10D4⋊7C2, (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

Derived series
C1 — C2×C4 — D8○Q16
Chief series
C1 — C2 — C4 — C2×C4 — C4○D4 — 2- 1+4 — Q8○D8 — D8○Q16
Lower central
C1 — C2 — C2×C4 — D8○Q16
Upper central
C1 — C2 — C2×C4 — D8○Q16
Jennings
C1 — C2 — C2 — C2×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, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C42, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, SD16, Q16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C4×C8, C4.10D4, C4≀C2, C8.C4, C4⋊Q8, C8○D4, C2×Q16, C2×Q16, C4○D8, C4○D8, C8.C22, C8.C22, 2- 1+4, C8○D8, D4.10D4, D4.5D4, C4⋊Q16, Q8○D8, D8○Q16
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 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
ρ264-4000000020000-200002√2-2√2-2√22√2000000    symplectic faithful, Schur index 2
ρ274-40000000-2000020000-2√22√2-2√22√2000000    symplectic faithful, Schur index 2
ρ284-4000000020000-20000-2√22√22√2-2√2000000    symplectic faithful, Schur index 2
ρ294-40000000-20000200002√2-2√22√2-2√2000000    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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(8 16)(17 25)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)

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

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