Copied to
clipboard

G = D8○D8order 128 = 27

Central product of D8 and D8

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

Aliases: D8D8, Q16Q16, D814D4, Q1614D4, SD164D4, C42.457C23, M4(2).17C23, 2+ 1+44C22, C22.52+ 1+4, C2.76D42, D4○D84C2, C8○D810C2, C8.13(C2×D4), D44D47C2, C84D422C2, (C4×C8)⋊35C22, D4.35(C2×D4), C8○D46C22, C4≀C214C22, Q8.35(C2×D4), D4.4D47C2, (C2×D8)⋊31C22, C8⋊C223C22, (C2×C4).25C24, C41D414C22, (C2×C8).289C23, C4○D4.14C23, C4○D8.29C22, (C2×D4).11C23, C4.D45C22, C4.106(C22×D4), C8.C421C22, 2-Sylow(Omega+(4,7)), SmallGroup(128,2024)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D8○D8
C1C2C4C2×C4C4○D42+ 1+4D4○D8 — D8○D8
C1C2C2×C4 — D8○D8
C1C2C2×C4 — D8○D8
C1C2C2C2×C4 — D8○D8

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

Subgroups: 572 in 238 conjugacy classes, 92 normal (8 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×6], C22, C22 [×16], C8 [×4], C8 [×4], C2×C4, C2×C4 [×9], D4 [×4], D4 [×24], Q8 [×4], C23 [×8], C42, C2×C8 [×2], C2×C8 [×4], M4(2) [×4], M4(2) [×4], D8 [×2], D8 [×16], SD16 [×4], SD16 [×8], Q16 [×2], C2×D4 [×4], C2×D4 [×14], C4○D4 [×4], C4○D4 [×8], C4×C8, C4.D4 [×4], C4≀C2 [×4], C8.C4 [×2], C41D4 [×2], C8○D4 [×4], C2×D8 [×4], C2×D8 [×4], C4○D8 [×2], C4○D8 [×4], C8⋊C22 [×8], C8⋊C22 [×8], 2+ 1+4 [×4], C8○D8 [×2], D44D4 [×4], D4.4D4 [×4], C84D4, D4○D8 [×4], D8○D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ 1+4, D42, D8○D8

Character table of D8○D8

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J
 size 11244448888224444442222448888
ρ111111111111111111111111111111    trivial
ρ21111-1-111-1-11111-111-11-1-1-1-1-1-111-1-1    linear of order 2
ρ3111-1-1-1-11111111-1-1-1-11111111-1-1-1-1    linear of order 2
ρ4111-111-11-1-111111-1-111-1-1-1-1-1-1-1-111    linear of order 2
ρ5111111-1-11-1111-111-11-111-1-1-111-1-1-1    linear of order 2
ρ61111-1-1-1-1-11111-1-11-1-1-1-1-1111-11-111    linear of order 2
ρ7111-1-1-11-11-1111-1-1-11-1-111-1-1-11-1111    linear of order 2
ρ8111-1111-1-11111-11-111-1-1-1111-1-11-1-1    linear of order 2
ρ911111-111-11-111-1-1111-111-1-1-11-1-11-1    linear of order 2
ρ101111-11111-1-111-1111-1-1-1-1111-1-1-1-11    linear of order 2
ρ11111-1-11-11-11-111-11-1-1-1-111-1-1-1111-11    linear of order 2
ρ12111-11-1-111-1-111-1-1-1-11-1-1-1111-1111-1    linear of order 2
ρ1311111-1-1-1-1-1-1111-11-111111111-11-11    linear of order 2
ρ141111-11-1-111-111111-1-11-1-1-1-1-1-1-111-1    linear of order 2
ρ15111-1-111-1-1-1-11111-11-111111111-11-1    linear of order 2
ρ16111-11-11-111-1111-1-1111-1-1-1-1-1-11-1-11    linear of order 2
ρ1722-2-22000000-220020-20-2-200020000    orthogonal lifted from D4
ρ1822-200-2200002-2020-20000-2-2200000    orthogonal lifted from D4
ρ1922-200-2-200002-20202000022-200000    orthogonal lifted from D4
ρ2022-22-2000000-2200-2020-2-200020000    orthogonal lifted from D4
ρ2122-2002200002-20-20-2000022-200000    orthogonal lifted from D4
ρ2222-222000000-2200-20-2022000-20000    orthogonal lifted from D4
ρ2322-2-2-2000000-2200202022000-20000    orthogonal lifted from D4
ρ2422-2002-200002-20-2020000-2-2200000    orthogonal lifted from D4
ρ2544400000000-4-40000000000000000    orthogonal lifted from 2+ 1+4
ρ264-400000000000-200002-2222-2222000000    orthogonal faithful
ρ274-40000000000020000-222-22-2222000000    orthogonal faithful
ρ284-40000000000020000-2-222222-22000000    orthogonal faithful
ρ294-400000000000-20000222-2222-22000000    orthogonal faithful

Permutation representations of D8○D8
On 16 points - transitive group 16T296
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 16)(8 15)
(1 2 3 4 5 6 7 8)(9 16 15 14 13 12 11 10)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,16)(8,15), (1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,16)(8,15), (1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,16),(8,15)], [(1,2,3,4,5,6,7,8),(9,16,15,14,13,12,11,10)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15)])

G:=TransitiveGroup(16,296);

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

2026
6256
6152
2266
,
5063
0245
5401
3640
,
2245
1242
1145
5210
,
4621
3413
6321
4234
G:=sub<GL(4,GF(7))| [2,6,6,2,0,2,1,2,2,5,5,6,6,6,2,6],[5,0,5,3,0,2,4,6,6,4,0,4,3,5,1,0],[2,1,1,5,2,2,1,2,4,4,4,1,5,2,5,0],[4,3,6,4,6,4,3,2,2,1,2,3,1,3,1,4] >;

D8○D8 in GAP, Magma, Sage, TeX

D_8\circ D_8
% in TeX

G:=Group("D8oD8");
// GroupNames label

G:=SmallGroup(128,2024);
// by ID

G=gap.SmallGroup(128,2024);
# by ID

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

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

Export

Character table of D8○D8 in TeX

׿
×
𝔽