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G = D8○D8  order 128 = 27

Central product of D8 and D8

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

Aliases: D8○D8, Q16○Q16, D8⋊14D4, Q16⋊14D4, SD16⋊4D4, C42.457C23, M4(2).17C23, 2+ (1+4)⋊4C22, C22.52+ (1+4), D4○D8⋊4C2, C8○D8⋊10C2, C2.76(D42), C8.13(C2×D4), D4⋊4D4⋊7C2, C8⋊4D4⋊22C2, (C4×C8)⋊35C22, D4.35(C2×D4), C8○D4⋊6C22, C4≀C2⋊14C22, Q8.35(C2×D4), D4.4D4⋊7C2, (C2×D8)⋊31C22, C8⋊C22⋊3C22, (C2×C4).25C24, C4⋊1D4⋊14C22, (C2×C8).289C23, C4○D4.14C23, C4○D8.29C22, (C2×D4).11C23, C4.106(C22×D4), C4.D4⋊5C22, C8.C4⋊21C22, 2-Sylow(Omega+(4,7)), SmallGroup(128,2024)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — D8○D8
Chief series
C1 — C2 — C4 — C2×C4 — C4○D4 — 2+ (1+4) — D4○D8 — D8○D8
Lower central
C1 — C2 — C2×C4 — D8○D8
Upper central
C1 — C2 — C2×C4 — D8○D8
Jennings
C1 — C2 — C2 — C2×C4 — D8○D8

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], C4⋊1D4 [×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], D4⋊4D4 [×4], D4.4D4 [×4], C8⋊4D4, 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

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

Permutation representations
►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 10)(2 9)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)

(1 2 3 4 5 6 7 8)(9 16 15 14 13 12 11 10)

(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,10)(2,9)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11), (1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,10)(2,9)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11), (1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11) );
 
G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,10),(2,9),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11)], [(1,2,3,4,5,6,7,8),(9,16,15,14,13,12,11,10)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11)])
 
G:=TransitiveGroup(16,296);
 

Matrix representation ►G ⊆ 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] >;
 

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)
ρ264-400000000000-2000022√22√22√22√2000000    orthogonal faithful
ρ274-40000000000020000-22√22√22√22√2000000    orthogonal faithful
ρ284-40000000000020000-22√22√22√22√2000000    orthogonal faithful
ρ294-400000000000-2000022√22√22√22√2000000    orthogonal faithful

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
 

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