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G = D83D4order 128 = 27

2nd semidirect product of D8 and D4 acting via D4/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D83D4, C8.4D8, Q163D4, C42.143D4, M5(2).4C22, C8○D84C2, C4.68(C2×D8), C8.77(C2×D4), C8.C82C2, C84D415C2, C16⋊C224C2, C8.2(C4○D4), (C2×C8).131D4, M5(2)⋊C24C2, C2.25(C4⋊D8), C4.56(C4⋊D4), (C2×C8).236C23, (C4×C8).163C22, C4○D8.19C22, (C2×D8).48C22, C22.25(C8⋊C22), C8.C4.19C22, (C2×C4).281(C2×D4), SmallGroup(128,945)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D83D4
C1C2C4C8C2×C8C4○D8C8○D8 — D83D4
C1C2C4C2×C8 — D83D4
C1C2C2×C4C4×C8 — D83D4
C1C2C2C2C2C4C4C2×C8 — D83D4

Generators and relations for D83D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >

Subgroups: 268 in 85 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×3], C22, C22 [×7], C8 [×4], C8, C2×C4, C2×C4 [×2], D4 [×10], Q8, C23 [×2], C16 [×2], C42, C2×C8 [×2], C2×C8, M4(2) [×2], D8, D8 [×6], SD16, Q16, C2×D4 [×4], C4○D4, C4×C8, C4≀C2, C8.C4, M5(2) [×2], D16 [×2], SD32 [×2], C41D4, C8○D4, C2×D8 [×2], C2×D8, C4○D8, M5(2)⋊C2 [×2], C8.C8, C8○D8, C84D4, C16⋊C22 [×2], D83D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, D83D4

Character table of D83D4

 class 12A2B2C2D2E4A4B4C4D4E8A8B8C8D8E8F8G8H16A16B16C16D
 size 1128161622448222244888888
ρ111111111111111111111111    trivial
ρ2111-1-1-11111-1111111-1-11111    linear of order 2
ρ3111-1111111-1111111-1-1-1-1-1-1    linear of order 2
ρ41111-1-11111111111111-1-1-1-1    linear of order 2
ρ5111-11-111-1-1-1-11-111-11111-1-1    linear of order 2
ρ61111-1111-1-11-11-111-1-1-111-1-1    linear of order 2
ρ711111-111-1-11-11-111-1-1-1-1-111    linear of order 2
ρ8111-1-1111-1-1-1-11-111-111-1-111    linear of order 2
ρ922-2200-2200-20202-20000000    orthogonal lifted from D4
ρ1022200022220-2-2-2-2-2-2000000    orthogonal lifted from D4
ρ1122200022-2-202-22-2-22000000    orthogonal lifted from D4
ρ1222-2-200-220020202-20000000    orthogonal lifted from D4
ρ1322-20002-2000-20-200200-22-22    orthogonal lifted from D8
ρ1422-20002-2000-20-2002002-22-2    orthogonal lifted from D8
ρ1522-20002-200020200-2002-2-22    orthogonal lifted from D8
ρ1622-20002-200020200-200-222-2    orthogonal lifted from D8
ρ1722-2000-220000-20-220-2i2i0000    complex lifted from C4○D4
ρ1822-2000-220000-20-2202i-2i0000    complex lifted from C4○D4
ρ19444000-4-4000000000000000    orthogonal lifted from C8⋊C22
ρ204-40000002-20-222222-2200000000    orthogonal faithful
ρ214-40000002-2022-22-222200000000    orthogonal faithful
ρ224-4000000-2202222-22-2200000000    orthogonal faithful
ρ234-4000000-220-22-22222200000000    orthogonal faithful

Permutation representations of D83D4
On 16 points - transitive group 16T379
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)
(1 7 5 3)(2 8 6 4)(9 13)(10 14)(11 15)(12 16)
(1 3)(4 8)(5 7)(9 14)(10 13)(11 12)(15 16)

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

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

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

G:=TransitiveGroup(16,379);

Matrix representation of D83D4 in GL4(𝔽7) generated by

5051
1521
1625
5511
,
3211
3455
5660
2121
,
6361
0254
0566
0235
,
1104
0344
0656
0665
G:=sub<GL(4,GF(7))| [5,1,1,5,0,5,6,5,5,2,2,1,1,1,5,1],[3,3,5,2,2,4,6,1,1,5,6,2,1,5,0,1],[6,0,0,0,3,2,5,2,6,5,6,3,1,4,6,5],[1,0,0,0,1,3,6,6,0,4,5,6,4,4,6,5] >;

D83D4 in GAP, Magma, Sage, TeX

D_8\rtimes_3D_4
% in TeX

G:=Group("D8:3D4");
// GroupNames label

G:=SmallGroup(128,945);
// by ID

G=gap.SmallGroup(128,945);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,512,422,2019,248,1684,438,242,4037,1027,124]);
// Polycyclic

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

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Character table of D83D4 in TeX

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