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G = D86D4order 128 = 27

5th semidirect product of D8 and D4 acting via D4/C4=C2

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

Aliases: D86D4, Q166D4, SD16SD16, SD1612D4, C42.456C23, M4(2).16C23, 2+ 1+43C22, C22.42+ 1+4, 2- 1+4.4C22, C2.75D42, C8○D89C2, C8.12(C2×D4), D44D46C2, C85D410C2, (C4×C8)⋊34C22, D4○SD164C2, D4.34(C2×D4), C8○D45C22, C4⋊Q819C22, C4≀C213C22, Q8.34(C2×D4), D4.3D46C2, (C2×C4).24C24, D4.10D46C2, (C2×Q8).8C23, (C2×C8).288C23, C4○D4.13C23, C4○D8.28C22, (C2×D4).10C23, C4.105(C22×D4), C4.D44C22, C8⋊C22.2C22, C8.C224C22, C8.C420C22, (C2×SD16)⋊32C22, C41D4.81C22, C4.10D45C22, 2-Sylow(CSO+(4,3)), SmallGroup(128,2023)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D86D4
C1C2C4C2×C4C4○D42+ 1+4D4○SD16 — D86D4
C1C2C2×C4 — D86D4
C1C2C2×C4 — D86D4
C1C2C2C2×C4 — D86D4

Generators and relations for D86D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, ac=ca, dad=a3, cbc-1=a6b, bd=db, dcd=c-1 >

Subgroups: 492 in 230 conjugacy classes, 92 normal (10 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×8], C22, C22 [×10], C8 [×4], C8 [×4], C2×C4, C2×C4 [×13], D4 [×4], D4 [×16], Q8 [×4], Q8 [×6], C23 [×4], C42, C4⋊C4, C2×C8 [×2], C2×C8 [×4], M4(2) [×4], M4(2) [×4], D8 [×2], D8 [×4], SD16 [×4], SD16 [×16], Q16 [×2], Q16 [×4], C2×D4 [×2], C2×D4 [×7], C2×Q8 [×2], C2×Q8 [×4], C4○D4 [×4], C4○D4 [×10], C4×C8, C4.D4 [×2], C4.10D4 [×2], C4≀C2 [×4], C8.C4 [×2], C41D4, C4⋊Q8, C8○D4 [×4], C2×SD16 [×4], C2×SD16 [×4], C4○D8 [×2], C4○D8 [×4], C8⋊C22 [×4], C8⋊C22 [×4], C8.C22 [×4], C8.C22 [×4], 2+ 1+4 [×2], 2- 1+4 [×2], C8○D8 [×2], D44D4 [×2], D4.10D4 [×2], D4.3D4 [×4], C85D4, D4○SD16 [×4], D86D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ 1+4, D42, D86D4

Character table of D86D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J
 size 11244448822444444882222448888
ρ111111111111111111111111111111    trivial
ρ2111-1-1-1-111111-1-1-1-1111111111-1-1-1-1    linear of order 2
ρ311111-111-111-1-1111-11-111-1-11-1-11-1-1    linear of order 2
ρ4111-1-11-11-111-11-1-1-1-11-111-1-11-11-111    linear of order 2
ρ5111111-1-1111-11-111-11-1-1-111-11-1-1-11    linear of order 2
ρ6111-1-1-11-1111-1-11-1-1-11-1-1-111-11111-1    linear of order 2
ρ711111-1-1-1-1111-1-111111-1-1-1-1-1-11-11-1    linear of order 2
ρ8111-1-111-1-111111-1-1111-1-1-1-1-1-1-11-11    linear of order 2
ρ91111-1-1111111-111-11-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ10111-111-1111111-1-111-1-1-1-1-1-1-1-111-1-1    linear of order 2
ρ111111-1111-111-1111-1-1-11-1-111-111-1-1-1    linear of order 2
ρ12111-11-1-11-111-1-1-1-11-1-11-1-111-11-1111    linear of order 2
ρ131111-1-1-1-1111-1-1-11-1-1-1111-1-11-111-11    linear of order 2
ρ14111-1111-1111-111-11-1-1111-1-11-1-1-11-1    linear of order 2
ρ151111-11-1-1-11111-11-11-1-1111111-111-1    linear of order 2
ρ16111-11-11-1-1111-11-111-1-11111111-1-11    linear of order 2
ρ1722-22020002-20-20-2000000-2-2020000    orthogonal lifted from D4
ρ1822-20-20-200-2200202000-2-200200000    orthogonal lifted from D4
ρ1922-2020-200-220020-20002200-200000    orthogonal lifted from D4
ρ2022-2-20-20002-20202000000-2-2020000    orthogonal lifted from D4
ρ2122-2-2020002-20-202000000220-20000    orthogonal lifted from D4
ρ2222-20-20200-2200-2020002200-200000    orthogonal lifted from D4
ρ2322-2020200-2200-20-2000-2-200200000    orthogonal lifted from D4
ρ2422-220-20002-2020-2000000220-20000    orthogonal lifted from D4
ρ25444000000-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-400000000020000-200-2-22-2-2-22-2000000    complex faithful
ρ274-400000000020000-2002-2-2-22-2-2-2000000    complex faithful
ρ284-4000000000-200002002-2-2-2-2-22-2000000    complex faithful
ρ294-4000000000-20000200-2-22-22-2-2-2000000    complex faithful

Permutation representations of D86D4
On 16 points - transitive group 16T328
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 5)(2 6)(3 7)(4 8)(9 11 13 15)(10 12 14 16)
(1 5)(2 8)(4 6)(9 15)(11 13)(12 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), (1,5)(2,6)(3,7)(4,8)(9,11,13,15)(10,12,14,16), (1,5)(2,8)(4,6)(9,15)(11,13)(12,16)>;

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

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

G:=TransitiveGroup(16,328);

Matrix representation of D86D4 in GL4(𝔽3) generated by

1010
2011
0120
2120
,
0012
2011
2221
1221
,
0121
0102
1220
2122
,
2210
1202
1112
1211
G:=sub<GL(4,GF(3))| [1,2,0,2,0,0,1,1,1,1,2,2,0,1,0,0],[0,2,2,1,0,0,2,2,1,1,2,2,2,1,1,1],[0,0,1,2,1,1,2,1,2,0,2,2,1,2,0,2],[2,1,1,1,2,2,1,2,1,0,1,1,0,2,2,1] >;

D86D4 in GAP, Magma, Sage, TeX

D_8\rtimes_6D_4
% in TeX

G:=Group("D8:6D4");
// GroupNames label

G:=SmallGroup(128,2023);
// by ID

G=gap.SmallGroup(128,2023);
# by ID

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

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^3,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of D86D4 in TeX

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