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G = D4.8D4order 64 = 26

3rd non-split extension by D4 of D4 acting via D4/C22=C2

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

Aliases: D4.8D4, Q8.8D4, C42.14C22, 2- 1+41C2, M4(2).1C22, C4≀C22C2, (C2×C4).5D4, C8⋊C222C2, C4.26(C2×D4), C4.4D41C2, (C2×C4).5C23, C2.15C22≀C2, C4.10D41C2, C4○D4.2C22, (C2×D4).7C22, C22.13(C2×D4), (C2×Q8).5C22, SmallGroup(64,135)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4.8D4
C1C2C22C2×C4C2×Q82- 1+4 — D4.8D4
C1C2C2×C4 — D4.8D4
C1C2C2×C4 — D4.8D4
C1C2C2C2×C4 — D4.8D4

Generators and relations for D4.8D4
 G = < a,b,c,d | a4=b2=d2=1, c4=a2, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a-1b, dcd=a2c3 >

Subgroups: 137 in 73 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×5], C22, C22 [×5], C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×7], D4 [×2], D4 [×6], Q8 [×2], Q8 [×4], C23, C42, C22⋊C4 [×2], M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C4○D4 [×2], C4○D4 [×4], C4.10D4, C4≀C2 [×2], C4.4D4, C8⋊C22 [×2], 2- 1+4, D4.8D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, D4.8D4

Character table of D4.8D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B
 size 1124482244444488
ρ11111111111111111    trivial
ρ2111-1-1111-11-1111-1-1    linear of order 2
ρ3111-1-1-111-11-1-11-111    linear of order 2
ρ411111-111111-11-1-1-1    linear of order 2
ρ51111-1-111-1-111-11-11    linear of order 2
ρ6111-11-1111-1-11-111-1    linear of order 2
ρ7111-111111-1-1-1-1-1-11    linear of order 2
ρ81111-1111-1-11-1-1-11-1    linear of order 2
ρ922-2200-2200-200000    orthogonal lifted from D4
ρ1022-20-202-220000000    orthogonal lifted from D4
ρ11222000-2-20200-2000    orthogonal lifted from D4
ρ1222-20202-2-20000000    orthogonal lifted from D4
ρ1322-2-200-2200200000    orthogonal lifted from D4
ρ14222000-2-20-2002000    orthogonal lifted from D4
ρ154-40000000002i0-2i00    complex faithful
ρ164-4000000000-2i02i00    complex faithful

Permutation representations of D4.8D4
On 16 points - transitive group 16T162
Generators in S16
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)
(1 11)(2 14)(3 13)(4 16)(5 15)(6 10)(7 9)(8 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)

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

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

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

G:=TransitiveGroup(16,162);

On 16 points - transitive group 16T164
Generators in S16
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)
(1 2)(3 4)(5 6)(7 8)(9 14)(10 13)(11 16)(12 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(8 16)

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

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

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

G:=TransitiveGroup(16,164);

D4.8D4 is a maximal subgroup of
C42.313C23  C42.12C23  D811D4  D4.3S4  2- 1+4⋊D5
 D4p.D4: D8.13D4  D8○SD16  D12.5D4  D4.10D12  D12.14D4  D12.38D4  D20.5D4  D4.10D20 ...
 (Cp×D4).D4: C42.3D4  C42.15D4  C42.16D4  C42.17D4  M4(2).C23  2- 1+44S3  2- 1+42D5  2- 1+4⋊D7 ...
D4.8D4 is a maximal quotient of
(C2×C4)⋊SD16  C4⋊C4.20D4  Q86SD16  Q8.Q16  C42.4C23  C42.9C23  C4.10D42C4  C4.4D413C4  M4(2)⋊D4  (C2×D4)⋊2Q8  C423Q8
 C42.D2p: C42.129D4  D4.10D12  D12.14D4  D4.10D20  D20.14D4  D4.10D28  D28.14D4 ...
 M4(2).D2p: M4(2).7D4  D12.5D4  D12.38D4  D20.5D4  D20.38D4  D28.5D4  D28.38D4 ...
 (Cp×D4).D4: C4⋊C4.D4  (C2×C4)⋊D8  C4⋊C4.18D4  C4⋊C4.19D4  D43D8  Q83D8  C42.189C23  D4.SD16 ...

Matrix representation of D4.8D4 in GL4(𝔽5) generated by

3000
0300
0020
0002
,
0023
0010
0100
2100
,
0010
0001
1300
1400
,
0040
0041
4000
4100
G:=sub<GL(4,GF(5))| [3,0,0,0,0,3,0,0,0,0,2,0,0,0,0,2],[0,0,0,2,0,0,1,1,2,1,0,0,3,0,0,0],[0,0,1,1,0,0,3,4,1,0,0,0,0,1,0,0],[0,0,4,4,0,0,0,1,4,4,0,0,0,1,0,0] >;

D4.8D4 in GAP, Magma, Sage, TeX

D_4._8D_4
% in TeX

G:=Group("D4.8D4");
// GroupNames label

G:=SmallGroup(64,135);
// by ID

G=gap.SmallGroup(64,135);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,362,158,963,489,255,117,730]);
// Polycyclic

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

Export

Character table of D4.8D4 in TeX

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