Copied to
clipboard

G = D4.4D4order 64 = 26

4th non-split extension by D4 of D4 acting via D4/C4=C2

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

Aliases: D4.4D4, C8.19D4, Q8.4D4, M4(2).4C22, C8○D42C2, (C2×D8)⋊8C2, C8⋊C223C2, C4.59(C2×D4), C8.C46C2, C4.D44C2, (C2×C4).9C23, (C2×C8).18C22, C2.24(C4⋊D4), C4○D4.10C22, (C2×D4).20C22, C22.7(C4○D4), SmallGroup(64,153)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4.4D4
C1C2C4C2×C4C4○D4C8○D4 — D4.4D4
C1C2C2×C4 — D4.4D4
C1C2C2×C4 — D4.4D4
C1C2C2C2×C4 — D4.4D4

Generators and relations for D4.4D4
 G = < a,b,c,d | a4=b2=d2=1, c4=a2, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=a2c3 >

2C2
4C2
8C2
8C2
2C22
2C4
4C22
4C22
8C22
8C22
2D4
2C8
2C23
2C8
2D4
2C8
2D4
2C23
2D4
2C2×C4
2D4
2D8
2SD16
2D8
2SD16
2D8
2D8
2M4(2)
2C2×C8

Character table of D4.4D4

 class 12A2B2C2D2E4A4B4C8A8B8C8D8E8F8G
 size 1124882242244488
ρ11111111111111111    trivial
ρ2111-11-111-1-1-1-111-11    linear of order 2
ρ3111-1-1111-1-1-1-1111-1    linear of order 2
ρ41111-1-111111111-1-1    linear of order 2
ρ5111-11111-1111-1-1-1-1    linear of order 2
ρ611111-1111-1-1-1-1-11-1    linear of order 2
ρ71111-11111-1-1-1-1-1-11    linear of order 2
ρ8111-1-1-111-1111-1-111    linear of order 2
ρ922-2-2002-220000000    orthogonal lifted from D4
ρ1022-2000-22022-20000    orthogonal lifted from D4
ρ1122-22002-2-20000000    orthogonal lifted from D4
ρ1222-2000-220-2-220000    orthogonal lifted from D4
ρ13222000-2-20000-2i2i00    complex lifted from C4○D4
ρ14222000-2-200002i-2i00    complex lifted from C4○D4
ρ154-40000000-222200000    orthogonal faithful
ρ164-4000000022-2200000    orthogonal faithful

Permutation representations of D4.4D4
On 16 points - transitive group 16T133
Generators in S16
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)

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

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

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

G:=TransitiveGroup(16,133);

D4.4D4 is a maximal subgroup of
M4(2).10C23  D811D4  D8○D8  C8.3S4
 M4(2).D2p: Q16.10D4  Q16.D4  D4.3D8  D4.5D8  M4(2).37D4  D12.3D4  C24.19D4  Q8.9D12 ...
 D4p.D4: D8○SD16  C24.23D4  C40.23D4  C56.23D4 ...
D4.4D4 is a maximal quotient of
 M4(2).D2p: M4(2).48D4  M4(2).32D4  M4(2).4D4  M4(2).10D4  M4(2).12D4  D12.3D4  C24.19D4  Q8.9D12 ...
 (C2×C8).D2p: C87D8  C813SD16  D4.1Q16  Q8.1Q16  D4.2SD16  Q8.2SD16  C82D8  C82SD16 ...

Matrix representation of D4.4D4 in GL4(𝔽7) generated by

0651
3056
3361
1631
,
0400
2000
6643
2523
,
5026
6556
6112
2262
,
1416
0254
0621
0422
G:=sub<GL(4,GF(7))| [0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[0,2,6,2,4,0,6,5,0,0,4,2,0,0,3,3],[5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[1,0,0,0,4,2,6,4,1,5,2,2,6,4,1,2] >;

D4.4D4 in GAP, Magma, Sage, TeX

D_4._4D_4
% in TeX

G:=Group("D4.4D4");
// GroupNames label

G:=SmallGroup(64,153);
// by ID

G=gap.SmallGroup(64,153);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,247,362,963,117,1444,376,88]);
// Polycyclic

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

Export

Subgroup lattice of D4.4D4 in TeX
Character table of D4.4D4 in TeX

׿
×
𝔽