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G = D4:4D4order 64 = 26

3rd semidirect product of D4 and D4 acting via D4/C22=C2

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

Aliases: D4:4D4, Q8:4D4, C23.5D4, C42:2C22, M4(2):1C22, 2+ 1+4:1C2, C4wrC2:1C2, C4:1D4:1C2, C8:C22:1C2, C4.25(C2xD4), C4.D4:1C2, (C2xD4):2C22, (C2xC4).4C23, C2.14C22wrC2, C4oD4.1C22, C22.12(C2xD4), 2-Sylow(SO+(4,3)), Hol(D4), SmallGroup(64,134)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — D4:4D4
C1C2C22C2xC4C2xD42+ 1+4 — D4:4D4
C1C2C2xC4 — D4:4D4
C1C2C2xC4 — D4:4D4
C1C2C2C2xC4 — D4:4D4

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

Subgroups: 185 in 84 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C42, M4(2), D8, SD16, C2xD4, C2xD4, C4oD4, C4oD4, C4.D4, C4wrC2, C4:1D4, C8:C22, 2+ 1+4, D4:4D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C22wrC2, D4:4D4

Character table of D4:4D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F8A8B
 size 1124444822444488
ρ11111111111111111    trivial
ρ21111-1-1-1-111111-1-11    linear of order 2
ρ3111-1-11-1-11111-111-1    linear of order 2
ρ4111-11-1111111-1-1-1-1    linear of order 2
ρ5111-1-11-1111-1-1-11-11    linear of order 2
ρ6111-11-11-111-1-1-1-111    linear of order 2
ρ71111111-111-1-111-1-1    linear of order 2
ρ81111-1-1-1111-1-11-11-1    linear of order 2
ρ922-2200002-200-2000    orthogonal lifted from D4
ρ1022-200-200-22000200    orthogonal lifted from D4
ρ1122-200200-22000-200    orthogonal lifted from D4
ρ1222-2-200002-2002000    orthogonal lifted from D4
ρ13222020-20-2-2000000    orthogonal lifted from D4
ρ142220-2020-2-2000000    orthogonal lifted from D4
ρ154-4000000002-20000    orthogonal faithful
ρ164-400000000-220000    orthogonal faithful

Permutation representations of D4:4D4
On 8 points - transitive group 8T26
Generators in S8
(1 2 3 4)(5 6 7 8)
(1 8)(2 7)(3 6)(4 5)
(1 4 3 2)
(2 4)(5 6)(7 8)

G:=sub<Sym(8)| (1,2,3,4)(5,6,7,8), (1,8)(2,7)(3,6)(4,5), (1,4,3,2), (2,4)(5,6)(7,8)>;

G:=Group( (1,2,3,4)(5,6,7,8), (1,8)(2,7)(3,6)(4,5), (1,4,3,2), (2,4)(5,6)(7,8) );

G=PermutationGroup([[(1,2,3,4),(5,6,7,8)], [(1,8),(2,7),(3,6),(4,5)], [(1,4,3,2)], [(2,4),(5,6),(7,8)]])

G:=TransitiveGroup(8,26);

On 16 points - transitive group 16T135
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3)(5 6)(7 8)(9 11)(13 14)(15 16)
(1 8 10 13)(2 5 11 14)(3 6 12 15)(4 7 9 16)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)

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

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

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

G:=TransitiveGroup(16,135);

On 16 points - transitive group 16T141
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 14)(6 13)(7 16)(8 15)
(9 12 11 10)(13 16 15 14)
(1 5)(2 8)(3 7)(4 6)(9 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(16,141);

On 16 points - transitive group 16T142
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 14)(6 13)(7 16)(8 15)
(1 5 3 7)(2 6 4 8)(9 14)(10 15)(11 16)(12 13)
(2 4)(5 7)(9 10)(11 12)(13 16)(14 15)

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

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

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

G:=TransitiveGroup(16,142);

On 16 points - transitive group 16T152
Generators in S16
(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 13)(10 16)(11 15)(12 14)
(1 10)(2 11)(3 12)(4 9)(5 16 7 14)(6 13 8 15)
(1 9)(2 12)(3 11)(4 10)(5 15)(6 14)(7 13)(8 16)

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

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

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

G:=TransitiveGroup(16,152);

D4:4D4 is a maximal subgroup of
C42.D4  C42.13D4  C42:D6  Q8.5S4  Q8:2S4
 D4p:D4: D8:11D4  D8:6D4  D8oD8  D12:1D4  Q8:5D12  C42:8D6  D12:18D4  D20:1D4 ...
 (CpxD4):D4: D4wrC2  C42:6D4  C42.313C23  M4(2):C23  C42.12C23  2+ 1+4:6S3  2+ 1+4:D5  2+ 1+4:D7 ...
D4:4D4 is a maximal quotient of
C23:SD16  C24.16D4  C24.17D4  C24.18D4  C42.181C23  Q8:D8  D4:SD16  Q8:SD16  C42.185C23  D4:2SD16  C42.191C23  Q8:2SD16  D4:Q16  Q8:Q16  C42.195C23  C42.C23  C42.2C23  C42.3C23  C42.5C23  C42.6C23  C42.7C23  2+ 1+4:3C4  C24.21D4  C4wrC2:C4  C8:C22:C4  C24.24D4  (C2xD4):2Q8  M4(2):Q8
 D4:D4p: D4:D8  Q8:5D12  D4:4D20  D4:4D28 ...
 C42:D2p: C42:9D4  C42:11D4  C42:8D6  D20:5D4  D28:5D4 ...
 (CpxD4):D4: C23:D8  C24.9D4  M4(2):D4  D12:18D4  2+ 1+4:6S3  D20:18D4  2+ 1+4:D5  D28:18D4 ...
 M4(2):D2p: M4(2):6D4  D12:1D4  D20:1D4  D28:1D4 ...

Polynomial with Galois group D4:4D4 over Q
actionf(x)Disc(f)
8T26x8-12x6+45x4-66x2+33216·37·113

Matrix representation of D4:4D4 in GL4(Z) generated by

0100
-1000
000-1
0010
,
000-1
0010
0100
-1000
,
-1000
0-100
000-1
0010
,
-1000
0100
000-1
00-10
G:=sub<GL(4,Integers())| [0,-1,0,0,1,0,0,0,0,0,0,1,0,0,-1,0],[0,0,0,-1,0,0,1,0,0,1,0,0,-1,0,0,0],[-1,0,0,0,0,-1,0,0,0,0,0,1,0,0,-1,0],[-1,0,0,0,0,1,0,0,0,0,0,-1,0,0,-1,0] >;

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

D_4\rtimes_4D_4
% in TeX

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

G:=SmallGroup(64,134);
// by ID

G=gap.SmallGroup(64,134);
# by ID

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

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

Export

Character table of D4:4D4 in TeX

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