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G = D44D4order 64 = 26

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

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

Aliases: D44D4, Q84D4, C23.5D4, C422C22, M4(2)⋊1C22, 2+ 1+41C2, C4≀C21C2, C41D41C2, C8⋊C221C2, C4.25(C2×D4), C4.D41C2, (C2×D4)⋊2C22, (C2×C4).4C23, C2.14C22≀C2, C4○D4.1C22, C22.12(C2×D4), 2-Sylow(SO+(4,3)), Hol(D4), SmallGroup(64,134)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D44D4
C1C2C22C2×C4C2×D42+ 1+4 — D44D4
C1C2C2×C4 — D44D4
C1C2C2×C4 — D44D4
C1C2C2C2×C4 — D44D4

Generators and relations for D44D4
 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 [×6], C4 [×2], C4 [×4], C22, C22 [×11], C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], D4 [×14], Q8 [×2], C23 [×2], C23 [×3], C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4 [×2], C2×D4 [×6], C4○D4 [×2], C4○D4 [×2], C4.D4, C4≀C2 [×2], C41D4, C8⋊C22 [×2], 2+ 1+4, D44D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, D44D4

Character table of D44D4

 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 D44D4
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 15)(2 5 11 16)(3 6 12 13)(4 7 9 14)
(1 14)(2 13)(3 16)(4 15)(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), (1,8,10,15)(2,5,11,16)(3,6,12,13)(4,7,9,14), (1,14)(2,13)(3,16)(4,15)(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), (1,8,10,15)(2,5,11,16)(3,6,12,13)(4,7,9,14), (1,14)(2,13)(3,16)(4,15)(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)], [(1,8,10,15),(2,5,11,16),(3,6,12,13),(4,7,9,14)], [(1,14),(2,13),(3,16),(4,15),(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);

D44D4 is a maximal subgroup of
C42.D4  C42.13D4  C42⋊D6  Q8.5S4  Q82S4
 D4p⋊D4: D811D4  D86D4  D8○D8  D121D4  Q85D12  C428D6  D1218D4  D201D4 ...
 (Cp×D4)⋊D4: D4≀C2  C426D4  C42.313C23  M4(2)⋊C23  C42.12C23  2+ 1+46S3  2+ 1+4⋊D5  2+ 1+4⋊D7 ...
D44D4 is a maximal quotient of
C23⋊SD16  C24.16D4  C24.17D4  C24.18D4  C42.181C23  Q8⋊D8  D4⋊SD16  Q8⋊SD16  C42.185C23  D42SD16  C42.191C23  Q82SD16  D4⋊Q16  Q8⋊Q16  C42.195C23  C42.C23  C42.2C23  C42.3C23  C42.5C23  C42.6C23  C42.7C23  2+ 1+43C4  C24.21D4  C4≀C2⋊C4  C8⋊C22⋊C4  C24.24D4  (C2×D4)⋊2Q8  M4(2)⋊Q8
 D4⋊D4p: D4⋊D8  Q85D12  D44D20  D44D28 ...
 C42⋊D2p: C429D4  C4211D4  C428D6  D205D4  D285D4 ...
 (Cp×D4)⋊D4: C23⋊D8  C24.9D4  M4(2)⋊D4  D1218D4  2+ 1+46S3  D2018D4  2+ 1+4⋊D5  D2818D4 ...
 M4(2)⋊D2p: M4(2)⋊6D4  D121D4  D201D4  D281D4 ...

Polynomial with Galois group D44D4 over ℚ
actionf(x)Disc(f)
8T26x8-12x6+45x4-66x2+33216·37·113

Matrix representation of D44D4 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] >;

D44D4 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 D44D4 in TeX

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