p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: D4⋊4D4, Q8⋊4D4, C23.5D4, C42⋊2C22, M4(2)⋊1C22, 2+ 1+4⋊1C2, C4≀C2⋊1C2, C4⋊1D4⋊1C2, C8⋊C22⋊1C2, C4.25(C2×D4), C4.D4⋊1C2, (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
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, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C4○D4, C4.D4, C4≀C2, C4⋊1D4, C8⋊C22, 2+ 1+4, D4⋊4D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, D4⋊4D4
Character table of D4⋊4D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | 2 | 0 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 8 points - transitive group 8T26
(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
(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
(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
(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
(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 D8○D8 D12⋊1D4 Q8⋊5D12 C42⋊8D6 D12⋊18D4 D20⋊1D4 ...
(Cp×D4)⋊D4: D4≀C2 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 C4≀C2⋊C4 C8⋊C22⋊C4 C24.24D4 (C2×D4)⋊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 ...
(Cp×D4)⋊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 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 8T26 | x8-12x6+45x4-66x2+33 | 216·37·113 |
Matrix representation of D4⋊4D4 ►in GL4(ℤ) generated by
| 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 |
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
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