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
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 | 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 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 ...
| 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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