p-group, metabelian, nilpotent (class 4), monomial
Aliases: D8:3D4, C8.4D8, Q16:3D4, C42.143D4, M5(2).4C22, C8oD8:4C2, C4.68(C2xD8), C8.77(C2xD4), C8.C8:2C2, C8:4D4:15C2, C16:C22:4C2, C8.2(C4oD4), (C2xC8).131D4, M5(2):C2:4C2, C2.25(C4:D8), C4.56(C4:D4), (C2xC8).236C23, (C4xC8).163C22, C4oD8.19C22, (C2xD8).48C22, C22.25(C8:C22), C8.C4.19C22, (C2xC4).281(C2xD4), SmallGroup(128,945)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8:3D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >
Subgroups: 268 in 85 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, Q8, C23, C16, C42, C2xC8, C2xC8, M4(2), D8, D8, SD16, Q16, C2xD4, C4oD4, C4xC8, C4wrC2, C8.C4, M5(2), D16, SD32, C4:1D4, C8oD4, C2xD8, C2xD8, C4oD8, M5(2):C2, C8.C8, C8oD8, C8:4D4, C16:C22, D8:3D4
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C4oD4, C4:D4, C2xD8, C8:C22, C4:D8, D8:3D4
Character table of D8:3D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 16A | 16B | 16C | 16D | |
| size | 1 | 1 | 2 | 8 | 16 | 16 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 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 | 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 | 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 | -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 | -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 | -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 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | -√2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ15 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ17 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ18 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ19 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2√2 | 2√2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2√2 | -2√2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2√2 | 2√2 | -2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -2√2 | -2√2 | 2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 16 points - transitive group 16T379
(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 16)(6 15)(7 14)(8 13)
(1 7 5 3)(2 8 6 4)(9 13)(10 14)(11 15)(12 16)
(1 3)(4 8)(5 7)(9 14)(10 13)(11 12)(15 16)
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)(5,16)(6,15)(7,14)(8,13), (1,7,5,3)(2,8,6,4)(9,13)(10,14)(11,15)(12,16), (1,3)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)>;
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)(5,16)(6,15)(7,14)(8,13), (1,7,5,3)(2,8,6,4)(9,13)(10,14)(11,15)(12,16), (1,3)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16) );
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),(5,16),(6,15),(7,14),(8,13)], [(1,7,5,3),(2,8,6,4),(9,13),(10,14),(11,15),(12,16)], [(1,3),(4,8),(5,7),(9,14),(10,13),(11,12),(15,16)]])
G:=TransitiveGroup(16,379);
Matrix representation of D8:3D4 ►in GL4(F7) generated by
| 5 | 0 | 5 | 1 |
| 1 | 5 | 2 | 1 |
| 1 | 6 | 2 | 5 |
| 5 | 5 | 1 | 1 |
| 3 | 2 | 1 | 1 |
| 3 | 4 | 5 | 5 |
| 5 | 6 | 6 | 0 |
| 2 | 1 | 2 | 1 |
| 6 | 3 | 6 | 1 |
| 0 | 2 | 5 | 4 |
| 0 | 5 | 6 | 6 |
| 0 | 2 | 3 | 5 |
| 1 | 1 | 0 | 4 |
| 0 | 3 | 4 | 4 |
| 0 | 6 | 5 | 6 |
| 0 | 6 | 6 | 5 |
G:=sub<GL(4,GF(7))| [5,1,1,5,0,5,6,5,5,2,2,1,1,1,5,1],[3,3,5,2,2,4,6,1,1,5,6,2,1,5,0,1],[6,0,0,0,3,2,5,2,6,5,6,3,1,4,6,5],[1,0,0,0,1,3,6,6,0,4,5,6,4,4,6,5] >;
D8:3D4 in GAP, Magma, Sage, TeX
D_8\rtimes_3D_4
% in TeX
G:=Group("D8:3D4"); // GroupNames label
G:=SmallGroup(128,945);
// by ID
G=gap.SmallGroup(128,945);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,512,422,2019,248,1684,438,242,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
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