p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.9D4, C23.C4, M4(2)⋊3C2, (C2×D4).2C2, (C2×C4).1C22, C22.3(C2×C4), C2.4(C22⋊C4), SmallGroup(32,7)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.D4
G = < a,b,c | a4=1, b4=a2, c2=a, bab-1=a-1, ac=ca, cbc-1=a-1b3 >
Character table of C4.D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 8 points - transitive group 8T16
(1 3 5 7)(2 8 6 4)
(1 2 3 4 5 6 7 8)
(1 4 3 2 5 8 7 6)
G:=sub<Sym(8)| (1,3,5,7)(2,8,6,4), (1,2,3,4,5,6,7,8), (1,4,3,2,5,8,7,6)>;
G:=Group( (1,3,5,7)(2,8,6,4), (1,2,3,4,5,6,7,8), (1,4,3,2,5,8,7,6) );
G=PermutationGroup([[(1,3,5,7),(2,8,6,4)], [(1,2,3,4,5,6,7,8)], [(1,4,3,2,5,8,7,6)]])
G:=TransitiveGroup(8,16);
►On 16 points - transitive group 16T36
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 12 3 10 5 16 7 14)(2 13 8 15 6 9 4 11)
G:=sub<Sym(16)| (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,12,3,10,5,16,7,14)(2,13,8,15,6,9,4,11)>;
G:=Group( (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,12,3,10,5,16,7,14)(2,13,8,15,6,9,4,11) );
G=PermutationGroup([[(1,3,5,7),(2,8,6,4),(9,11,13,15),(10,16,14,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,12,3,10,5,16,7,14),(2,13,8,15,6,9,4,11)]])
G:=TransitiveGroup(16,36);
►On 16 points - transitive group 16T41
(1 12 5 16)(2 9 6 13)(3 14 7 10)(4 11 8 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 15 12 4 5 11 16 8)(2 7 9 10 6 3 13 14)
G:=sub<Sym(16)| (1,12,5,16)(2,9,6,13)(3,14,7,10)(4,11,8,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15,12,4,5,11,16,8)(2,7,9,10,6,3,13,14)>;
G:=Group( (1,12,5,16)(2,9,6,13)(3,14,7,10)(4,11,8,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15,12,4,5,11,16,8)(2,7,9,10,6,3,13,14) );
G=PermutationGroup([[(1,12,5,16),(2,9,6,13),(3,14,7,10),(4,11,8,15)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,15,12,4,5,11,16,8),(2,7,9,10,6,3,13,14)]])
G:=TransitiveGroup(16,41);
C4.D4 is a maximal subgroup of
C2≀C4 C23.D4 M4(2).8C22 D4⋊4D4 D4.9D4 C23.F5 C4.S3≀C2 (C2×C62).C4 Dic13.4D4
C4p.D4: D4.3D4 D4.4D4 C12.46D4 C12.D4 C20.46D4 C20.D4 C28.46D4 C28.D4 ...
C4.D4 is a maximal quotient of
C23.F5 C4.S3≀C2 (C2×C62).C4 Dic13.4D4
C4.D4p: C4.D8 C12.46D4 C20.46D4 C28.46D4 C44.46D4 C52.46D4 ...
(C2×C4).D2p: C23⋊C8 C42.C22 C4.6Q16 C22.C42 C12.D4 C20.D4 C28.D4 C44.D4 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 8T16 | x8-2x7-12x6+21x5+35x4-39x3-37x2+3x+1 | 57·118·192 |
Matrix representation of C4.D4 ►in GL4(ℤ) generated by
| 0 | 1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
G:=sub<GL(4,Integers())| [0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,1,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,-1,0,0],[0,0,0,-1,0,0,1,0,1,0,0,0,0,1,0,0] >;
C4.D4 in GAP, Magma, Sage, TeX
C_4.D_4
% in TeX
G:=Group("C4.D4"); // GroupNames label
G:=SmallGroup(32,7);
// by ID
G=gap.SmallGroup(32,7);
# by ID
G:=PCGroup([5,-2,2,-2,2,-2,40,61,302,248,58]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^4=a^2,c^2=a,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^3>;
// generators/relations
Export
Subgroup lattice of C4.D4 in TeX
Character table of C4.D4 in TeX