p-group, non-abelian, nilpotent (class 5), monomial
Aliases: C42.3D4, 2- 1+4:C4, C2.10C2wrC4, C4.10D4:C4, (C2xD4).2D4, C42:3C4:1C2, D4.8D4.1C2, C22.3(C23:C4), C4.4D4.2C22, C42.C22:6C2, (C2xQ8).1(C2xC4), (C2xC4).7(C22:C4), SmallGroup(128,136)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.3D4
G = < a,b,c,d | a4=b4=c4=1, d2=a-1, ab=ba, cac-1=a-1b-1, ad=da, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a-1c-1 >
Quotients: C1, C2, C4, C22, C2xC4, D4, C22:C4, C23:C4, C2wrC4, C42.3D4
2C2
8C2
8C2
2C4
4C4
4C22
4C22
4C4
4C4
8C22
16C4
2Q8
2C23
2D4
4C8
4D4
4C8
4Q8
4D4
4D4
4C8
4Q8
4D8
4SD16
Character table of C42.3D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | |
| size | 1 | 1 | 2 | 8 | 8 | 4 | 4 | 4 | 8 | 8 | 16 | 16 | 8 | 8 | 8 | 8 | 16 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | i | -i | -i | i | i | -i | 1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | i | -i | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | -i | i | -1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -i | i | i | -i | -i | i | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 0 | -2 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2wrC4 |
| ρ12 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23:C4 |
| ρ13 | 4 | 4 | -4 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2wrC4 |
| ρ14 | 4 | -4 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1-i | -1-i | 1+i | -1+i | 0 | complex faithful |
| ρ15 | 4 | -4 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -1+i | 1+i | -1-i | 1-i | 0 | complex faithful |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -1-i | 1-i | -1+i | 1+i | 0 | complex faithful |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1+i | -1+i | 1-i | -1-i | 0 | complex faithful |
►On 16 points - transitive group 16T397
Generators in S16
(5 7)(6 8)(9 15 13 11)(10 16 14 12)
(1 4 2 3)(5 8 7 6)(9 11 13 15)(10 16 14 12)
(1 12 8 15)(2 16 6 11)(3 10 7 13)(4 14 5 9)
(1 2)(3 4)(5 6 7 8)(9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (5,7)(6,8)(9,15,13,11)(10,16,14,12), (1,4,2,3)(5,8,7,6)(9,11,13,15)(10,16,14,12), (1,12,8,15)(2,16,6,11)(3,10,7,13)(4,14,5,9), (1,2)(3,4)(5,6,7,8)(9,10,11,12,13,14,15,16)>;
G:=Group( (5,7)(6,8)(9,15,13,11)(10,16,14,12), (1,4,2,3)(5,8,7,6)(9,11,13,15)(10,16,14,12), (1,12,8,15)(2,16,6,11)(3,10,7,13)(4,14,5,9), (1,2)(3,4)(5,6,7,8)(9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(5,7),(6,8),(9,15,13,11),(10,16,14,12)], [(1,4,2,3),(5,8,7,6),(9,11,13,15),(10,16,14,12)], [(1,12,8,15),(2,16,6,11),(3,10,7,13),(4,14,5,9)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,397);
Matrix representation of C42.3D4 ►in GL4(F5) generated by
| 1 | 3 | 1 | 3 |
| 3 | 3 | 2 | 1 |
| 1 | 4 | 0 | 2 |
| 4 | 1 | 2 | 0 |
| 1 | 0 | 4 | 0 |
| 1 | 3 | 3 | 0 |
| 2 | 0 | 4 | 0 |
| 3 | 0 | 3 | 2 |
| 0 | 0 | 3 | 2 |
| 1 | 0 | 1 | 1 |
| 0 | 1 | 2 | 2 |
| 0 | 0 | 0 | 3 |
| 0 | 4 | 1 | 0 |
| 0 | 3 | 1 | 4 |
| 0 | 4 | 3 | 3 |
| 1 | 3 | 2 | 2 |
G:=sub<GL(4,GF(5))| [1,3,1,4,3,3,4,1,1,2,0,2,3,1,2,0],[1,1,2,3,0,3,0,0,4,3,4,3,0,0,0,2],[0,1,0,0,0,0,1,0,3,1,2,0,2,1,2,3],[0,0,0,1,4,3,4,3,1,1,3,2,0,4,3,2] >;
C42.3D4 in GAP, Magma, Sage, TeX
C_4^2._3D_4
% in TeX
G:=Group("C4^2.3D4"); // GroupNames label
G:=SmallGroup(128,136);
// by ID
G=gap.SmallGroup(128,136);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,723,352,346,745,1684,1411,718,375,172,4037,2028]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^-1,a*b=b*a,c*a*c^-1=a^-1*b^-1,a*d=d*a,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=a^-1*c^-1>;
// generators/relations
Export
Subgroup lattice of C42.3D4 in TeX
Character table of C42.3D4 in TeX