direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4xD4, C42:4C2, C22.7C23, C23.8C22, C4:C4:7C2, C4:1(C2xC4), C4o2(C4:C4), C2.3(C2xD4), C22:C4:6C2, (C22xC4):2C2, C22:1(C2xC4), (C2xD4).7C2, C4o2(C22:C4), C2.2(C4oD4), C2.4(C22xC4), (C2xC4).11C22, (C2xC4)o(C2xD4), (C2xC4)o(C4:C4), (C2xC4)o(C22:C4), 2-Sylow(CO(3,5)), SmallGroup(32,25)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4xD4
G = < a,b,c | a4=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, C4xD4
Character table of C4xD4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | -i | i | -i | i | -i | -i | i | 1 | i | -1 | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | -i | -i | i | i | i | -1 | -i | 1 | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | -i | -i | -i | -i | i | i | -1 | i | 1 | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | i | -i | i | -i | -i | i | i | -i | i | 1 | -i | -1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | i | i | i | i | -i | -i | -1 | -i | 1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | i | -i | i | i | -i | -i | i | -i | 1 | i | -1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | i | -i | i | -i | i | i | -i | 1 | -i | -1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | i | i | -i | -i | -i | -1 | i | 1 | linear of order 4 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
►On 16 points - transitive group 16T19
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 14 11 7)(2 15 12 8)(3 16 9 5)(4 13 10 6)
(5 16)(6 13)(7 14)(8 15)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,11,7)(2,15,12,8)(3,16,9,5)(4,13,10,6), (5,16)(6,13)(7,14)(8,15)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,11,7)(2,15,12,8)(3,16,9,5)(4,13,10,6), (5,16)(6,13)(7,14)(8,15) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,14,11,7),(2,15,12,8),(3,16,9,5),(4,13,10,6)], [(5,16),(6,13),(7,14),(8,15)]])
G:=TransitiveGroup(16,19);
C4xD4 is a maximal subgroup of
D4:C8 C8:9D4 C8:6D4 SD16:C4 C4:D8 D4.D4 D4.2D4 D4:Q8 D4:2Q8 D4.Q8 C22.19C24 C23.36C23 C22.26C24 C22.32C24 C22.33C24 C22.34C24 C22.36C24 D4:5D4 D4:6D4 Q8:5D4 Q8:6D4 C22.45C24 C22.46C24 C22.47C24 D4:3Q8 C22.49C24 C22.50C24 C22.53C24
D4p:C4: D8:C4 Dic3:5D4 D20:8C4 D28:C4 D44:C4 D52:8C4 ...
D2p:(C2xC4): C22.11C24 C23.33C23 Dic3:4D4 Dic5:4D4 Dic7:4D4 Dic11:4D4 Dic13:4D4 ...
C4xD4 is a maximal quotient of
C23.63C23 C24.C22 C23.65C23 C24.3C22 C8:9D4 C8:6D4 SD16:C4 Q16:C4
D4p:C4: D8:C4 C8oD8 C8.26D4 Dic3:5D4 D20:8C4 D28:C4 D44:C4 D52:8C4 ...
C23.D2p: C23.8Q8 C23.23D4 Dic3:4D4 Dic5:4D4 Dic7:4D4 Dic11:4D4 Dic13:4D4 ...
Matrix representation of C4xD4 ►in GL3(F5) generated by
| 2 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 0 | 4 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 4 |
G:=sub<GL(3,GF(5))| [2,0,0,0,4,0,0,0,4],[1,0,0,0,0,1,0,4,0],[1,0,0,0,1,0,0,0,4] >;
C4xD4 in GAP, Magma, Sage, TeX
C_4\times D_4
% in TeX
G:=Group("C4xD4"); // GroupNames label
G:=SmallGroup(32,25);
// by ID
G=gap.SmallGroup(32,25);
# by ID
G:=PCGroup([5,-2,2,2,-2,2,80,101,72]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C4xD4 in TeX
Character table of C4xD4 in TeX