direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×D4, C4⋊C10, C20⋊3C2, C22⋊C10, C10.6C22, (C2×C10)⋊1C2, C2.1(C2×C10), SmallGroup(40,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4
G = < a,b,c | a5=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C5×D4
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 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 | 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 | -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 | -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 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | -1 | ζ5 | ζ54 | ζ52 | ζ53 | ζ52 | ζ54 | ζ53 | ζ5 | -ζ5 | -ζ52 | -ζ53 | ζ53 | ζ5 | ζ54 | ζ52 | -ζ54 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | linear of order 10 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | ζ5 | ζ54 | ζ52 | ζ53 | ζ52 | ζ54 | ζ53 | ζ5 | ζ5 | ζ52 | ζ53 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | ζ54 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | linear of order 10 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | ζ54 | ζ5 | ζ53 | ζ52 | ζ53 | ζ5 | ζ52 | ζ54 | ζ54 | ζ53 | ζ52 | ζ52 | ζ54 | ζ5 | ζ53 | ζ5 | ζ52 | ζ54 | ζ5 | ζ53 | linear of order 5 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | ζ54 | ζ5 | ζ53 | ζ52 | ζ53 | ζ5 | ζ52 | ζ54 | -ζ54 | -ζ53 | -ζ52 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | -ζ5 | ζ52 | ζ54 | ζ5 | ζ53 | linear of order 10 |
| ρ9 | 1 | 1 | 1 | -1 | -1 | ζ53 | ζ52 | ζ5 | ζ54 | ζ5 | ζ52 | ζ54 | ζ53 | ζ53 | ζ5 | ζ54 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | ζ52 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | linear of order 10 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | ζ53 | ζ52 | ζ5 | ζ54 | ζ5 | ζ52 | ζ54 | ζ53 | -ζ53 | -ζ5 | -ζ54 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | -ζ52 | ζ54 | ζ53 | ζ52 | ζ5 | linear of order 10 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | ζ53 | ζ52 | ζ5 | ζ54 | ζ5 | ζ52 | ζ54 | ζ53 | ζ53 | ζ5 | ζ54 | ζ54 | ζ53 | ζ52 | ζ5 | ζ52 | ζ54 | ζ53 | ζ52 | ζ5 | linear of order 5 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | ζ5 | ζ54 | ζ52 | ζ53 | ζ52 | ζ54 | ζ53 | ζ5 | -ζ5 | -ζ52 | -ζ53 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | -ζ54 | ζ53 | ζ5 | ζ54 | ζ52 | linear of order 10 |
| ρ13 | 1 | 1 | -1 | 1 | -1 | ζ53 | ζ52 | ζ5 | ζ54 | ζ5 | ζ52 | ζ54 | ζ53 | -ζ53 | -ζ5 | -ζ54 | ζ54 | ζ53 | ζ52 | ζ5 | -ζ52 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | linear of order 10 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | ζ52 | ζ53 | ζ54 | ζ5 | ζ54 | ζ53 | ζ5 | ζ52 | ζ52 | ζ54 | ζ5 | ζ5 | ζ52 | ζ53 | ζ54 | ζ53 | ζ5 | ζ52 | ζ53 | ζ54 | linear of order 5 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | ζ5 | ζ54 | ζ52 | ζ53 | ζ52 | ζ54 | ζ53 | ζ5 | ζ5 | ζ52 | ζ53 | ζ53 | ζ5 | ζ54 | ζ52 | ζ54 | ζ53 | ζ5 | ζ54 | ζ52 | linear of order 5 |
| ρ16 | 1 | 1 | 1 | -1 | -1 | ζ52 | ζ53 | ζ54 | ζ5 | ζ54 | ζ53 | ζ5 | ζ52 | ζ52 | ζ54 | ζ5 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | ζ53 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | linear of order 10 |
| ρ17 | 1 | 1 | 1 | -1 | -1 | ζ54 | ζ5 | ζ53 | ζ52 | ζ53 | ζ5 | ζ52 | ζ54 | ζ54 | ζ53 | ζ52 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | ζ5 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | linear of order 10 |
| ρ18 | 1 | 1 | -1 | 1 | -1 | ζ52 | ζ53 | ζ54 | ζ5 | ζ54 | ζ53 | ζ5 | ζ52 | -ζ52 | -ζ54 | -ζ5 | ζ5 | ζ52 | ζ53 | ζ54 | -ζ53 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | linear of order 10 |
| ρ19 | 1 | 1 | -1 | 1 | -1 | ζ54 | ζ5 | ζ53 | ζ52 | ζ53 | ζ5 | ζ52 | ζ54 | -ζ54 | -ζ53 | -ζ52 | ζ52 | ζ54 | ζ5 | ζ53 | -ζ5 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | linear of order 10 |
| ρ20 | 1 | 1 | -1 | -1 | 1 | ζ52 | ζ53 | ζ54 | ζ5 | ζ54 | ζ53 | ζ5 | ζ52 | -ζ52 | -ζ54 | -ζ5 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | -ζ53 | ζ5 | ζ52 | ζ53 | ζ54 | linear of order 10 |
| ρ21 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | -2 | 0 | 0 | 0 | 2ζ54 | 2ζ5 | 2ζ53 | 2ζ52 | -2ζ53 | -2ζ5 | -2ζ52 | -2ζ54 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 2 | -2 | 0 | 0 | 0 | 2ζ5 | 2ζ54 | 2ζ52 | 2ζ53 | -2ζ52 | -2ζ54 | -2ζ53 | -2ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 2 | -2 | 0 | 0 | 0 | 2ζ52 | 2ζ53 | 2ζ54 | 2ζ5 | -2ζ54 | -2ζ53 | -2ζ5 | -2ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ25 | 2 | -2 | 0 | 0 | 0 | 2ζ53 | 2ζ52 | 2ζ5 | 2ζ54 | -2ζ5 | -2ζ52 | -2ζ54 | -2ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 20 points - transitive group 20T12
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 8 13 17)(2 9 14 18)(3 10 15 19)(4 6 11 20)(5 7 12 16)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 11)(7 12)(8 13)(9 14)(10 15)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,8,13,17)(2,9,14,18)(3,10,15,19)(4,6,11,20)(5,7,12,16), (1,17)(2,18)(3,19)(4,20)(5,16)(6,11)(7,12)(8,13)(9,14)(10,15)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,8,13,17)(2,9,14,18)(3,10,15,19)(4,6,11,20)(5,7,12,16), (1,17)(2,18)(3,19)(4,20)(5,16)(6,11)(7,12)(8,13)(9,14)(10,15) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,8,13,17),(2,9,14,18),(3,10,15,19),(4,6,11,20),(5,7,12,16)], [(1,17),(2,18),(3,19),(4,20),(5,16),(6,11),(7,12),(8,13),(9,14),(10,15)]])
G:=TransitiveGroup(20,12);
C5×D4 is a maximal subgroup of
D4⋊D5 D4.D5 D4⋊2D5 D44⋊C5 C22⋊F11
C5×D4 is a maximal quotient of D44⋊C5 C22⋊F11
Matrix representation of C5×D4 ►in GL2(𝔽11) generated by
| 9 | 0 |
| 0 | 9 |
| 0 | 7 |
| 3 | 0 |
| 0 | 4 |
| 3 | 0 |
G:=sub<GL(2,GF(11))| [9,0,0,9],[0,3,7,0],[0,3,4,0] >;
C5×D4 in GAP, Magma, Sage, TeX
C_5\times D_4
% in TeX
G:=Group("C5xD4"); // GroupNames label
G:=SmallGroup(40,10);
// by ID
G=gap.SmallGroup(40,10);
# by ID
G:=PCGroup([4,-2,-2,-5,-2,177]);
// Polycyclic
G:=Group<a,b,c|a^5=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C5×D4 in TeX
Character table of C5×D4 in TeX