metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2D4, C22⋊D5, Dic5⋊C2, D10⋊2C2, C2.5D10, C10.5C22, (C2×C10)⋊2C2, SmallGroup(40,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊D4
G = < a,b,c | a5=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Character table of C5⋊D4
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 10A | 10B | 10C | 10D | 10E | 10F | |
| size | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ7 | 2 | 2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ8 | 2 | 2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ10 | 2 | -2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ53-ζ52 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | 1-√5/2 | 1+√5/2 | complex faithful |
| ρ11 | 2 | -2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ53+ζ52 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | 1-√5/2 | 1+√5/2 | complex faithful |
| ρ12 | 2 | -2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ54-ζ5 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | 1+√5/2 | 1-√5/2 | complex faithful |
| ρ13 | 2 | -2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ54+ζ5 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | 1+√5/2 | 1-√5/2 | complex faithful |
►On 20 points - transitive group 20T7
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 14 9 19)(2 13 10 18)(3 12 6 17)(4 11 7 16)(5 15 8 20)
(2 5)(3 4)(6 7)(8 10)(11 17)(12 16)(13 20)(14 19)(15 18)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,14,9,19)(2,13,10,18)(3,12,6,17)(4,11,7,16)(5,15,8,20), (2,5)(3,4)(6,7)(8,10)(11,17)(12,16)(13,20)(14,19)(15,18)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,14,9,19)(2,13,10,18)(3,12,6,17)(4,11,7,16)(5,15,8,20), (2,5)(3,4)(6,7)(8,10)(11,17)(12,16)(13,20)(14,19)(15,18) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,14,9,19),(2,13,10,18),(3,12,6,17),(4,11,7,16),(5,15,8,20)], [(2,5),(3,4),(6,7),(8,10),(11,17),(12,16),(13,20),(14,19),(15,18)]])
G:=TransitiveGroup(20,7);
►On 20 points - transitive group 20T11
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 16 9 11)(2 20 10 15)(3 19 6 14)(4 18 7 13)(5 17 8 12)
(1 11)(2 15)(3 14)(4 13)(5 12)(6 19)(7 18)(8 17)(9 16)(10 20)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16,9,11)(2,20,10,15)(3,19,6,14)(4,18,7,13)(5,17,8,12), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16,9,11)(2,20,10,15)(3,19,6,14)(4,18,7,13)(5,17,8,12), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,16,9,11),(2,20,10,15),(3,19,6,14),(4,18,7,13),(5,17,8,12)], [(1,11),(2,15),(3,14),(4,13),(5,12),(6,19),(7,18),(8,17),(9,16),(10,20)]])
G:=TransitiveGroup(20,11);
C5⋊D4 is a maximal subgroup of
C4○D20 D4×D5 D4⋊2D5 C15⋊D4 C5⋊D12 C15⋊7D4 C5⋊S4 C25⋊D4 C52⋊2D4 C5⋊D20 C52⋊7D4 C35⋊D4 C5⋊D28 C35⋊7D4 S32⋊D5 C55⋊D4 C5⋊D44 C55⋊7D4
C5⋊D4 is a maximal quotient of
C10.D4 D10⋊C4 D4⋊D5 D4.D5 Q8⋊D5 C5⋊Q16 C23.D5 C15⋊D4 C5⋊D12 C15⋊7D4 C25⋊D4 C52⋊2D4 C5⋊D20 C52⋊7D4 C35⋊D4 C5⋊D28 C35⋊7D4 S32⋊D5 C55⋊D4 C5⋊D44 C55⋊7D4
Matrix representation of C5⋊D4 ►in GL2(𝔽11) generated by
| 5 | 8 |
| 8 | 2 |
| 8 | 2 |
| 6 | 3 |
| 1 | 1 |
| 0 | 10 |
G:=sub<GL(2,GF(11))| [5,8,8,2],[8,6,2,3],[1,0,1,10] >;
C5⋊D4 in GAP, Magma, Sage, TeX
C_5\rtimes D_4
% in TeX
G:=Group("C5:D4"); // GroupNames label
G:=SmallGroup(40,8);
// by ID
G=gap.SmallGroup(40,8);
# by ID
G:=PCGroup([4,-2,-2,-2,-5,49,515]);
// Polycyclic
G:=Group<a,b,c|a^5=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C5⋊D4 in TeX
Character table of C5⋊D4 in TeX