non-abelian, soluble, monomial
Aliases: C2.2D5≀C2, C5⋊D5.2Q8, C52⋊3(C4⋊C4), C52⋊C4⋊3C4, (C5×C10).2D4, Dic5⋊2D5.2C2, C5⋊D5.6(C2×C4), (C2×C52⋊C4).3C2, (C2×C5⋊D5).7C22, SmallGroup(400,130)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.D5≀C2
G = < a,b,c,d,e | a2=b5=c5=d4=1, e2=a, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ece-1=b2, ebe-1=dcd-1=c3, ede-1=d-1 >
25C2
25C2
2C5
2C5
2C5
10C4
10C4
25C4
25C22
25C4
2C10
2C10
2C10
10D5
10D5
10D5
10D5
10D5
10D5
25C2×C4
25C2×C4
25C2×C4
2Dic5
2Dic5
10F5
10C20
10F5
10C20
10D10
10D10
10D10
25C4⋊C4
10C2×F5
10C4×D5
10C4×D5
Character table of C2.D5≀C2
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 5E | 10A | 10B | 10C | 10D | 10E | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 1 | 25 | 25 | 10 | 10 | 10 | 10 | 50 | 50 | 4 | 4 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | |
| 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 | 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 | 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 | -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 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | i | -i | -i | i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -i | -i | -i | i | i | i | i | -i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | -i | i | -i | i | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | i | -i | -i | -i | i | i | -i | i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | i | -i | i | -i | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -i | i | i | i | -i | -i | i | -i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | -i | i | i | -i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | i | i | i | -i | -i | -i | -i | i | linear of order 4 |
| ρ9 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 4 | 4 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1+√5 | -1-√5 | -1 | -1-√5 | 3-√5/2 | -1+√5 | 3+√5/2 | -1 | 0 | -1-√5/2 | -1+√5/2 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ12 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1+√5 | -1-√5 | -1 | -1-√5 | 3-√5/2 | -1+√5 | 3+√5/2 | -1 | 0 | 1+√5/2 | 1-√5/2 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ13 | 4 | 4 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 3+√5/2 | 3-√5/2 | -1 | 3-√5/2 | -1+√5 | 3+√5/2 | -1-√5 | -1 | 1+√5/2 | 0 | 0 | 1+√5/2 | 0 | 0 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D5≀C2 |
| ρ14 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1-√5 | -1+√5 | -1 | -1+√5 | 3+√5/2 | -1-√5 | 3-√5/2 | -1 | 0 | 1-√5/2 | 1+√5/2 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ15 | 4 | 4 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 3-√5/2 | 3+√5/2 | -1 | 3+√5/2 | -1-√5 | 3-√5/2 | -1+√5 | -1 | 1-√5/2 | 0 | 0 | 1-√5/2 | 0 | 0 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D5≀C2 |
| ρ16 | 4 | 4 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1-√5 | -1+√5 | -1 | -1+√5 | 3+√5/2 | -1-√5 | 3-√5/2 | -1 | 0 | -1+√5/2 | -1-√5/2 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ17 | 4 | 4 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 3-√5/2 | 3+√5/2 | -1 | 3+√5/2 | -1-√5 | 3-√5/2 | -1+√5 | -1 | -1+√5/2 | 0 | 0 | -1+√5/2 | 0 | 0 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5≀C2 |
| ρ18 | 4 | 4 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 3+√5/2 | 3-√5/2 | -1 | 3-√5/2 | -1+√5 | 3+√5/2 | -1-√5 | -1 | -1-√5/2 | 0 | 0 | -1-√5/2 | 0 | 0 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5≀C2 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 3-√5/2 | 3+√5/2 | -1-√5 | -1+√5 | -1 | 1-√5 | -3-√5/2 | 1+√5 | -3+√5/2 | 1 | 0 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | 0 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | 0 | 0 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 3-√5/2 | 3+√5/2 | -1 | -3-√5/2 | 1+√5 | -3+√5/2 | 1-√5 | 1 | ζ4ζ54+ζ4ζ5 | 0 | 0 | ζ43ζ54+ζ43ζ5 | 0 | 0 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 3-√5/2 | 3+√5/2 | -1-√5 | -1+√5 | -1 | 1-√5 | -3-√5/2 | 1+√5 | -3+√5/2 | 1 | 0 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | 0 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 3+√5/2 | 3-√5/2 | -1 | -3+√5/2 | 1-√5 | -3-√5/2 | 1+√5 | 1 | ζ43ζ53+ζ43ζ52 | 0 | 0 | ζ4ζ53+ζ4ζ52 | 0 | 0 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 3+√5/2 | 3-√5/2 | -1+√5 | -1-√5 | -1 | 1+√5 | -3+√5/2 | 1-√5 | -3-√5/2 | 1 | 0 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | 0 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | 0 | 0 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 3+√5/2 | 3-√5/2 | -1 | -3+√5/2 | 1-√5 | -3-√5/2 | 1+√5 | 1 | ζ4ζ53+ζ4ζ52 | 0 | 0 | ζ43ζ53+ζ43ζ52 | 0 | 0 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 3+√5/2 | 3-√5/2 | -1+√5 | -1-√5 | -1 | 1+√5 | -3+√5/2 | 1-√5 | -3-√5/2 | 1 | 0 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | 0 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | 0 | 0 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 3-√5/2 | 3+√5/2 | -1 | -3-√5/2 | 1+√5 | -3+√5/2 | 1-√5 | 1 | ζ43ζ54+ζ43ζ5 | 0 | 0 | ζ4ζ54+ζ4ζ5 | 0 | 0 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | complex faithful |
| ρ27 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 3 | 2 | 2 | 2 | 2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ28 | 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 3 | -2 | -2 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D5≀C2 |
►On 20 points - transitive group 20T105
Generators in S20
(1 6)(2 7)(3 8)(4 9)(5 10)(11 19)(12 20)(13 16)(14 17)(15 18)
(11 12 13 14 15)(16 17 18 19 20)
(1 3 5 2 4)(6 8 10 7 9)
(1 6)(2 8 5 9)(3 10 4 7)(11 19)(12 17 15 16)(13 20 14 18)
(1 19 6 11)(2 20 7 12)(3 16 8 13)(4 17 9 14)(5 18 10 15)
G:=sub<Sym(20)| (1,6)(2,7)(3,8)(4,9)(5,10)(11,19)(12,20)(13,16)(14,17)(15,18), (11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,8,10,7,9), (1,6)(2,8,5,9)(3,10,4,7)(11,19)(12,17,15,16)(13,20,14,18), (1,19,6,11)(2,20,7,12)(3,16,8,13)(4,17,9,14)(5,18,10,15)>;
G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(11,19)(12,20)(13,16)(14,17)(15,18), (11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,8,10,7,9), (1,6)(2,8,5,9)(3,10,4,7)(11,19)(12,17,15,16)(13,20,14,18), (1,19,6,11)(2,20,7,12)(3,16,8,13)(4,17,9,14)(5,18,10,15) );
G=PermutationGroup([[(1,6),(2,7),(3,8),(4,9),(5,10),(11,19),(12,20),(13,16),(14,17),(15,18)], [(11,12,13,14,15),(16,17,18,19,20)], [(1,3,5,2,4),(6,8,10,7,9)], [(1,6),(2,8,5,9),(3,10,4,7),(11,19),(12,17,15,16),(13,20,14,18)], [(1,19,6,11),(2,20,7,12),(3,16,8,13),(4,17,9,14),(5,18,10,15)]])
G:=TransitiveGroup(20,105);
Matrix representation of C2.D5≀C2 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 0 | 7 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 36 |
| 0 | 0 | 0 | 0 | 40 | 35 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 36 |
| 0 | 0 | 0 | 0 | 40 | 35 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,7,0,0,0,0,34,40,0,0,0,0,0,0,40,40,0,0,0,0,36,35],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,40,0,0,0,0,36,35],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,35,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,6,1,0,0],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,0,35,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,6,0,0] >;
C2.D5≀C2 in GAP, Magma, Sage, TeX
C_2.D_5\wr C_2
% in TeX
G:=Group("C2.D5wrC2"); // GroupNames label
G:=SmallGroup(400,130);
// by ID
G=gap.SmallGroup(400,130);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,5,48,73,55,7204,1210,496,1157,299,2897]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^5=d^4=1,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*c*e^-1=b^2,e*b*e^-1=d*c*d^-1=c^3,e*d*e^-1=d^-1>;
// generators/relations
Export
Subgroup lattice of C2.D5≀C2 in TeX
Character table of C2.D5≀C2 in TeX