direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×D5, C5⋊C23, C10⋊C22, (C2×C10)⋊3C2, SmallGroup(40,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C22×D5 |
Generators and relations for C22×D5
G = < a,b,c,d | a2=b2=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Character table of C22×D5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 5A | 5B | 10A | 10B | 10C | 10D | 10E | 10F | |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 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 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 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 | linear of order 2 |
| ρ6 | 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 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 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 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 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 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 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 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | 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 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 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 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 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 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 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 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 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 |
►On 20 points - transitive group 20T8
Generators in S20
(1 19)(2 20)(3 16)(4 17)(5 18)(6 11)(7 12)(8 13)(9 14)(10 15)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 13)(2 12)(3 11)(4 15)(5 14)(6 16)(7 20)(8 19)(9 18)(10 17)
G:=sub<Sym(20)| (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,13)(2,12)(3,11)(4,15)(5,14)(6,16)(7,20)(8,19)(9,18)(10,17)>;
G:=Group( (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,13)(2,12)(3,11)(4,15)(5,14)(6,16)(7,20)(8,19)(9,18)(10,17) );
G=PermutationGroup([[(1,19),(2,20),(3,16),(4,17),(5,18),(6,11),(7,12),(8,13),(9,14),(10,15)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,13),(2,12),(3,11),(4,15),(5,14),(6,16),(7,20),(8,19),(9,18),(10,17)]])
G:=TransitiveGroup(20,8);
C22×D5 is a maximal subgroup of
D10⋊C4 C22⋊F5
C22×D5 is a maximal quotient of C4○D20 D4⋊2D5 Q8⋊2D5
Matrix representation of C22×D5 ►in GL3(𝔽11) generated by
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 10 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 10 | 7 |
| 1 | 0 | 0 |
| 0 | 0 | 10 |
| 0 | 10 | 0 |
G:=sub<GL(3,GF(11))| [10,0,0,0,10,0,0,0,10],[10,0,0,0,1,0,0,0,1],[1,0,0,0,0,10,0,1,7],[1,0,0,0,0,10,0,10,0] >;
C22×D5 in GAP, Magma, Sage, TeX
C_2^2\times D_5
% in TeX
G:=Group("C2^2xD5"); // GroupNames label
G:=SmallGroup(40,13);
// by ID
G=gap.SmallGroup(40,13);
# by ID
G:=PCGroup([4,-2,-2,-2,-5,515]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C22×D5 in TeX
Character table of C22×D5 in TeX