direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2xC52:C4, C10:2F5, C5:D5:4C4, C5:3(C2xF5), (C5xC10):4C4, C52:6(C2xC4), C5:D5.5C22, (C2xC5:D5).3C2, SmallGroup(200,48)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C2xC52:C4 |
Generators and relations for C2xC52:C4
G = < a,b,c,d | a2=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2, dcd-1=c3 >
Quotients: C1, C2, C4, C22, C2xC4, F5, C2xF5, C52:C4, C2xC52:C4
25C2
25C2
2C5
2C5
25C22
25C4
25C4
2C10
2C10
5D5
5D5
5D5
5D5
10D5
10D5
10D5
10D5
25C2xC4
5D10
5D10
5F5
5F5
5F5
5F5
10D10
10D10
Character table of C2xC52:C4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | 5F | 10A | 10B | 10C | 10D | 10E | 10F | |
| size | 1 | 1 | 25 | 25 | 25 | 25 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 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 | 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 | 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 | linear of order 4 |
| ρ9 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 4 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ10 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -4 | 1 | 1 | 1 | orthogonal lifted from C2xF5 |
| ρ11 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 4 | 1 | -4 | 1 | 1 | 1 | 1 | orthogonal lifted from C2xF5 |
| ρ12 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 4 | -1 | 4 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ13 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1-√5 | 3+√5/2 | 3-√5/2 | -1+√5 | -1 | 1-√5 | 1 | 1 | 1+√5 | -3-√5/2 | -3+√5/2 | orthogonal faithful |
| ρ14 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 3-√5/2 | -1-√5 | -1+√5 | 3+√5/2 | -1 | 3+√5/2 | -1 | -1 | 3-√5/2 | -1-√5 | -1+√5 | orthogonal lifted from C52:C4 |
| ρ15 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1-√5 | 3+√5/2 | 3-√5/2 | -1+√5 | -1 | -1+√5 | -1 | -1 | -1-√5 | 3+√5/2 | 3-√5/2 | orthogonal lifted from C52:C4 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 3+√5/2 | -1+√5 | -1-√5 | 3-√5/2 | -1 | -3+√5/2 | 1 | 1 | -3-√5/2 | 1-√5 | 1+√5 | orthogonal faithful |
| ρ17 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 3+√5/2 | -1+√5 | -1-√5 | 3-√5/2 | -1 | 3-√5/2 | -1 | -1 | 3+√5/2 | -1+√5 | -1-√5 | orthogonal lifted from C52:C4 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1+√5 | 3-√5/2 | 3+√5/2 | -1-√5 | -1 | 1+√5 | 1 | 1 | 1-√5 | -3+√5/2 | -3-√5/2 | orthogonal faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 3-√5/2 | -1-√5 | -1+√5 | 3+√5/2 | -1 | -3-√5/2 | 1 | 1 | -3+√5/2 | 1+√5 | 1-√5 | orthogonal faithful |
| ρ20 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1+√5 | 3-√5/2 | 3+√5/2 | -1-√5 | -1 | -1-√5 | -1 | -1 | -1+√5 | 3-√5/2 | 3+√5/2 | orthogonal lifted from C52:C4 |
►On 20 points - transitive group 20T49
Generators in S20
(1 6)(2 7)(3 8)(4 9)(5 10)(11 20)(12 16)(13 17)(14 18)(15 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)
(1 13 6 17)(2 11 10 19)(3 14 9 16)(4 12 8 18)(5 15 7 20)
G:=sub<Sym(20)| (1,6)(2,7)(3,8)(4,9)(5,10)(11,20)(12,16)(13,17)(14,18)(15,19), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,13,6,17)(2,11,10,19)(3,14,9,16)(4,12,8,18)(5,15,7,20)>;
G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(11,20)(12,16)(13,17)(14,18)(15,19), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,13,6,17)(2,11,10,19)(3,14,9,16)(4,12,8,18)(5,15,7,20) );
G=PermutationGroup([[(1,6),(2,7),(3,8),(4,9),(5,10),(11,20),(12,16),(13,17),(14,18),(15,19)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,3,5,2,4),(6,8,10,7,9),(11,14,12,15,13),(16,19,17,20,18)], [(1,13,6,17),(2,11,10,19),(3,14,9,16),(4,12,8,18),(5,15,7,20)]])
G:=TransitiveGroup(20,49);
►On 20 points - transitive group 20T52
Generators in S20
(1 8)(2 9)(3 10)(4 6)(5 7)(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 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)
(1 16)(2 19 5 18)(3 17 4 20)(6 15 10 12)(7 13 9 14)(8 11)
G:=sub<Sym(20)| (1,8)(2,9)(3,10)(4,6)(5,7)(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), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (1,16)(2,19,5,18)(3,17,4,20)(6,15,10,12)(7,13,9,14)(8,11)>;
G:=Group( (1,8)(2,9)(3,10)(4,6)(5,7)(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), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (1,16)(2,19,5,18)(3,17,4,20)(6,15,10,12)(7,13,9,14)(8,11) );
G=PermutationGroup([[(1,8),(2,9),(3,10),(4,6),(5,7),(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)], [(1,5,4,3,2),(6,10,9,8,7),(11,12,13,14,15),(16,17,18,19,20)], [(1,16),(2,19,5,18),(3,17,4,20),(6,15,10,12),(7,13,9,14),(8,11)]])
G:=TransitiveGroup(20,52);
C2xC52:C4 is a maximal subgroup of
C52:3C42 D10:F5 Dic5:F5 D52:C4 C2.D5wrC2 (C5xC10).Q8 C20:2F5 C102:4C4
C2xC52:C4 is a maximal quotient of
C20.11F5 C52:8M4(2) C20:2F5 C52:14M4(2) C102:4C4
Matrix representation of C2xC52:C4 ►in GL4(F41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 |
| 40 | 34 | 0 | 0 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 34 | 40 |
| 40 | 34 | 0 | 0 |
| 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,40,0,0,1,34,0,0,0,0,7,34,0,0,7,40],[40,7,0,0,34,7,0,0,0,0,34,1,0,0,40,0],[0,0,40,7,0,0,0,1,40,0,0,0,0,40,0,0] >;
C2xC52:C4 in GAP, Magma, Sage, TeX
C_2\times C_5^2\rtimes C_4
% in TeX
G:=Group("C2xC5^2:C4"); // GroupNames label
G:=SmallGroup(200,48);
// by ID
G=gap.SmallGroup(200,48);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,20,483,173,2004,1014]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^2,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of C2xC52:C4 in TeX
Character table of C2xC52:C4 in TeX