direct product, metabelian, supersoluble, monomial, A-group
Aliases: D52, C5⋊1D10, C52⋊C22, C5⋊D5⋊C2, (C5×D5)⋊C2, SmallGroup(100,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — D52 |
Generators and relations for D52
G = < a,b,c,d | a5=b2=c5=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Character table of D52
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 10A | 10B | 10C | 10D | |
| size | 1 | 5 | 5 | 25 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | |
| 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 | 2 | 0 | 2 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 0 | -1-√5/2 | -1+√5/2 | 0 | orthogonal lifted from D5 |
| ρ6 | 2 | 0 | -2 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 0 | 1-√5/2 | 1+√5/2 | 0 | orthogonal lifted from D10 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1-√5/2 | orthogonal lifted from D5 |
| ρ8 | 2 | 0 | 2 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 0 | -1+√5/2 | -1-√5/2 | 0 | orthogonal lifted from D5 |
| ρ9 | 2 | -2 | 0 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 0 | 0 | 1+√5/2 | orthogonal lifted from D10 |
| ρ10 | 2 | 0 | -2 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 0 | 1+√5/2 | 1-√5/2 | 0 | orthogonal lifted from D10 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1+√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 0 | 0 | 1-√5/2 | orthogonal lifted from D10 |
| ρ13 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | -1-√5 | -1+√5 | -1 | 3-√5/2 | 3+√5/2 | -1 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ14 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | -1-√5 | -1+√5 | 3+√5/2 | -1 | -1 | 3-√5/2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ15 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | -1+√5 | -1-√5 | -1 | 3+√5/2 | 3-√5/2 | -1 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ16 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | -1+√5 | -1-√5 | 3-√5/2 | -1 | -1 | 3+√5/2 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 10 points - transitive group 10T9
Generators in S10
(1 2 3 4 5)(6 7 8 9 10)
(1 8)(2 7)(3 6)(4 10)(5 9)
(1 5 4 3 2)(6 7 8 9 10)
(1 8)(2 9)(3 10)(4 6)(5 7)
G:=sub<Sym(10)| (1,2,3,4,5)(6,7,8,9,10), (1,8)(2,7)(3,6)(4,10)(5,9), (1,5,4,3,2)(6,7,8,9,10), (1,8)(2,9)(3,10)(4,6)(5,7)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10), (1,8)(2,7)(3,6)(4,10)(5,9), (1,5,4,3,2)(6,7,8,9,10), (1,8)(2,9)(3,10)(4,6)(5,7) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10)], [(1,8),(2,7),(3,6),(4,10),(5,9)], [(1,5,4,3,2),(6,7,8,9,10)], [(1,8),(2,9),(3,10),(4,6),(5,7)]])
G:=TransitiveGroup(10,9);
►On 20 points - transitive group 20T28
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)(2 7)(3 6)(4 10)(5 9)(11 18)(12 17)(13 16)(14 20)(15 19)
(1 5 4 3 2)(6 7 8 9 10)(11 12 13 14 15)(16 20 19 18 17)
(1 15)(2 11)(3 12)(4 13)(5 14)(6 17)(7 18)(8 19)(9 20)(10 16)
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)(2,7)(3,6)(4,10)(5,9)(11,18)(12,17)(13,16)(14,20)(15,19), (1,5,4,3,2)(6,7,8,9,10)(11,12,13,14,15)(16,20,19,18,17), (1,15)(2,11)(3,12)(4,13)(5,14)(6,17)(7,18)(8,19)(9,20)(10,16)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,8)(2,7)(3,6)(4,10)(5,9)(11,18)(12,17)(13,16)(14,20)(15,19), (1,5,4,3,2)(6,7,8,9,10)(11,12,13,14,15)(16,20,19,18,17), (1,15)(2,11)(3,12)(4,13)(5,14)(6,17)(7,18)(8,19)(9,20)(10,16) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,8),(2,7),(3,6),(4,10),(5,9),(11,18),(12,17),(13,16),(14,20),(15,19)], [(1,5,4,3,2),(6,7,8,9,10),(11,12,13,14,15),(16,20,19,18,17)], [(1,15),(2,11),(3,12),(4,13),(5,14),(6,17),(7,18),(8,19),(9,20),(10,16)]])
G:=TransitiveGroup(20,28);
►On 25 points - transitive group 25T12
Generators in S25
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 23)(24 25)
(1 25 19 14 9)(2 21 20 15 10)(3 22 16 11 6)(4 23 17 12 7)(5 24 18 13 8)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 22)(12 23)(13 24)(14 25)(15 21)
G:=sub<Sym(25)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,23)(24,25), (1,25,19,14,9)(2,21,20,15,10)(3,22,16,11,6)(4,23,17,12,7)(5,24,18,13,8), (1,9)(2,10)(3,6)(4,7)(5,8)(11,22)(12,23)(13,24)(14,25)(15,21)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,23)(24,25), (1,25,19,14,9)(2,21,20,15,10)(3,22,16,11,6)(4,23,17,12,7)(5,24,18,13,8), (1,9)(2,10)(3,6)(4,7)(5,8)(11,22)(12,23)(13,24)(14,25)(15,21) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(21,23),(24,25)], [(1,25,19,14,9),(2,21,20,15,10),(3,22,16,11,6),(4,23,17,12,7),(5,24,18,13,8)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,22),(12,23),(13,24),(14,25),(15,21)]])
G:=TransitiveGroup(25,12);
D52 is a maximal subgroup of
D5⋊F5 D5≀C2 C52⋊D6 D15⋊D5 C52⋊D10 C52⋊5D10
D52 is a maximal quotient of Dic5⋊2D5 C52⋊2D4 C5⋊D20 C52⋊2Q8 D15⋊D5 C52⋊D10 C52⋊5D10
| action | f(x) | Disc(f) |
|---|---|---|
| 10T9 | x10+6x9-572x8-2326x7+739x6+5561x5+2493x4-1164x3-582x2+2x+1 | 114·195·435·4015·573674 |
Matrix representation of D52 ►in GL4(𝔽11) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 10 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 10 |
| 0 | 0 | 4 | 4 |
| 7 | 10 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 10 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(11))| [1,0,0,0,0,1,0,0,0,0,7,1,0,0,10,0],[1,0,0,0,0,1,0,0,0,0,7,4,0,0,10,4],[7,1,0,0,10,0,0,0,0,0,1,0,0,0,0,1],[7,4,0,0,10,4,0,0,0,0,1,0,0,0,0,1] >;
D52 in GAP, Magma, Sage, TeX
D_5^2
% in TeX
G:=Group("D5^2"); // GroupNames label
G:=SmallGroup(100,13);
// by ID
G=gap.SmallGroup(100,13);
# by ID
G:=PCGroup([4,-2,-2,-5,-5,102,1283]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^5=d^2=1,b*a*b=a^-1,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 D52 in TeX
Character table of D52 in TeX