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## G = D52order 100 = 22·52

### Direct product of D5 and D5

Aliases: D52, C51D10, C52⋊C22, C5⋊D5⋊C2, (C5×D5)⋊C2, SmallGroup(100,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D52
 Chief series C1 — C5 — C52 — C5×D5 — D52
 Lower central C52 — D52
 Upper central C1

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 >

5C2
5C2
25C2
2C5
2C5
25C22
5C10
5C10
5D5
5D5
10D5
10D5
5D10
5D10

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

Permutation representations of D52
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  C525D10
D52 is a maximal quotient of   Dic52D5  C522D4  C5⋊D20  C522Q8  D15⋊D5  C52⋊D10  C525D10

Polynomial with Galois group D52 over ℚ
actionf(x)Disc(f)
10T9x10+6x9-572x8-2326x7+739x6+5561x5+2493x4-1164x3-582x2+2x+1114·195·435·4015·573674

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`

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