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## G = F52order 400 = 24·52

### Direct product of F5 and F5

Aliases: F52, C52⋊C42, (C5×F5)⋊C4, D5.D5⋊C4, C52⋊C4⋊C4, C5⋊F5⋊C4, C54(C4×F5), (D5×F5).C2, D5.(C2×F5), D5⋊F5.1C2, D52.1C22, (C5×D5).(C2×C4), C5⋊D5.1(C2×C4), Hol(F5), SmallGroup(400,205)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — F52
 Chief series C1 — C5 — C52 — C5×D5 — D52 — D5×F5 — F52
 Lower central C52 — F52
 Upper central C1

Generators and relations for F52
G = < a,b,c,d | a5=b4=c5=d4=1, bab-1=a3, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 456 in 63 conjugacy classes, 22 normal (8 characteristic)
C1, C2 [×3], C4 [×6], C22, C5 [×2], C5, C2×C4 [×3], D5 [×2], D5 [×3], C10 [×2], C42, Dic5 [×2], C20 [×2], F5 [×2], F5 [×7], D10 [×2], C52, C4×D5 [×2], C2×F5 [×4], C5×D5 [×2], C5⋊D5, C4×F5 [×2], C5×F5 [×2], D5.D5 [×2], C5⋊F5, C52⋊C4, D52, D5×F5 [×2], D5⋊F5, F52
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], C42, F5 [×2], C2×F5 [×2], C4×F5 [×2], F52

Character table of F52

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 5C 10A 10B 20A 20B 20C 20D size 1 5 5 25 5 5 5 5 25 25 25 25 25 25 25 25 4 4 16 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 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 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 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 linear of order 2 ρ5 1 -1 1 -1 -1 -i -1 i i 1 -i 1 -i i -i i 1 1 1 -1 1 -i i -1 -1 linear of order 4 ρ6 1 -1 1 -1 -1 i -1 -i -i 1 i 1 i -i i -i 1 1 1 -1 1 i -i -1 -1 linear of order 4 ρ7 1 -1 -1 1 -i -i i i -1 -i i i -1 1 1 -i 1 1 1 -1 -1 -i i -i i linear of order 4 ρ8 1 1 -1 -1 i -1 -i -1 -i -i 1 i i i -i 1 1 1 1 1 -1 -1 -1 i -i linear of order 4 ρ9 1 1 -1 -1 i 1 -i 1 i -i -1 i -i -i i -1 1 1 1 1 -1 1 1 i -i linear of order 4 ρ10 1 -1 -1 1 -i i i -i 1 -i -i i 1 -1 -1 i 1 1 1 -1 -1 i -i -i i linear of order 4 ρ11 1 -1 1 -1 1 i 1 -i i -1 i -1 -i i -i -i 1 1 1 -1 1 i -i 1 1 linear of order 4 ρ12 1 -1 1 -1 1 -i 1 i -i -1 -i -1 i -i i i 1 1 1 -1 1 -i i 1 1 linear of order 4 ρ13 1 1 -1 -1 -i 1 i 1 -i i -1 -i i i -i -1 1 1 1 1 -1 1 1 -i i linear of order 4 ρ14 1 -1 -1 1 i i -i -i -1 i -i -i -1 1 1 i 1 1 1 -1 -1 i -i i -i linear of order 4 ρ15 1 1 -1 -1 -i -1 i -1 i i 1 -i -i -i i 1 1 1 1 1 -1 -1 -1 -i i linear of order 4 ρ16 1 -1 -1 1 i -i -i i 1 i i -i 1 -1 -1 -i 1 1 1 -1 -1 -i i i -i linear of order 4 ρ17 4 0 4 0 -4 0 -4 0 0 0 0 0 0 0 0 0 4 -1 -1 0 -1 0 0 1 1 orthogonal lifted from C2×F5 ρ18 4 4 0 0 0 -4 0 -4 0 0 0 0 0 0 0 0 -1 4 -1 -1 0 1 1 0 0 orthogonal lifted from C2×F5 ρ19 4 0 4 0 4 0 4 0 0 0 0 0 0 0 0 0 4 -1 -1 0 -1 0 0 -1 -1 orthogonal lifted from F5 ρ20 4 4 0 0 0 4 0 4 0 0 0 0 0 0 0 0 -1 4 -1 -1 0 -1 -1 0 0 orthogonal lifted from F5 ρ21 4 0 -4 0 -4i 0 4i 0 0 0 0 0 0 0 0 0 4 -1 -1 0 1 0 0 i -i complex lifted from C4×F5 ρ22 4 -4 0 0 0 -4i 0 4i 0 0 0 0 0 0 0 0 -1 4 -1 1 0 i -i 0 0 complex lifted from C4×F5 ρ23 4 0 -4 0 4i 0 -4i 0 0 0 0 0 0 0 0 0 4 -1 -1 0 1 0 0 -i i complex lifted from C4×F5 ρ24 4 -4 0 0 0 4i 0 -4i 0 0 0 0 0 0 0 0 -1 4 -1 1 0 -i i 0 0 complex lifted from C4×F5 ρ25 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -4 -4 1 0 0 0 0 0 0 orthogonal faithful

Permutation representations of F52
On 20 points - transitive group 20T102
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 18 6 12)(2 20 10 15)(3 17 9 13)(4 19 8 11)(5 16 7 14)
(1 5 4 3 2)(6 7 8 9 10)(11 13 15 12 14)(16 19 17 20 18)
(1 12 6 18)(2 13 7 19)(3 14 8 20)(4 15 9 16)(5 11 10 17)```

`G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,18,6,12)(2,20,10,15)(3,17,9,13)(4,19,8,11)(5,16,7,14), (1,5,4,3,2)(6,7,8,9,10)(11,13,15,12,14)(16,19,17,20,18), (1,12,6,18)(2,13,7,19)(3,14,8,20)(4,15,9,16)(5,11,10,17)>;`

`G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,18,6,12)(2,20,10,15)(3,17,9,13)(4,19,8,11)(5,16,7,14), (1,5,4,3,2)(6,7,8,9,10)(11,13,15,12,14)(16,19,17,20,18), (1,12,6,18)(2,13,7,19)(3,14,8,20)(4,15,9,16)(5,11,10,17) );`

`G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,18,6,12),(2,20,10,15),(3,17,9,13),(4,19,8,11),(5,16,7,14)], [(1,5,4,3,2),(6,7,8,9,10),(11,13,15,12,14),(16,19,17,20,18)], [(1,12,6,18),(2,13,7,19),(3,14,8,20),(4,15,9,16),(5,11,10,17)])`

`G:=TransitiveGroup(20,102);`

On 25 points - transitive group 25T32
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)
(2 3 5 4)(6 10 8 9)(11 13 12 15)(16 19 20 17)(21 24 25 22)
(1 18 23 14 7)(2 19 24 15 8)(3 20 25 11 9)(4 16 21 12 10)(5 17 22 13 6)
(6 13 17 22)(7 14 18 23)(8 15 19 24)(9 11 20 25)(10 12 16 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), (2,3,5,4)(6,10,8,9)(11,13,12,15)(16,19,20,17)(21,24,25,22), (1,18,23,14,7)(2,19,24,15,8)(3,20,25,11,9)(4,16,21,12,10)(5,17,22,13,6), (6,13,17,22)(7,14,18,23)(8,15,19,24)(9,11,20,25)(10,12,16,21)>;`

`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), (2,3,5,4)(6,10,8,9)(11,13,12,15)(16,19,20,17)(21,24,25,22), (1,18,23,14,7)(2,19,24,15,8)(3,20,25,11,9)(4,16,21,12,10)(5,17,22,13,6), (6,13,17,22)(7,14,18,23)(8,15,19,24)(9,11,20,25)(10,12,16,21) );`

`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)], [(2,3,5,4),(6,10,8,9),(11,13,12,15),(16,19,20,17),(21,24,25,22)], [(1,18,23,14,7),(2,19,24,15,8),(3,20,25,11,9),(4,16,21,12,10),(5,17,22,13,6)], [(6,13,17,22),(7,14,18,23),(8,15,19,24),(9,11,20,25),(10,12,16,21)])`

`G:=TransitiveGroup(25,32);`

Matrix representation of F52 in GL8(𝔽41)

 40 1 0 0 0 0 0 0 40 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 40 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 40 40 40 40 0 0 0 0 0 0 1 0
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0

`G:=sub<GL(8,GF(41))| [40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,40,1,0,0,0,0,0,1,40,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0] >;`

F52 in GAP, Magma, Sage, TeX

`F_5^2`
`% in TeX`

`G:=Group("F5^2");`
`// GroupNames label`

`G:=SmallGroup(400,205);`
`// by ID`

`G=gap.SmallGroup(400,205);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,490,262,5765,2897]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^4=c^5=d^4=1,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;`
`// generators/relations`

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