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## G = D52⋊C4order 400 = 24·52

### 1st semidirect product of D52 and C4 acting via C4/C2=C2

Aliases: D521C4, C2.1D5≀C2, C5⋊D5.3D4, (C5×C10).1D4, C523(C22⋊C4), Dic52D54C2, (C2×D52).3C2, C5⋊D5.5(C2×C4), (C2×C52⋊C4)⋊2C2, (C2×C5⋊D5).6C22, SmallGroup(400,129)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C5⋊D5 — D52⋊C4
 Chief series C1 — C52 — C5⋊D5 — C2×C5⋊D5 — C2×D52 — D52⋊C4
 Lower central C52 — C5⋊D5 — D52⋊C4
 Upper central C1 — C2

Generators and relations for D52⋊C4
G = < a,b,c,d,e | a5=b2=c5=d2=e4=1, bab=a-1, ac=ca, ad=da, eae-1=c, bc=cb, bd=db, ebe-1=d, dcd=c-1, ece-1=a, ede-1=b >

Subgroups: 630 in 66 conjugacy classes, 13 normal (11 characteristic)
C1, C2, C2 [×4], C4 [×2], C22 [×5], C5 [×3], C2×C4 [×2], C23, D5 [×8], C10 [×5], C22⋊C4, Dic5, C20, F5 [×2], D10 [×8], C2×C10, C52, C4×D5, C2×F5, C22×D5, C5×D5 [×2], C5⋊D5 [×2], C5×C10, C5×Dic5, C52⋊C4, D52 [×2], D52, D5×C10, C2×C5⋊D5, Dic52D5, C2×C52⋊C4, C2×D52, D52⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D5≀C2, D52⋊C4

Character table of D52⋊C4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 5E 10A 10B 10C 10D 10E 10F 10G 10H 10I 20A 20B 20C 20D size 1 1 10 10 25 25 10 10 50 50 4 4 4 4 8 4 4 4 4 8 20 20 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 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 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 -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 -1 -1 -1 linear of order 2 ρ5 1 -1 -1 1 1 -1 i -i -i i 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 -i i i -i linear of order 4 ρ6 1 -1 1 -1 1 -1 i -i i -i 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -i i i -i linear of order 4 ρ7 1 -1 -1 1 1 -1 -i i i -i 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 i -i -i i linear of order 4 ρ8 1 -1 1 -1 1 -1 -i i -i i 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 i -i -i i linear of order 4 ρ9 2 -2 0 0 -2 2 0 0 0 0 2 2 2 2 2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 0 -2 -2 0 0 0 0 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 -4 2 -2 0 0 0 0 0 0 3+√5/2 3-√5/2 -1+√5 -1-√5 -1 1+√5 -3+√5/2 1-√5 -3-√5/2 1 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 0 0 0 0 orthogonal faithful ρ12 4 4 2 2 0 0 0 0 0 0 3-√5/2 3+√5/2 -1-√5 -1+√5 -1 -1+√5 3+√5/2 -1-√5 3-√5/2 -1 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 0 0 0 0 orthogonal lifted from D5≀C2 ρ13 4 -4 2 -2 0 0 0 0 0 0 3-√5/2 3+√5/2 -1-√5 -1+√5 -1 1-√5 -3-√5/2 1+√5 -3+√5/2 1 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 0 0 0 0 orthogonal faithful ρ14 4 4 -2 -2 0 0 0 0 0 0 3+√5/2 3-√5/2 -1+√5 -1-√5 -1 -1-√5 3-√5/2 -1+√5 3+√5/2 -1 1+√5/2 1-√5/2 1-√5/2 1+√5/2 0 0 0 0 orthogonal lifted from D5≀C2 ρ15 4 4 2 2 0 0 0 0 0 0 3+√5/2 3-√5/2 -1+√5 -1-√5 -1 -1-√5 3-√5/2 -1+√5 3+√5/2 -1 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 0 0 0 0 orthogonal lifted from D5≀C2 ρ16 4 4 -2 -2 0 0 0 0 0 0 3-√5/2 3+√5/2 -1-√5 -1+√5 -1 -1+√5 3+√5/2 -1-√5 3-√5/2 -1 1-√5/2 1+√5/2 1+√5/2 1-√5/2 0 0 0 0 orthogonal lifted from D5≀C2 ρ17 4 -4 -2 2 0 0 0 0 0 0 3-√5/2 3+√5/2 -1-√5 -1+√5 -1 1-√5 -3-√5/2 1+√5 -3+√5/2 1 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 0 0 0 0 orthogonal faithful ρ18 4 4 0 0 0 0 2 2 0 0 -1+√5 -1-√5 3-√5/2 3+√5/2 -1 3+√5/2 -1-√5 3-√5/2 -1+√5 -1 0 0 0 0 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5≀C2 ρ19 4 4 0 0 0 0 -2 -2 0 0 -1+√5 -1-√5 3-√5/2 3+√5/2 -1 3+√5/2 -1-√5 3-√5/2 -1+√5 -1 0 0 0 0 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D5≀C2 ρ20 4 4 0 0 0 0 2 2 0 0 -1-√5 -1+√5 3+√5/2 3-√5/2 -1 3-√5/2 -1+√5 3+√5/2 -1-√5 -1 0 0 0 0 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5≀C2 ρ21 4 4 0 0 0 0 -2 -2 0 0 -1-√5 -1+√5 3+√5/2 3-√5/2 -1 3-√5/2 -1+√5 3+√5/2 -1-√5 -1 0 0 0 0 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D5≀C2 ρ22 4 -4 -2 2 0 0 0 0 0 0 3+√5/2 3-√5/2 -1+√5 -1-√5 -1 1+√5 -3+√5/2 1-√5 -3-√5/2 1 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 0 0 0 0 orthogonal faithful ρ23 4 -4 0 0 0 0 -2i 2i 0 0 -1-√5 -1+√5 3+√5/2 3-√5/2 -1 -3+√5/2 1-√5 -3-√5/2 1+√5 1 0 0 0 0 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 complex faithful ρ24 4 -4 0 0 0 0 -2i 2i 0 0 -1+√5 -1-√5 3-√5/2 3+√5/2 -1 -3-√5/2 1+√5 -3+√5/2 1-√5 1 0 0 0 0 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 complex faithful ρ25 4 -4 0 0 0 0 2i -2i 0 0 -1-√5 -1+√5 3+√5/2 3-√5/2 -1 -3+√5/2 1-√5 -3-√5/2 1+√5 1 0 0 0 0 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 complex faithful ρ26 4 -4 0 0 0 0 2i -2i 0 0 -1+√5 -1-√5 3-√5/2 3+√5/2 -1 -3-√5/2 1+√5 -3+√5/2 1-√5 1 0 0 0 0 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 complex faithful ρ27 8 -8 0 0 0 0 0 0 0 0 -2 -2 -2 -2 3 2 2 2 2 -3 0 0 0 0 0 0 0 0 orthogonal faithful ρ28 8 8 0 0 0 0 0 0 0 0 -2 -2 -2 -2 3 -2 -2 -2 -2 3 0 0 0 0 0 0 0 0 orthogonal lifted from D5≀C2

Permutation representations of D52⋊C4
On 20 points - transitive group 20T93
Generators in S20
```(11 12 13 14 15)(16 17 18 19 20)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 20)(12 19)(13 18)(14 17)(15 16)
(1 2 3 4 5)(6 7 8 9 10)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)```

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

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

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

`G:=TransitiveGroup(20,93);`

On 20 points - transitive group 20T94
Generators in S20
```(11 12 13 14 15)(16 17 18 19 20)
(11 20)(12 19)(13 18)(14 17)(15 16)
(1 2 3 4 5)(6 7 8 9 10)
(1 10)(2 9)(3 8)(4 7)(5 6)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)```

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

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

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

`G:=TransitiveGroup(20,94);`

On 20 points - transitive group 20T95
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 16)(7 20)(8 19)(9 18)(10 17)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)
(1 18)(2 19)(3 20)(4 16)(5 17)(6 12)(7 13)(8 14)(9 15)(10 11)
(1 16 7 11)(2 19 8 14)(3 17 9 12)(4 20 10 15)(5 18 6 13)```

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

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

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

`G:=TransitiveGroup(20,95);`

Matrix representation of D52⋊C4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 40 35 5 6
,
 40 39 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 6 34 0 0 0 0 6 0 0 0 0 0 0 1 1
,
 1 2 0 0 0 0 0 40 0 0 0 0 0 0 1 0 1 0 0 0 0 1 40 0 0 0 0 0 40 0 0 0 0 0 1 1
,
 40 39 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 34 40 35 35 0 0 40 0 0 1 0 0 1 0 1 0

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,1,0,35,0,0,0,0,1,5,0,0,1,0,0,6],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,1,34,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,2,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,40,40,1,0,0,0,0,0,1],[40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,34,40,1,0,0,0,40,0,0,0,0,0,35,0,1,0,0,0,35,1,0] >;`

D52⋊C4 in GAP, Magma, Sage, TeX

`D_5^2\rtimes C_4`
`% in TeX`

`G:=Group("D5^2:C4");`
`// GroupNames label`

`G:=SmallGroup(400,129);`
`// by ID`

`G=gap.SmallGroup(400,129);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,5,73,55,7204,1210,496,1157,299,2897]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^5=b^2=c^5=d^2=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=c,b*c=c*b,b*d=d*b,e*b*e^-1=d,d*c*d=c^-1,e*c*e^-1=a,e*d*e^-1=b>;`
`// generators/relations`

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