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## G = C2.D5≀C2order 400 = 24·52

### 2nd central extension by C2 of D5≀C2

Aliases: C2.2D5≀C2, C5⋊D5.2Q8, C523(C4⋊C4), C52⋊C43C4, (C5×C10).2D4, Dic52D5.2C2, C5⋊D5.6(C2×C4), (C2×C52⋊C4).3C2, (C2×C5⋊D5).7C22, SmallGroup(400,130)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C5⋊D5 — C2.D5≀C2
 Chief series C1 — C52 — C5⋊D5 — C2×C5⋊D5 — Dic5⋊2D5 — C2.D5≀C2
 Lower central C52 — C5⋊D5 — C2.D5≀C2
 Upper central C1 — C2

Generators and relations for C2.D5≀C2
G = < a,b,c,d,e | a2=b5=c5=d4=1, e2=a, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ece-1=b2, ebe-1=dcd-1=c3, ede-1=d-1 >

25C2
25C2
2C5
2C5
2C5
10C4
10C4
25C4
25C22
25C4
2C10
2C10
2C10
10D5
10D5
10D5
10D5
10D5
10D5
25C2×C4
25C2×C4
25C2×C4
2Dic5
2Dic5
10F5
10C20
10F5
10C20
10D10
10D10
10D10
25C4⋊C4
10C2×F5
10C4×D5
10C4×D5

Character table of C2.D5≀C2

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 5E 10A 10B 10C 10D 10E 20A 20B 20C 20D 20E 20F 20G 20H size 1 1 25 25 10 10 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 i -i -i i -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -i -i -i i i i i -i 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 i -i -i -i i i -i i 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 -i i i i -i -i i -i 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 i i i -i -i -i -i i linear of order 4 ρ9 2 2 -2 -2 0 0 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 -2 2 0 0 0 0 0 0 2 2 2 2 2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ11 4 4 0 0 0 0 2 2 0 0 3+√5/2 3-√5/2 -1+√5 -1-√5 -1 -1-√5 3-√5/2 -1+√5 3+√5/2 -1 0 -1-√5/2 -1+√5/2 0 -1-√5/2 -1+√5/2 0 0 orthogonal lifted from D5≀C2 ρ12 4 4 0 0 0 0 -2 -2 0 0 3+√5/2 3-√5/2 -1+√5 -1-√5 -1 -1-√5 3-√5/2 -1+√5 3+√5/2 -1 0 1+√5/2 1-√5/2 0 1+√5/2 1-√5/2 0 0 orthogonal lifted from D5≀C2 ρ13 4 4 0 0 -2 -2 0 0 0 0 -1-√5 -1+√5 3+√5/2 3-√5/2 -1 3-√5/2 -1+√5 3+√5/2 -1-√5 -1 1+√5/2 0 0 1+√5/2 0 0 1-√5/2 1-√5/2 orthogonal lifted from D5≀C2 ρ14 4 4 0 0 0 0 -2 -2 0 0 3-√5/2 3+√5/2 -1-√5 -1+√5 -1 -1+√5 3+√5/2 -1-√5 3-√5/2 -1 0 1-√5/2 1+√5/2 0 1-√5/2 1+√5/2 0 0 orthogonal lifted from D5≀C2 ρ15 4 4 0 0 -2 -2 0 0 0 0 -1+√5 -1-√5 3-√5/2 3+√5/2 -1 3+√5/2 -1-√5 3-√5/2 -1+√5 -1 1-√5/2 0 0 1-√5/2 0 0 1+√5/2 1+√5/2 orthogonal lifted from D5≀C2 ρ16 4 4 0 0 0 0 2 2 0 0 3-√5/2 3+√5/2 -1-√5 -1+√5 -1 -1+√5 3+√5/2 -1-√5 3-√5/2 -1 0 -1+√5/2 -1-√5/2 0 -1+√5/2 -1-√5/2 0 0 orthogonal lifted from D5≀C2 ρ17 4 4 0 0 2 2 0 0 0 0 -1+√5 -1-√5 3-√5/2 3+√5/2 -1 3+√5/2 -1-√5 3-√5/2 -1+√5 -1 -1+√5/2 0 0 -1+√5/2 0 0 -1-√5/2 -1-√5/2 orthogonal lifted from D5≀C2 ρ18 4 4 0 0 2 2 0 0 0 0 -1-√5 -1+√5 3+√5/2 3-√5/2 -1 3-√5/2 -1+√5 3+√5/2 -1-√5 -1 -1-√5/2 0 0 -1-√5/2 0 0 -1+√5/2 -1+√5/2 orthogonal lifted from D5≀C2 ρ19 4 -4 0 0 0 0 2i -2i 0 0 3-√5/2 3+√5/2 -1-√5 -1+√5 -1 1-√5 -3-√5/2 1+√5 -3+√5/2 1 0 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 0 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 0 0 complex faithful ρ20 4 -4 0 0 -2i 2i 0 0 0 0 -1+√5 -1-√5 3-√5/2 3+√5/2 -1 -3-√5/2 1+√5 -3+√5/2 1-√5 1 ζ4ζ54+ζ4ζ5 0 0 ζ43ζ54+ζ43ζ5 0 0 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 complex faithful ρ21 4 -4 0 0 0 0 -2i 2i 0 0 3-√5/2 3+√5/2 -1-√5 -1+√5 -1 1-√5 -3-√5/2 1+√5 -3+√5/2 1 0 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 0 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 0 0 complex faithful ρ22 4 -4 0 0 2i -2i 0 0 0 0 -1-√5 -1+√5 3+√5/2 3-√5/2 -1 -3+√5/2 1-√5 -3-√5/2 1+√5 1 ζ43ζ53+ζ43ζ52 0 0 ζ4ζ53+ζ4ζ52 0 0 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 complex faithful ρ23 4 -4 0 0 0 0 2i -2i 0 0 3+√5/2 3-√5/2 -1+√5 -1-√5 -1 1+√5 -3+√5/2 1-√5 -3-√5/2 1 0 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 0 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 0 0 complex faithful ρ24 4 -4 0 0 -2i 2i 0 0 0 0 -1-√5 -1+√5 3+√5/2 3-√5/2 -1 -3+√5/2 1-√5 -3-√5/2 1+√5 1 ζ4ζ53+ζ4ζ52 0 0 ζ43ζ53+ζ43ζ52 0 0 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 complex faithful ρ25 4 -4 0 0 0 0 -2i 2i 0 0 3+√5/2 3-√5/2 -1+√5 -1-√5 -1 1+√5 -3+√5/2 1-√5 -3-√5/2 1 0 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 0 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 0 0 complex faithful ρ26 4 -4 0 0 2i -2i 0 0 0 0 -1+√5 -1-√5 3-√5/2 3+√5/2 -1 -3-√5/2 1+√5 -3+√5/2 1-√5 1 ζ43ζ54+ζ43ζ5 0 0 ζ4ζ54+ζ4ζ5 0 0 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 C2.D5≀C2
On 20 points - transitive group 20T105
Generators in S20
```(1 6)(2 7)(3 8)(4 9)(5 10)(11 19)(12 20)(13 16)(14 17)(15 18)
(11 12 13 14 15)(16 17 18 19 20)
(1 3 5 2 4)(6 8 10 7 9)
(1 6)(2 8 5 9)(3 10 4 7)(11 19)(12 17 15 16)(13 20 14 18)
(1 19 6 11)(2 20 7 12)(3 16 8 13)(4 17 9 14)(5 18 10 15)```

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

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

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

`G:=TransitiveGroup(20,105);`

Matrix representation of C2.D5≀C2 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 34 0 0 0 0 7 40 0 0 0 0 0 0 40 36 0 0 0 0 40 35
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 7 0 0 0 0 34 7 0 0 0 0 0 0 40 36 0 0 0 0 40 35
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 6 0 0 0 0 0 1 0 0 35 1 0 0 0 0 1 0 0 0
,
 0 9 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 6 0 0 35 1 0 0 0 0 1 0 0 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,7,0,0,0,0,34,40,0,0,0,0,0,0,40,40,0,0,0,0,36,35],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,40,0,0,0,0,36,35],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,35,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,6,1,0,0],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,0,35,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,6,0,0] >;`

C2.D5≀C2 in GAP, Magma, Sage, TeX

`C_2.D_5\wr C_2`
`% in TeX`

`G:=Group("C2.D5wrC2");`
`// GroupNames label`

`G:=SmallGroup(400,130);`
`// by ID`

`G=gap.SmallGroup(400,130);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,5,48,73,55,7204,1210,496,1157,299,2897]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^5=c^5=d^4=1,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*c*e^-1=b^2,e*b*e^-1=d*c*d^-1=c^3,e*d*e^-1=d^-1>;`
`// generators/relations`

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