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## G = C25.D5order 320 = 26·5

### The non-split extension by C25 of D5 acting faithfully

Aliases: C25.D5, C24⋊Dic5, C24⋊C52C4, C2.1(C24⋊D5), (C2×C24⋊C5).C2, SmallGroup(320,1583)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24⋊C5 — C25.D5
 Chief series C1 — C24 — C24⋊C5 — C2×C24⋊C5 — C25.D5
 Lower central C24⋊C5 — C25.D5
 Upper central C1 — C2

Generators and relations for C25.D5
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f5=1, g2=a, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, geg-1=bc=cb, bd=db, fcf-1=gcg-1=be=eb, fbf-1=e, bg=gb, cd=dc, ce=ec, de=ed, fdf-1=bce, gdg-1=cde, fef-1=bcde, gfg-1=f-1 >

Subgroups: 800 in 111 conjugacy classes, 7 normal (all characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×25], C5, C2×C4 [×12], C23 [×25], C10, C22⋊C4 [×12], C22×C4 [×4], C24, C24 [×6], Dic5, C2×C22⋊C4 [×6], C25, C243C4, C24⋊C5, C2×C24⋊C5, C25.D5
Quotients: C1, C2, C4, D5, Dic5, C24⋊D5, C25.D5

Character table of C25.D5

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 10A 10B size 1 1 5 5 5 5 5 5 20 20 20 20 20 20 20 20 32 32 32 32 ρ1 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 linear of order 2 ρ3 1 -1 1 1 1 -1 -1 -1 -i i i i -i -i -i i 1 1 -1 -1 linear of order 4 ρ4 1 -1 1 1 1 -1 -1 -1 i -i -i -i i i i -i 1 1 -1 -1 linear of order 4 ρ5 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ6 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ7 2 -2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ8 2 -2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ9 5 5 -3 1 1 1 -3 1 1 -1 -1 1 1 -1 -1 1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ10 5 5 -3 1 1 1 -3 1 -1 1 1 -1 -1 1 1 -1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ11 5 5 1 1 -3 -3 1 1 1 -1 1 1 -1 -1 1 -1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ12 5 5 1 -3 1 1 1 -3 1 1 -1 1 -1 1 -1 -1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ13 5 5 1 -3 1 1 1 -3 -1 -1 1 -1 1 -1 1 1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ14 5 5 1 1 -3 -3 1 1 -1 1 -1 -1 1 1 -1 1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ15 5 -5 1 -3 1 -1 -1 3 -i i -i i i -i i -i 0 0 0 0 complex faithful ρ16 5 -5 1 1 -3 3 -1 -1 -i -i i i i i -i -i 0 0 0 0 complex faithful ρ17 5 -5 -3 1 1 -1 3 -1 i i i -i i -i -i -i 0 0 0 0 complex faithful ρ18 5 -5 1 -3 1 -1 -1 3 i -i i -i -i i -i i 0 0 0 0 complex faithful ρ19 5 -5 1 1 -3 3 -1 -1 i i -i -i -i -i i i 0 0 0 0 complex faithful ρ20 5 -5 -3 1 1 -1 3 -1 -i -i -i i -i i i i 0 0 0 0 complex faithful

Permutation representations of C25.D5
On 20 points - transitive group 20T82
Generators in S20
```(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)
(1 6)(3 13)(5 20)(8 18)(10 15)(11 16)
(1 11)(2 7)(3 13)(4 14)(5 20)(6 16)(8 18)(9 19)(10 15)(12 17)
(1 6)(2 17)(3 18)(5 10)(7 12)(8 13)(11 16)(15 20)
(2 12)(4 19)(5 10)(7 17)(9 14)(15 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 10 11 20)(2 9 12 19)(3 8 13 18)(4 7 14 17)(5 6 15 16)```

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

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

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

`G:=TransitiveGroup(20,82);`

On 20 points - transitive group 20T84
Generators in S20
```(1 11)(2 12)(3 13)(4 14)(5 15)(6 19)(7 20)(8 16)(9 17)(10 18)
(2 12)(3 13)(7 20)(8 16)
(2 12)(4 14)(7 20)(9 17)
(2 12)(3 13)(4 14)(5 15)(7 20)(8 16)(9 17)(10 18)
(1 11)(2 12)(6 19)(7 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 9 11 17)(2 8 12 16)(3 7 13 20)(4 6 14 19)(5 10 15 18)```

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

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

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

`G:=TransitiveGroup(20,84);`

Matrix representation of C25.D5 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 40 2 0 0 0 0 1 0 0 0 0 23 40 0 0 0 21 0 40 0 0 26 0 0 40
,
 40 0 0 0 0 0 40 0 0 0 0 18 1 0 0 1 19 0 1 0 34 22 0 0 1
,
 1 39 0 0 0 0 40 0 0 0 0 18 1 0 0 0 20 0 1 0 39 6 25 23 40
,
 40 0 0 0 0 0 40 0 0 0 0 18 1 0 0 0 18 2 40 0 0 22 7 0 40
,
 0 0 1 0 0 0 9 1 0 0 40 0 34 39 0 0 0 0 7 1 24 5 30 35 32
,
 0 23 40 0 0 0 32 0 0 0 1 39 0 0 0 33 12 31 1 32 39 3 18 23 40

`G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,2,1,23,21,26,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,1,34,0,40,18,19,22,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,39,39,40,18,20,6,0,0,1,0,25,0,0,0,1,23,0,0,0,0,40],[40,0,0,0,0,0,40,18,18,22,0,0,1,2,7,0,0,0,40,0,0,0,0,0,40],[0,0,40,0,24,0,9,0,0,5,1,1,34,0,30,0,0,39,7,35,0,0,0,1,32],[0,0,1,33,39,23,32,39,12,3,40,0,0,31,18,0,0,0,1,23,0,0,0,32,40] >;`

C25.D5 in GAP, Magma, Sage, TeX

`C_2^5.D_5`
`% in TeX`

`G:=Group("C2^5.D5");`
`// GroupNames label`

`G:=SmallGroup(320,1583);`
`// by ID`

`G=gap.SmallGroup(320,1583);`
`# by ID`

`G:=PCGroup([7,-2,-2,-5,-2,2,2,2,14,338,1683,437,1068,9245,2539,4906,265]);`
`// Polycyclic`

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

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