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## G = C20.D4order 160 = 25·5

### 8th non-split extension by C20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20.D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4.Dic5 — C20.D4
 Lower central C5 — C10 — C2×C10 — C20.D4
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for C20.D4
G = < a,b,c | a20=1, b4=a10, c2=a5, bab-1=a-1, cac-1=a9, cbc-1=a15b3 >

Smallest permutation representation of C20.D4
On 40 points
Generators in S40
```(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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)
(1 28 6 23 11 38 16 33)(2 27 7 22 12 37 17 32)(3 26 8 21 13 36 18 31)(4 25 9 40 14 35 19 30)(5 24 10 39 15 34 20 29)
(1 38 6 23 11 28 16 33)(2 27 7 32 12 37 17 22)(3 36 8 21 13 26 18 31)(4 25 9 30 14 35 19 40)(5 34 10 39 15 24 20 29)```

`G:=sub<Sym(40)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (1,28,6,23,11,38,16,33)(2,27,7,22,12,37,17,32)(3,26,8,21,13,36,18,31)(4,25,9,40,14,35,19,30)(5,24,10,39,15,34,20,29), (1,38,6,23,11,28,16,33)(2,27,7,32,12,37,17,22)(3,36,8,21,13,26,18,31)(4,25,9,30,14,35,19,40)(5,34,10,39,15,24,20,29)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (1,28,6,23,11,38,16,33)(2,27,7,22,12,37,17,32)(3,26,8,21,13,36,18,31)(4,25,9,40,14,35,19,30)(5,24,10,39,15,34,20,29), (1,38,6,23,11,28,16,33)(2,27,7,32,12,37,17,22)(3,36,8,21,13,26,18,31)(4,25,9,30,14,35,19,40)(5,34,10,39,15,24,20,29) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)], [(1,28,6,23,11,38,16,33),(2,27,7,22,12,37,17,32),(3,26,8,21,13,36,18,31),(4,25,9,40,14,35,19,30),(5,24,10,39,15,34,20,29)], [(1,38,6,23,11,28,16,33),(2,27,7,32,12,37,17,22),(3,36,8,21,13,26,18,31),(4,25,9,30,14,35,19,40),(5,34,10,39,15,24,20,29)]])`

C20.D4 is a maximal subgroup of
C53C2≀C4  (C2×C20).D4  C242Dic5  (C22×C20)⋊C4  D5×C4.D4  M4(2).19D10  C425D10  D205D4  C40.23D4  C40.44D4  M4(2).D10  M4(2).13D10  (D4×C10).29C4  2+ 1+4⋊D5  2+ 1+4.D5  C20.5D12  C60.8D4
C20.D4 is a maximal quotient of
C24.Dic5  (C2×C20).Q8  C42.7D10  C20.9D8  C20.5Q16  C20.5D12  C60.8D4

31 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D order 1 2 2 2 2 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 20 20 20 size 1 1 2 4 4 2 2 2 2 20 20 20 20 2 ··· 2 4 ··· 4 4 4 4 4

31 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + image C1 C2 C2 C4 D4 D5 D10 Dic5 C5⋊D4 C4.D4 C20.D4 kernel C20.D4 C4.Dic5 D4×C10 C22×C10 C20 C2×D4 C2×C4 C23 C4 C5 C1 # reps 1 2 1 4 2 2 2 4 8 1 4

Matrix representation of C20.D4 in GL6(𝔽41)

 24 1 0 0 0 0 1 24 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 9 11 0 0 0 0 30 32 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 1 0 0 0 0 1 0 0 0
,
 11 9 0 0 0 0 32 30 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 40 0 0 0

`G:=sub<GL(6,GF(41))| [24,1,0,0,0,0,1,24,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[9,30,0,0,0,0,11,32,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,40,0,0],[11,32,0,0,0,0,9,30,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C20.D4 in GAP, Magma, Sage, TeX

`C_{20}.D_4`
`% in TeX`

`G:=Group("C20.D4");`
`// GroupNames label`

`G:=SmallGroup(160,40);`
`// by ID`

`G=gap.SmallGroup(160,40);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,188,86,579,4613]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^4=a^10,c^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^15*b^3>;`
`// generators/relations`

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