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## G = D20⋊7C4order 160 = 25·5

### 4th semidirect product of D20 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D20⋊7C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D20⋊7C4
 Lower central C5 — C10 — C20 — D20⋊7C4
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for D207C4
G = < a,b,c | a20=b2=c4=1, bab=a-1, cac-1=a9, cbc-1=a3b >

Smallest permutation representation of D207C4
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 22)(2 21)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)
(1 11)(2 20)(3 9)(4 18)(5 7)(6 16)(8 14)(10 12)(13 19)(15 17)(21 30 31 40)(22 39 32 29)(23 28 33 38)(24 37 34 27)(25 26 35 36)```

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

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

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

D207C4 is a maximal subgroup of
D20.1D4  D201D4  D20.4D4  D20.5D4  D5×C4≀C2  C42⋊D10  D4016C4  D4013C4  C23.20D20  C40.93D4  C40.50D4  D2018D4  D20.38D4  D20.39D4  D20.40D4  C60.96D4  D6016C4  D6010C4
D207C4 is a maximal quotient of
C42.D10  C42.2D10  C23.30D20  C22.2D40  D204C8  Dic104C8  C20.33C42  C60.96D4  D6016C4  D6010C4

34 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 10A 10B 10C 10D 20A 20B 20C 20D 20E 20F 40A ··· 40H order 1 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 10 10 10 10 20 20 20 20 20 20 40 ··· 40 size 1 1 2 20 1 1 2 10 10 10 10 20 2 2 4 4 2 2 4 4 2 2 2 2 4 4 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D4 D5 D10 C4≀C2 C4×D5 C5⋊D4 D20 D20⋊7C4 kernel D20⋊7C4 C4×Dic5 C5×M4(2) C4○D20 Dic10 D20 C20 C2×C10 M4(2) C2×C4 C5 C4 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 2 2 4 4 4 4 4

Matrix representation of D207C4 in GL4(𝔽41) generated by

 7 1 0 0 33 40 0 0 0 0 32 0 0 0 0 9
,
 1 1 0 0 0 40 0 0 0 0 0 32 0 0 9 0
,
 34 35 0 0 8 7 0 0 0 0 40 0 0 0 0 9
`G:=sub<GL(4,GF(41))| [7,33,0,0,1,40,0,0,0,0,32,0,0,0,0,9],[1,0,0,0,1,40,0,0,0,0,0,9,0,0,32,0],[34,8,0,0,35,7,0,0,0,0,40,0,0,0,0,9] >;`

D207C4 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_7C_4`
`% in TeX`

`G:=Group("D20:7C4");`
`// GroupNames label`

`G:=SmallGroup(160,32);`
`// by ID`

`G=gap.SmallGroup(160,32);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,86,579,297,69,4613]);`
`// Polycyclic`

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

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