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## G = D20⋊4C4order 160 = 25·5

### 1st semidirect product of D20 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D20⋊4C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4○D20 — D20⋊4C4
 Lower central C5 — C10 — C20 — D20⋊4C4
 Upper central C1 — C4 — C2×C4 — C42

Generators and relations for D204C4
G = < a,b,c | a20=b2=c4=1, bab=a-1, ac=ca, cbc-1=a15b >

Smallest permutation representation of D204C4
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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 40)(17 39)(18 38)(19 37)(20 36)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)```

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

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

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

D204C4 is a maximal subgroup of
D4017C4  D4010C4  D5×C4≀C2  C42⋊D10  D44D20  M4(2).22D10  C42.196D10  M4(2)⋊D10  D4.9D20  D4.10D20  C424D10  C425D10  D20.14D4  D205D4  D20.15D4  C60.97D4  D6013C4  D607C4  C42⋊D15
D204C4 is a maximal quotient of
(C2×D20)⋊C4  C4⋊Dic5⋊C4  Dic103C8  D203C8  C42.D10  C42.2D10  C426Dic5  C60.97D4  D6013C4  D607C4

46 conjugacy classes

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

46 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D4 D5 D10 C4≀C2 C4×D5 D20 C5⋊D4 D20⋊4C4 kernel D20⋊4C4 C4.Dic5 C4×C20 C4○D20 Dic10 D20 C20 C2×C10 C42 C2×C4 C5 C4 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 2 2 4 4 4 4 16

Matrix representation of D204C4 in GL2(𝔽41) generated by

 5 0 0 33
,
 0 33 5 0
,
 40 0 0 32
`G:=sub<GL(2,GF(41))| [5,0,0,33],[0,5,33,0],[40,0,0,32] >;`

D204C4 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_4C_4`
`% in TeX`

`G:=Group("D20:4C4");`
`// GroupNames label`

`G:=SmallGroup(160,12);`
`// by ID`

`G=gap.SmallGroup(160,12);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,362,579,69,4613]);`
`// Polycyclic`

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

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