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## G = D20⋊C4order 160 = 25·5

### 1st semidirect product of D20 and C4 acting faithfully

Aliases: D41F5, D201C4, D5.2D8, D10.18D4, Dic5.2D4, D5.2SD16, C4⋊F51C2, D5⋊C81C2, C5⋊(D4⋊C4), (C5×D4)⋊1C4, C4.1(C2×F5), C20.1(C2×C4), (D4×D5).2C2, (C4×D5).7C22, C2.6(C22⋊F5), C10.5(C22⋊C4), SmallGroup(160,82)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D20⋊C4
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — D20⋊C4
 Lower central C5 — C10 — C20 — D20⋊C4
 Upper central C1 — C2 — C4 — D4

Generators and relations for D20⋊C4
G = < a,b,c | a20=b2=c4=1, bab=a-1, cac-1=a3, cbc-1=a17b >

Character table of D20⋊C4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5 8A 8B 8C 8D 10A 10B 10C 20 size 1 1 4 5 5 20 2 10 20 20 4 10 10 10 10 4 8 8 8 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 -1 -1 -1 1 -1 i -i 1 i -i -i i 1 1 1 1 linear of order 4 ρ6 1 1 -1 -1 -1 1 1 -1 i -i 1 -i i i -i 1 -1 -1 1 linear of order 4 ρ7 1 1 1 -1 -1 -1 1 -1 -i i 1 -i i i -i 1 1 1 1 linear of order 4 ρ8 1 1 -1 -1 -1 1 1 -1 -i i 1 i -i -i i 1 -1 -1 1 linear of order 4 ρ9 2 2 0 -2 -2 0 -2 2 0 0 2 0 0 0 0 2 0 0 -2 orthogonal lifted from D4 ρ10 2 2 0 2 2 0 -2 -2 0 0 2 0 0 0 0 2 0 0 -2 orthogonal lifted from D4 ρ11 2 -2 0 2 -2 0 0 0 0 0 2 √2 -√2 √2 -√2 -2 0 0 0 orthogonal lifted from D8 ρ12 2 -2 0 2 -2 0 0 0 0 0 2 -√2 √2 -√2 √2 -2 0 0 0 orthogonal lifted from D8 ρ13 2 -2 0 -2 2 0 0 0 0 0 2 -√-2 -√-2 √-2 √-2 -2 0 0 0 complex lifted from SD16 ρ14 2 -2 0 -2 2 0 0 0 0 0 2 √-2 √-2 -√-2 -√-2 -2 0 0 0 complex lifted from SD16 ρ15 4 4 4 0 0 0 4 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F5 ρ16 4 4 -4 0 0 0 4 0 0 0 -1 0 0 0 0 -1 1 1 -1 orthogonal lifted from C2×F5 ρ17 4 4 0 0 0 0 -4 0 0 0 -1 0 0 0 0 -1 √5 -√5 1 orthogonal lifted from C22⋊F5 ρ18 4 4 0 0 0 0 -4 0 0 0 -1 0 0 0 0 -1 -√5 √5 1 orthogonal lifted from C22⋊F5 ρ19 8 -8 0 0 0 0 0 0 0 0 -2 0 0 0 0 2 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of D20⋊C4
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 27)(22 26)(23 25)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)
(1 37)(2 24 10 40)(3 31 19 23)(4 38 8 26)(5 25 17 29)(6 32)(7 39 15 35)(9 33 13 21)(11 27)(12 34 20 30)(14 28 18 36)(16 22)```

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

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

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

D20⋊C4 is a maximal subgroup of
D8×F5  D40⋊C4  SD16×F5  SD16⋊F5  (D4×C10)⋊C4  C4○D4⋊F5  C4○D20⋊C4  D60⋊C4  D12⋊F5  D20⋊Dic3
D20⋊C4 is a maximal quotient of
D10.1D8  D20⋊C8  Dic5.D8  D10.18D8  D10.D8  D5.D16  D8.F5  D40.C4  D401C4  D5.Q32  Q16.F5  Dic20.C4  D10.SD16  Dic5.23D8  Dic5.SD16  D60⋊C4  D12⋊F5  D20⋊Dic3

Matrix representation of D20⋊C4 in GL6(𝔽41)

 1 39 0 0 0 0 1 40 0 0 0 0 0 0 6 40 0 0 0 0 36 1 0 0 0 0 0 0 0 7 0 0 0 0 35 35
,
 40 2 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 5 40
,
 0 30 0 0 0 0 15 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 3 24 0 0 0 0 18 38 0 0

`G:=sub<GL(6,GF(41))| [1,1,0,0,0,0,39,40,0,0,0,0,0,0,6,36,0,0,0,0,40,1,0,0,0,0,0,0,0,35,0,0,0,0,7,35],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,1,40,0,0,0,0,0,0,1,5,0,0,0,0,0,40],[0,15,0,0,0,0,30,0,0,0,0,0,0,0,0,0,3,18,0,0,0,0,24,38,0,0,2,0,0,0,0,0,0,2,0,0] >;`

D20⋊C4 in GAP, Magma, Sage, TeX

`D_{20}\rtimes C_4`
`% in TeX`

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

`G:=SmallGroup(160,82);`
`// by ID`

`G=gap.SmallGroup(160,82);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,579,297,69,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^17*b>;`
`// generators/relations`

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