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## G = C4○D20order 80 = 24·5

### Central product of C4 and D20

Aliases: C4D20, D205C2, C4Dic10, C4.16D10, Dic105C2, C10.4C23, C22.2D10, C20.16C22, D10.1C22, Dic5.2C22, (C2×C4)⋊3D5, (C2×C20)⋊4C2, (C4×D5)⋊4C2, C51(C4○D4), C4(C5⋊D4), C5⋊D43C2, C2.5(C22×D5), (C2×C10).11C22, SmallGroup(80,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4○D20
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4○D20
 Lower central C5 — C10 — C4○D20
 Upper central C1 — C4 — C2×C4

Generators and relations for C4○D20
G = < a,b,c | a4=c2=1, b10=a2, ab=ba, ac=ca, cbc=a2b9 >

Character table of C4○D20

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 10A 10B 10C 10D 10E 10F 20A 20B 20C 20D 20E 20F 20G 20H size 1 1 2 10 10 1 1 2 10 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 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 -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 -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 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 -1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ7 1 1 -1 1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 -1 1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ9 2 2 2 0 0 -2 -2 -2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ10 2 2 2 0 0 2 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ11 2 2 2 0 0 2 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ12 2 2 -2 0 0 2 2 -2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D10 ρ13 2 2 2 0 0 -2 -2 -2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ14 2 2 -2 0 0 -2 -2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ15 2 2 -2 0 0 2 2 -2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D10 ρ16 2 2 -2 0 0 -2 -2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ17 2 -2 0 0 0 2i -2i 0 0 0 2 2 -2 0 0 0 0 -2 0 0 0 0 2i -2i -2i 2i complex lifted from C4○D4 ρ18 2 -2 0 0 0 -2i 2i 0 0 0 2 2 -2 0 0 0 0 -2 0 0 0 0 -2i 2i 2i -2i complex lifted from C4○D4 ρ19 2 -2 0 0 0 -2i 2i 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 -ζ54+ζ5 ζ53-ζ52 -ζ53+ζ52 ζ54-ζ5 1+√5/2 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 complex faithful ρ20 2 -2 0 0 0 -2i 2i 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 -ζ53+ζ52 1-√5/2 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 complex faithful ρ21 2 -2 0 0 0 2i -2i 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 ζ53-ζ52 1-√5/2 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 complex faithful ρ22 2 -2 0 0 0 2i -2i 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 -ζ54+ζ5 ζ53-ζ52 -ζ53+ζ52 ζ54-ζ5 1+√5/2 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 complex faithful ρ23 2 -2 0 0 0 -2i 2i 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 -ζ54+ζ5 1+√5/2 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 complex faithful ρ24 2 -2 0 0 0 2i -2i 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 -ζ54+ζ5 1+√5/2 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 complex faithful ρ25 2 -2 0 0 0 -2i 2i 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 ζ53-ζ52 1-√5/2 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 complex faithful ρ26 2 -2 0 0 0 2i -2i 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 -ζ53+ζ52 1-√5/2 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 complex faithful

Smallest permutation representation of C4○D20
On 40 points
Generators in S40
```(1 31 11 21)(2 32 12 22)(3 33 13 23)(4 34 14 24)(5 35 15 25)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)```

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

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

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

Matrix representation of C4○D20 in GL2(𝔽41) generated by

 9 0 0 9
,
 16 2 39 28
,
 35 1 6 6
`G:=sub<GL(2,GF(41))| [9,0,0,9],[16,39,2,28],[35,6,1,6] >;`

C4○D20 in GAP, Magma, Sage, TeX

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

`G:=Group("C4oD20");`
`// GroupNames label`

`G:=SmallGroup(80,38);`
`// by ID`

`G=gap.SmallGroup(80,38);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-5,46,182,1604]);`
`// Polycyclic`

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

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