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## G = D10⋊C4order 80 = 24·5

### 1st semidirect product of D10 and C4 acting via C4/C2=C2

Aliases: D101C4, C10.6D4, C2.2D20, C22.6D10, (C2×C4)⋊1D5, (C2×C20)⋊1C2, C2.5(C4×D5), C52(C22⋊C4), C10.12(C2×C4), (C2×Dic5)⋊1C2, C2.2(C5⋊D4), (C2×C10).6C22, (C22×D5).1C2, SmallGroup(80,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D10⋊C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — D10⋊C4
 Lower central C5 — C10 — D10⋊C4
 Upper central C1 — C22 — C2×C4

Generators and relations for D10⋊C4
G = < a,b,c | a10=b2=c4=1, bab=a-1, ac=ca, cbc-1=a5b >

Character table of D10⋊C4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 10A 10B 10C 10D 10E 10F 20A 20B 20C 20D 20E 20F 20G 20H size 1 1 1 1 10 10 2 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 -i i i -i 1 1 -1 1 -1 -1 -1 1 i i i i -i -i -i -i linear of order 4 ρ6 1 1 -1 -1 1 -1 -i i -i i 1 1 -1 1 -1 -1 -1 1 i i i i -i -i -i -i linear of order 4 ρ7 1 1 -1 -1 1 -1 i -i i -i 1 1 -1 1 -1 -1 -1 1 -i -i -i -i i i i i linear of order 4 ρ8 1 1 -1 -1 -1 1 i -i -i i 1 1 -1 1 -1 -1 -1 1 -i -i -i -i i i i i linear of order 4 ρ9 2 -2 2 -2 0 0 0 0 0 0 2 2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 0 0 0 0 0 0 2 2 2 -2 2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 -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 ρ12 2 -2 -2 2 0 0 0 0 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 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 orthogonal lifted from D20 ρ13 2 2 2 2 0 0 -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 2 0 0 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 ρ15 2 2 2 2 0 0 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 ρ16 2 -2 -2 2 0 0 0 0 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 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 orthogonal lifted from D20 ρ17 2 -2 -2 2 0 0 0 0 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 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 orthogonal lifted from D20 ρ18 2 -2 -2 2 0 0 0 0 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 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 orthogonal lifted from D20 ρ19 2 -2 2 -2 0 0 0 0 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 ζ54-ζ5 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 complex lifted from C5⋊D4 ρ20 2 -2 2 -2 0 0 0 0 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 ζ53-ζ52 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 complex lifted from C5⋊D4 ρ21 2 2 -2 -2 0 0 -2i 2i 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 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ54+ζ43ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ43ζ53+ζ43ζ52 complex lifted from C4×D5 ρ22 2 -2 2 -2 0 0 0 0 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 -ζ54+ζ5 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 complex lifted from C5⋊D4 ρ23 2 2 -2 -2 0 0 2i -2i 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 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ54+ζ4ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ4ζ53+ζ4ζ52 complex lifted from C4×D5 ρ24 2 -2 2 -2 0 0 0 0 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 -ζ53+ζ52 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 -ζ53+ζ52 complex lifted from C5⋊D4 ρ25 2 2 -2 -2 0 0 -2i 2i 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 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ43ζ53+ζ43ζ52 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ43ζ54+ζ43ζ5 complex lifted from C4×D5 ρ26 2 2 -2 -2 0 0 2i -2i 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 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ53+ζ4ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ4ζ54+ζ4ζ5 complex lifted from C4×D5

Smallest permutation representation of D10⋊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 12)(2 11)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(30 40)
(1 33 13 23)(2 34 14 24)(3 35 15 25)(4 36 16 26)(5 37 17 27)(6 38 18 28)(7 39 19 29)(8 40 20 30)(9 31 11 21)(10 32 12 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), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(30,40), (1,33,13,23)(2,34,14,24)(3,35,15,25)(4,36,16,26)(5,37,17,27)(6,38,18,28)(7,39,19,29)(8,40,20,30)(9,31,11,21)(10,32,12,22)>;`

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

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

Matrix representation of D10⋊C4 in GL3(𝔽41) generated by

 1 0 0 0 34 34 0 7 1
,
 1 0 0 0 34 34 0 1 7
,
 9 0 0 0 17 40 0 1 24
`G:=sub<GL(3,GF(41))| [1,0,0,0,34,7,0,34,1],[1,0,0,0,34,1,0,34,7],[9,0,0,0,17,1,0,40,24] >;`

D10⋊C4 in GAP, Magma, Sage, TeX

`D_{10}\rtimes C_4`
`% in TeX`

`G:=Group("D10:C4");`
`// GroupNames label`

`G:=SmallGroup(80,14);`
`// by ID`

`G=gap.SmallGroup(80,14);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-5,101,26,1604]);`
`// Polycyclic`

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

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