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## G = C10.D4order 80 = 24·5

### 1st non-split extension by C10 of D4 acting via D4/C22=C2

Aliases: C10.5D4, C10.1Q8, Dic51C4, C2.1Dic10, C22.4D10, C52(C4⋊C4), C2.4(C4×D5), (C2×C4).1D5, (C2×C20).1C2, C10.11(C2×C4), C2.1(C5⋊D4), (C2×C10).4C22, (C2×Dic5).1C2, SmallGroup(80,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C10.D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C10.D4
 Lower central C5 — C10 — C10.D4
 Upper central C1 — C22 — C2×C4

Generators and relations for C10.D4
G = < a,b,c | a10=b4=1, c2=a5, bab-1=cac-1=a-1, cbc-1=b-1 >

Character table of C10.D4

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 10A 10B 10C 10D 10E 10F 20A 20B 20C 20D 20E 20F 20G 20H size 1 1 1 1 2 2 10 10 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 -i i i -1 -i 1 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 i -i -i -1 i 1 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 i -i i 1 -i -1 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 -i i -i 1 i -1 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 2 2 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 -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 2 -2 -2 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 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 -2 -2 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 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 2 2 2 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 -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 ρ14 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 symplectic lifted from Q8, Schur index 2 ρ15 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 symplectic lifted from Dic10, Schur index 2 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 symplectic lifted from Dic10, Schur index 2 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 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 symplectic lifted from Dic10, Schur index 2 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 symplectic lifted from Dic10, Schur index 2 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 ζ53-ζ52 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 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 -ζ54+ζ5 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 complex lifted from C5⋊D4 ρ21 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 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 2i -2i 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 ζ43ζ53+ζ43ζ52 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ53+ζ4ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ4ζ54+ζ4ζ5 complex lifted from C4×D5 ρ24 2 -2 -2 2 -2i 2i 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 ζ4ζ54+ζ4ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ54+ζ43ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ43ζ53+ζ43ζ52 complex lifted from C4×D5 ρ25 2 -2 -2 2 -2i 2i 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 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 2i -2i 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 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ54+ζ4ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ4ζ53+ζ4ζ52 complex lifted from C4×D5

Smallest permutation representation of C10.D4
Regular action on 80 points
Generators in S80
```(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)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)
(1 65 30 58)(2 64 21 57)(3 63 22 56)(4 62 23 55)(5 61 24 54)(6 70 25 53)(7 69 26 52)(8 68 27 51)(9 67 28 60)(10 66 29 59)(11 50 79 39)(12 49 80 38)(13 48 71 37)(14 47 72 36)(15 46 73 35)(16 45 74 34)(17 44 75 33)(18 43 76 32)(19 42 77 31)(20 41 78 40)
(1 73 6 78)(2 72 7 77)(3 71 8 76)(4 80 9 75)(5 79 10 74)(11 29 16 24)(12 28 17 23)(13 27 18 22)(14 26 19 21)(15 25 20 30)(31 57 36 52)(32 56 37 51)(33 55 38 60)(34 54 39 59)(35 53 40 58)(41 65 46 70)(42 64 47 69)(43 63 48 68)(44 62 49 67)(45 61 50 66)```

`G:=sub<Sym(80)| (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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,65,30,58)(2,64,21,57)(3,63,22,56)(4,62,23,55)(5,61,24,54)(6,70,25,53)(7,69,26,52)(8,68,27,51)(9,67,28,60)(10,66,29,59)(11,50,79,39)(12,49,80,38)(13,48,71,37)(14,47,72,36)(15,46,73,35)(16,45,74,34)(17,44,75,33)(18,43,76,32)(19,42,77,31)(20,41,78,40), (1,73,6,78)(2,72,7,77)(3,71,8,76)(4,80,9,75)(5,79,10,74)(11,29,16,24)(12,28,17,23)(13,27,18,22)(14,26,19,21)(15,25,20,30)(31,57,36,52)(32,56,37,51)(33,55,38,60)(34,54,39,59)(35,53,40,58)(41,65,46,70)(42,64,47,69)(43,63,48,68)(44,62,49,67)(45,61,50,66)>;`

`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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,65,30,58)(2,64,21,57)(3,63,22,56)(4,62,23,55)(5,61,24,54)(6,70,25,53)(7,69,26,52)(8,68,27,51)(9,67,28,60)(10,66,29,59)(11,50,79,39)(12,49,80,38)(13,48,71,37)(14,47,72,36)(15,46,73,35)(16,45,74,34)(17,44,75,33)(18,43,76,32)(19,42,77,31)(20,41,78,40), (1,73,6,78)(2,72,7,77)(3,71,8,76)(4,80,9,75)(5,79,10,74)(11,29,16,24)(12,28,17,23)(13,27,18,22)(14,26,19,21)(15,25,20,30)(31,57,36,52)(32,56,37,51)(33,55,38,60)(34,54,39,59)(35,53,40,58)(41,65,46,70)(42,64,47,69)(43,63,48,68)(44,62,49,67)(45,61,50,66) );`

`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),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,65,30,58),(2,64,21,57),(3,63,22,56),(4,62,23,55),(5,61,24,54),(6,70,25,53),(7,69,26,52),(8,68,27,51),(9,67,28,60),(10,66,29,59),(11,50,79,39),(12,49,80,38),(13,48,71,37),(14,47,72,36),(15,46,73,35),(16,45,74,34),(17,44,75,33),(18,43,76,32),(19,42,77,31),(20,41,78,40)], [(1,73,6,78),(2,72,7,77),(3,71,8,76),(4,80,9,75),(5,79,10,74),(11,29,16,24),(12,28,17,23),(13,27,18,22),(14,26,19,21),(15,25,20,30),(31,57,36,52),(32,56,37,51),(33,55,38,60),(34,54,39,59),(35,53,40,58),(41,65,46,70),(42,64,47,69),(43,63,48,68),(44,62,49,67),(45,61,50,66)])`

Matrix representation of C10.D4 in GL4(𝔽41) generated by

 1 34 0 0 7 34 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 7 40 0 0 0 0 0 1 0 0 40 0
,
 32 0 0 0 19 9 0 0 0 0 40 0 0 0 0 1
`G:=sub<GL(4,GF(41))| [1,7,0,0,34,34,0,0,0,0,1,0,0,0,0,1],[1,7,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[32,19,0,0,0,9,0,0,0,0,40,0,0,0,0,1] >;`

C10.D4 in GAP, Magma, Sage, TeX

`C_{10}.D_4`
`% in TeX`

`G:=Group("C10.D4");`
`// GroupNames label`

`G:=SmallGroup(80,12);`
`// by ID`

`G=gap.SmallGroup(80,12);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-5,40,101,26,1604]);`
`// Polycyclic`

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

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