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## G = C52⋊2D4order 200 = 23·52

### 1st semidirect product of C52 and D4 acting via D4/C2=C22

Aliases: C522D4, D101D5, C10.3D10, C2.3D52, (D5×C10)⋊1C2, C52(C5⋊D4), C526C42C2, (C5×C10).3C22, SmallGroup(200,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C52⋊2D4
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — C52⋊2D4
 Lower central C52 — C5×C10 — C52⋊2D4
 Upper central C1 — C2

Generators and relations for C522D4
G = < a,b,c,d | a5=b5=c4=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Character table of C522D4

 class 1 2A 2B 2C 4 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M 10N 10O 10P size 1 1 10 10 50 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 10 10 10 10 10 10 10 10 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 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 -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 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 -1 -1 -1 linear of order 2 ρ5 2 -2 0 0 0 2 2 2 2 2 2 2 2 -2 -2 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 0 2 0 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 0 orthogonal lifted from D5 ρ7 2 2 -2 0 0 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 2 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 0 0 0 0 1+√5/2 orthogonal lifted from D10 ρ8 2 2 0 -2 0 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 0 0 0 1-√5/2 1+√5/2 1+√5/2 1-√5/2 0 orthogonal lifted from D10 ρ9 2 2 -2 0 0 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 2 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 0 0 0 0 1-√5/2 orthogonal lifted from D10 ρ10 2 2 0 2 0 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 0 orthogonal lifted from D5 ρ11 2 2 2 0 0 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 2 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 0 0 0 0 -1+√5/2 orthogonal lifted from D5 ρ12 2 2 0 -2 0 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 0 0 0 1+√5/2 1-√5/2 1-√5/2 1+√5/2 0 orthogonal lifted from D10 ρ13 2 2 2 0 0 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 2 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 0 0 0 0 -1-√5/2 orthogonal lifted from D5 ρ14 2 -2 0 0 0 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -2 -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 0 0 0 0 ζ54-ζ5 complex lifted from C5⋊D4 ρ15 2 -2 0 0 0 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -2 -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 0 0 0 0 -ζ54+ζ5 complex lifted from C5⋊D4 ρ16 2 -2 0 0 0 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -2 -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 0 0 0 0 ζ53-ζ52 complex lifted from C5⋊D4 ρ17 2 -2 0 0 0 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 -2 -2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 0 0 0 -ζ53+ζ52 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 0 complex lifted from C5⋊D4 ρ18 2 -2 0 0 0 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 -2 -2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 0 0 0 ζ54-ζ5 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 0 complex lifted from C5⋊D4 ρ19 2 -2 0 0 0 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 -2 -2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 0 0 0 -ζ54+ζ5 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 0 complex lifted from C5⋊D4 ρ20 2 -2 0 0 0 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -2 -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 0 0 0 0 -ζ53+ζ52 complex lifted from C5⋊D4 ρ21 2 -2 0 0 0 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 -2 -2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 0 0 0 ζ53-ζ52 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 0 complex lifted from C5⋊D4 ρ22 4 4 0 0 0 -1+√5 -1-√5 -1+√5 -1-√5 -1 3+√5/2 -1 3-√5/2 -1+√5 -1-√5 -1+√5 -1-√5 3-√5/2 -1 -1 3+√5/2 0 0 0 0 0 0 0 0 orthogonal lifted from D52 ρ23 4 4 0 0 0 -1-√5 -1+√5 -1-√5 -1+√5 -1 3-√5/2 -1 3+√5/2 -1-√5 -1+√5 -1-√5 -1+√5 3+√5/2 -1 -1 3-√5/2 0 0 0 0 0 0 0 0 orthogonal lifted from D52 ρ24 4 4 0 0 0 -1+√5 -1-√5 -1-√5 -1+√5 3-√5/2 -1 3+√5/2 -1 -1+√5 -1-√5 -1-√5 -1+√5 -1 3-√5/2 3+√5/2 -1 0 0 0 0 0 0 0 0 orthogonal lifted from D52 ρ25 4 4 0 0 0 -1-√5 -1+√5 -1+√5 -1-√5 3+√5/2 -1 3-√5/2 -1 -1-√5 -1+√5 -1+√5 -1-√5 -1 3+√5/2 3-√5/2 -1 0 0 0 0 0 0 0 0 orthogonal lifted from D52 ρ26 4 -4 0 0 0 -1-√5 -1+√5 -1-√5 -1+√5 -1 3-√5/2 -1 3+√5/2 1+√5 1-√5 1+√5 1-√5 -3-√5/2 1 1 -3+√5/2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ27 4 -4 0 0 0 -1+√5 -1-√5 -1-√5 -1+√5 3-√5/2 -1 3+√5/2 -1 1-√5 1+√5 1+√5 1-√5 1 -3+√5/2 -3-√5/2 1 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ28 4 -4 0 0 0 -1+√5 -1-√5 -1+√5 -1-√5 -1 3+√5/2 -1 3-√5/2 1-√5 1+√5 1-√5 1+√5 -3+√5/2 1 1 -3-√5/2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ29 4 -4 0 0 0 -1-√5 -1+√5 -1+√5 -1-√5 3+√5/2 -1 3-√5/2 -1 1+√5 1-√5 1-√5 1+√5 1 -3-√5/2 -3+√5/2 1 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C522D4
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 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 35 34 33 32)(36 40 39 38 37)
(1 39 6 34)(2 38 7 33)(3 37 8 32)(4 36 9 31)(5 40 10 35)(11 29 16 24)(12 28 17 23)(13 27 18 22)(14 26 19 21)(15 30 20 25)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 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), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,35,34,33,32)(36,40,39,38,37), (1,39,6,34)(2,38,7,33)(3,37,8,32)(4,36,9,31)(5,40,10,35)(11,29,16,24)(12,28,17,23)(13,27,18,22)(14,26,19,21)(15,30,20,25), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)>;`

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

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

C522D4 is a maximal subgroup of   C52⋊D8  C52⋊SD16  D205D5  D10.9D10  C20⋊D10  D10.4D10  D5×C5⋊D4
C522D4 is a maximal quotient of   C522D8  D20.D5  C522Q16  D10⋊Dic5  C10.Dic10

Matrix representation of C522D4 in GL4(𝔽41) generated by

 6 40 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 40 0 0 1 34
,
 23 6 0 0 21 18 0 0 0 0 0 1 0 0 1 0
,
 23 6 0 0 35 18 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(41))| [6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,34],[23,21,0,0,6,18,0,0,0,0,0,1,0,0,1,0],[23,35,0,0,6,18,0,0,0,0,0,1,0,0,1,0] >;`

C522D4 in GAP, Magma, Sage, TeX

`C_5^2\rtimes_2D_4`
`% in TeX`

`G:=Group("C5^2:2D4");`
`// GroupNames label`

`G:=SmallGroup(200,24);`
`// by ID`

`G=gap.SmallGroup(200,24);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-5,-5,61,328,4004]);`
`// Polycyclic`

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

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