Copied to
clipboard

G = C52⋊2Q8order 200 = 23·52

The semidirect product of C52 and Q8 acting via Q8/C2=C22

Aliases: C522Q8, C51Dic10, C10.5D10, Dic5.1D5, C2.5D52, (C5×C10).5C22, C526C4.1C2, (C5×Dic5).2C2, SmallGroup(200,26)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C52⋊2Q8
 Chief series C1 — C5 — C52 — C5×C10 — C5×Dic5 — C52⋊2Q8
 Lower central C52 — C5×C10 — C52⋊2Q8
 Upper central C1 — C2

Generators and relations for C522Q8
G = < a,b,c,d | a5=b5=c4=1, d2=c2, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Character table of C522Q8

 class 1 2 4A 4B 4C 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E 10F 10G 10H 20A 20B 20C 20D 20E 20F 20G 20H 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 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 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 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 ρ8 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 ρ9 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 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 -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 ρ13 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 symplectic lifted from Q8, Schur index 2 ρ14 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 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 0 symplectic lifted from Dic10, Schur index 2 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 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 0 0 0 0 ζ43ζ54-ζ43ζ5 symplectic lifted from Dic10, Schur index 2 ρ16 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 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 0 symplectic lifted from Dic10, Schur index 2 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 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 0 symplectic lifted from Dic10, Schur index 2 ρ18 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 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 0 0 0 0 ζ4ζ53-ζ4ζ52 symplectic lifted from Dic10, Schur index 2 ρ19 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 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 0 0 0 0 -ζ4ζ53+ζ4ζ52 symplectic lifted from Dic10, Schur index 2 ρ20 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 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 0 symplectic lifted from Dic10, Schur index 2 ρ21 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 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 0 0 0 0 -ζ43ζ54+ζ43ζ5 symplectic lifted from Dic10, Schur index 2 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 C522Q8
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 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)(21 24 22 25 23)(26 29 27 30 28)(31 33 35 32 34)(36 38 40 37 39)
(1 17 8 12)(2 16 9 11)(3 20 10 15)(4 19 6 14)(5 18 7 13)(21 34 26 39)(22 33 27 38)(23 32 28 37)(24 31 29 36)(25 35 30 40)
(1 28 8 23)(2 29 9 24)(3 30 10 25)(4 26 6 21)(5 27 7 22)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)```

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

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

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

C522Q8 is a maximal subgroup of   C52⋊SD16  C52⋊Q16  D5×Dic10  Dic10⋊D5  D10.9D10  Dic5.D10  D10.4D10
C522Q8 is a maximal quotient of   Dic5⋊Dic5  C10.Dic10

Matrix representation of C522Q8 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 34 40 0 0 1 0
,
 6 40 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 2 13 0 0 28 39 0 0 0 0 1 0 0 0 34 40
,
 18 21 0 0 6 23 0 0 0 0 1 0 0 0 0 1
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,34,1,0,0,40,0],[6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[2,28,0,0,13,39,0,0,0,0,1,34,0,0,0,40],[18,6,0,0,21,23,0,0,0,0,1,0,0,0,0,1] >;`

C522Q8 in GAP, Magma, Sage, TeX

`C_5^2\rtimes_2Q_8`
`% in TeX`

`G:=Group("C5^2:2Q8");`
`// GroupNames label`

`G:=SmallGroup(200,26);`
`// by ID`

`G=gap.SmallGroup(200,26);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-5,-5,20,61,26,328,4004]);`
`// Polycyclic`

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

Export

׿
×
𝔽