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## G = C52⋊D8order 400 = 24·52

### The semidirect product of C52 and D8 acting via D8/C2=D4

Aliases: C52⋊D8, C2.3D5≀C2, (C5×C10).3D4, C525C83C2, C522D45C2, C526C4.6C22, SmallGroup(400,131)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊6C4 — C52⋊D8
 Chief series C1 — C52 — C5×C10 — C52⋊6C4 — C52⋊2D4 — C52⋊D8
 Lower central C52 — C5×C10 — C52⋊6C4 — C52⋊D8
 Upper central C1 — C2

Generators and relations for C52⋊D8
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=dbd=a2, dad=cbc-1=b3, dcd=c-1 >

20C2
20C2
2C5
2C5
2C5
10C22
10C22
25C4
2C10
2C10
2C10
4D5
4D5
20C10
20C10
25D4
25D4
25C8
2D10
2D10
10Dic5
10C2×C10
10Dic5
10C2×C10
10Dic5
25D8
10C5⋊D4
10C5⋊D4
10C5⋊C8

Character table of C52⋊D8

 class 1 2A 2B 2C 4 5A 5B 5C 5D 5E 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M size 1 1 20 20 50 4 4 4 4 8 50 50 4 4 4 4 8 20 20 20 20 20 20 20 20 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 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 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 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 linear of order 2 ρ5 2 2 0 0 -2 2 2 2 2 2 0 0 2 2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 -2 0 0 0 2 2 2 2 2 √2 -√2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ7 2 -2 0 0 0 2 2 2 2 2 -√2 √2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ8 4 4 0 -2 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 1-√5/2 0 1+√5/2 0 0 1+√5/2 1-√5/2 orthogonal lifted from D5≀C2 ρ9 4 4 -2 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 1+√5/2 0 1-√5/2 0 1+√5/2 1-√5/2 0 0 orthogonal lifted from D5≀C2 ρ10 4 4 2 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 -1-√5/2 0 -1+√5/2 0 -1-√5/2 -1+√5/2 0 0 orthogonal lifted from D5≀C2 ρ11 4 4 0 2 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 -1+√5/2 0 -1-√5/2 0 0 -1-√5/2 -1+√5/2 orthogonal lifted from D5≀C2 ρ12 4 4 -2 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 1-√5/2 0 1+√5/2 0 1-√5/2 1+√5/2 0 0 orthogonal lifted from D5≀C2 ρ13 4 4 0 2 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 -1-√5/2 0 -1+√5/2 0 0 -1+√5/2 -1-√5/2 orthogonal lifted from D5≀C2 ρ14 4 4 0 -2 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 1+√5/2 0 1-√5/2 0 0 1-√5/2 1+√5/2 orthogonal lifted from D5≀C2 ρ15 4 4 2 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 -1+√5/2 0 -1-√5/2 0 -1+√5/2 -1-√5/2 0 0 orthogonal lifted from D5≀C2 ρ16 4 -4 0 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 0 0 1+√5 1-√5 -3+√5/2 -3-√5/2 1 ζ53-ζ52 0 ζ54-ζ5 0 -ζ53+ζ52 -ζ54+ζ5 0 0 complex faithful ρ17 4 -4 0 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 0 -3+√5/2 -3-√5/2 1-√5 1+√5 1 0 ζ53-ζ52 0 ζ54-ζ5 0 0 -ζ54+ζ5 -ζ53+ζ52 complex faithful ρ18 4 -4 0 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 0 0 1-√5 1+√5 -3-√5/2 -3+√5/2 1 -ζ54+ζ5 0 ζ53-ζ52 0 ζ54-ζ5 -ζ53+ζ52 0 0 complex faithful ρ19 4 -4 0 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 0 0 1+√5 1-√5 -3+√5/2 -3-√5/2 1 -ζ53+ζ52 0 -ζ54+ζ5 0 ζ53-ζ52 ζ54-ζ5 0 0 complex faithful ρ20 4 -4 0 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 0 -3-√5/2 -3+√5/2 1+√5 1-√5 1 0 ζ54-ζ5 0 -ζ53+ζ52 0 0 ζ53-ζ52 -ζ54+ζ5 complex faithful ρ21 4 -4 0 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 0 -3-√5/2 -3+√5/2 1+√5 1-√5 1 0 -ζ54+ζ5 0 ζ53-ζ52 0 0 -ζ53+ζ52 ζ54-ζ5 complex faithful ρ22 4 -4 0 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 0 0 1-√5 1+√5 -3-√5/2 -3+√5/2 1 ζ54-ζ5 0 -ζ53+ζ52 0 -ζ54+ζ5 ζ53-ζ52 0 0 complex faithful ρ23 4 -4 0 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 0 -3+√5/2 -3-√5/2 1-√5 1+√5 1 0 -ζ53+ζ52 0 -ζ54+ζ5 0 0 ζ54-ζ5 ζ53-ζ52 complex faithful ρ24 8 8 0 0 0 -2 -2 -2 -2 3 0 0 -2 -2 -2 -2 3 0 0 0 0 0 0 0 0 orthogonal lifted from D5≀C2 ρ25 8 -8 0 0 0 -2 -2 -2 -2 3 0 0 2 2 2 2 -3 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C52⋊D8
On 40 points
Generators in S40
```(1 33 18 9 30)(2 19 31 34 10)(3 32 11 20 35)(4 12 36 25 21)(5 37 22 13 26)(6 23 27 38 14)(7 28 15 24 39)(8 16 40 29 17)
(1 18 30 33 9)(2 34 19 10 31)(3 11 35 32 20)(4 25 12 21 36)(5 22 26 37 13)(6 38 23 14 27)(7 15 39 28 24)(8 29 16 17 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)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 19)(20 24)(21 23)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)```

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

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

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

Matrix representation of C52⋊D8 in GL4(𝔽41) generated by

 16 0 0 0 0 18 0 0 0 0 10 0 0 0 0 37
,
 10 0 0 0 0 37 0 0 0 0 16 0 0 0 0 18
,
 0 0 0 1 0 0 1 0 1 0 0 0 0 40 0 0
,
 1 0 0 0 0 40 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(41))| [16,0,0,0,0,18,0,0,0,0,10,0,0,0,0,37],[10,0,0,0,0,37,0,0,0,0,16,0,0,0,0,18],[0,0,1,0,0,0,0,40,0,1,0,0,1,0,0,0],[1,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;`

C52⋊D8 in GAP, Magma, Sage, TeX

`C_5^2\rtimes D_8`
`% in TeX`

`G:=Group("C5^2:D8");`
`// GroupNames label`

`G:=SmallGroup(400,131);`
`// by ID`

`G=gap.SmallGroup(400,131);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,5,73,218,116,50,7204,1210,496,1157,299,2897]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*b*d=a^2,d*a*d=c*b*c^-1=b^3,d*c*d=c^-1>;`
`// generators/relations`

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