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## G = C52⋊Q16order 400 = 24·52

### The semidirect product of C52 and Q16 acting via Q16/C2=D4

Aliases: C52⋊Q16, C2.5D5≀C2, (C5×C10).5D4, C525C8.C2, C522Q8.C2, C526C4.8C22, SmallGroup(400,133)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊6C4 — C52⋊Q16
 Chief series C1 — C52 — C5×C10 — C52⋊6C4 — C52⋊2Q8 — C52⋊Q16
 Lower central C52 — C5×C10 — C52⋊6C4 — C52⋊Q16
 Upper central C1 — C2

Generators and relations for C52⋊Q16
G = < a,b,c,d | a5=b5=c8=1, d2=c4, ab=ba, cac-1=a2, dad-1=b, cbc-1=b3, dbd-1=a, dcd-1=c-1 >

2C5
2C5
2C5
10C4
10C4
25C4
2C10
2C10
2C10
25C8
25Q8
25Q8
2Dic5
2Dic5
10Dic5
10Dic5
10C20
10C20
10Dic5
25Q16
10Dic10
10Dic10
10C5⋊C8

Character table of C52⋊Q16

 class 1 2 4A 4B 4C 5A 5B 5C 5D 5E 8A 8B 10A 10B 10C 10D 10E 20A 20B 20C 20D 20E 20F 20G 20H 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 symplectic lifted from Q16, Schur index 2 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 symplectic lifted from Q16, Schur index 2 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 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 ρ13 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 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 -ζ43ζ54+ζ43ζ5 0 -ζ4ζ53+ζ4ζ52 0 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 0 0 symplectic faithful, Schur index 2 ρ17 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 ζ43ζ54-ζ43ζ5 0 ζ4ζ53-ζ4ζ52 0 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 0 0 symplectic faithful, Schur index 2 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 ζ4ζ53-ζ4ζ52 0 -ζ43ζ54+ζ43ζ5 0 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 0 0 symplectic faithful, Schur index 2 ρ19 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 -ζ43ζ54+ζ43ζ5 0 -ζ4ζ53+ζ4ζ52 0 0 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 symplectic faithful, Schur index 2 ρ20 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 -ζ4ζ53+ζ4ζ52 0 ζ43ζ54-ζ43ζ5 0 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 0 0 symplectic faithful, Schur index 2 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 ζ4ζ53-ζ4ζ52 0 -ζ43ζ54+ζ43ζ5 0 0 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 symplectic faithful, Schur index 2 ρ22 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 -ζ4ζ53+ζ4ζ52 0 ζ43ζ54-ζ43ζ5 0 0 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 symplectic faithful, Schur index 2 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 ζ43ζ54-ζ43ζ5 0 ζ4ζ53-ζ4ζ52 0 0 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 symplectic faithful, Schur index 2 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⋊Q16
On 80 points
Generators in S80
```(1 41 66 75 12)(2 67 13 42 76)(3 14 77 68 43)(4 78 44 15 69)(5 45 70 79 16)(6 71 9 46 80)(7 10 73 72 47)(8 74 48 11 65)(17 29 60 50 40)(18 61 33 30 51)(19 34 52 62 31)(20 53 32 35 63)(21 25 64 54 36)(22 57 37 26 55)(23 38 56 58 27)(24 49 28 39 59)
(1 66 12 41 75)(2 42 67 76 13)(3 77 43 14 68)(4 15 78 69 44)(5 70 16 45 79)(6 46 71 80 9)(7 73 47 10 72)(8 11 74 65 48)(17 50 29 40 60)(18 33 51 61 30)(19 62 34 31 52)(20 32 63 53 35)(21 54 25 36 64)(22 37 55 57 26)(23 58 38 27 56)(24 28 59 49 39)
(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 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 45 21 41)(18 44 22 48)(19 43 23 47)(20 42 24 46)(25 75 29 79)(26 74 30 78)(27 73 31 77)(28 80 32 76)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)```

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

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

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

Matrix representation of C52⋊Q16 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 35 0 0 0 0 7 34 0 0 0 0 0 0 40 1 0 0 0 0 33 7
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 1 0 0 0 0 33 7 0 0 0 0 0 0 0 35 0 0 0 0 7 34
,
 27 7 0 0 0 0 0 38 0 0 0 0 0 0 0 0 26 18 0 0 0 0 24 15 0 0 40 0 0 0 0 0 0 40 0 0
,
 5 28 0 0 0 0 2 36 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,0,0,0,35,34,0,0,0,0,0,0,40,33,0,0,0,0,1,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,1,7,0,0,0,0,0,0,0,7,0,0,0,0,35,34],[27,0,0,0,0,0,7,38,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,26,24,0,0,0,0,18,15,0,0],[5,2,0,0,0,0,28,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C52⋊Q16 in GAP, Magma, Sage, TeX

`C_5^2\rtimes Q_{16}`
`% in TeX`

`G:=Group("C5^2:Q16");`
`// GroupNames label`

`G:=SmallGroup(400,133);`
`// by ID`

`G=gap.SmallGroup(400,133);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,5,48,73,55,218,116,50,7204,1210,496,1157,299,2897]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^5=c^8=1,d^2=c^4,a*b=b*a,c*a*c^-1=a^2,d*a*d^-1=b,c*b*c^-1=b^3,d*b*d^-1=a,d*c*d^-1=c^-1>;`
`// generators/relations`

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