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G = C52⋊C16order 400 = 24·52

The semidirect product of C52 and C16 acting via C16/C2=C8

Aliases: C52⋊C16, (C5×C10).C8, C2.(C52⋊C8), C524C8.C2, C526C4.1C4, SmallGroup(400,116)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊C16
 Chief series C1 — C52 — C5×C10 — C52⋊6C4 — C52⋊4C8 — C52⋊C16
 Lower central C52 — C52⋊C16
 Upper central C1 — C2

Generators and relations for C52⋊C16
G = < a,b,c | a5=b5=c16=1, ab=ba, cac-1=ab2, cbc-1=ab-1 >

2C5
2C5
2C5
25C4
2C10
2C10
2C10
25C8
10Dic5
10Dic5
10Dic5
25C16
10C5⋊C8
10C5⋊C8
10C5⋊C8

Character table of C52⋊C16

 class 1 2 4A 4B 5A 5B 5C 8A 8B 8C 8D 10A 10B 10C 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 25 25 8 8 8 25 25 25 25 8 8 8 25 25 25 25 25 25 25 25 ρ1 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 linear of order 2 ρ3 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -i i i i -i -i -i i linear of order 4 ρ4 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 i -i -i -i i i i -i linear of order 4 ρ5 1 1 -1 -1 1 1 1 -i -i i i 1 1 1 ζ8 ζ87 ζ83 ζ83 ζ85 ζ85 ζ8 ζ87 linear of order 8 ρ6 1 1 -1 -1 1 1 1 -i -i i i 1 1 1 ζ85 ζ83 ζ87 ζ87 ζ8 ζ8 ζ85 ζ83 linear of order 8 ρ7 1 1 -1 -1 1 1 1 i i -i -i 1 1 1 ζ87 ζ8 ζ85 ζ85 ζ83 ζ83 ζ87 ζ8 linear of order 8 ρ8 1 1 -1 -1 1 1 1 i i -i -i 1 1 1 ζ83 ζ85 ζ8 ζ8 ζ87 ζ87 ζ83 ζ85 linear of order 8 ρ9 1 -1 i -i 1 1 1 ζ166 ζ1614 ζ1610 ζ162 -1 -1 -1 ζ16 ζ167 ζ1611 ζ163 ζ1613 ζ165 ζ169 ζ1615 linear of order 16 ρ10 1 -1 i -i 1 1 1 ζ166 ζ1614 ζ1610 ζ162 -1 -1 -1 ζ169 ζ1615 ζ163 ζ1611 ζ165 ζ1613 ζ16 ζ167 linear of order 16 ρ11 1 -1 i -i 1 1 1 ζ1614 ζ166 ζ162 ζ1610 -1 -1 -1 ζ165 ζ163 ζ167 ζ1615 ζ16 ζ169 ζ1613 ζ1611 linear of order 16 ρ12 1 -1 i -i 1 1 1 ζ1614 ζ166 ζ162 ζ1610 -1 -1 -1 ζ1613 ζ1611 ζ1615 ζ167 ζ169 ζ16 ζ165 ζ163 linear of order 16 ρ13 1 -1 -i i 1 1 1 ζ1610 ζ162 ζ166 ζ1614 -1 -1 -1 ζ167 ζ16 ζ1613 ζ165 ζ1611 ζ163 ζ1615 ζ169 linear of order 16 ρ14 1 -1 -i i 1 1 1 ζ1610 ζ162 ζ166 ζ1614 -1 -1 -1 ζ1615 ζ169 ζ165 ζ1613 ζ163 ζ1611 ζ167 ζ16 linear of order 16 ρ15 1 -1 -i i 1 1 1 ζ162 ζ1610 ζ1614 ζ166 -1 -1 -1 ζ1611 ζ1613 ζ169 ζ16 ζ1615 ζ167 ζ163 ζ165 linear of order 16 ρ16 1 -1 -i i 1 1 1 ζ162 ζ1610 ζ1614 ζ166 -1 -1 -1 ζ163 ζ165 ζ16 ζ169 ζ167 ζ1615 ζ1611 ζ1613 linear of order 16 ρ17 8 8 0 0 3 -2 -2 0 0 0 0 3 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C52⋊C8 ρ18 8 8 0 0 -2 -2 3 0 0 0 0 -2 -2 3 0 0 0 0 0 0 0 0 orthogonal lifted from C52⋊C8 ρ19 8 8 0 0 -2 3 -2 0 0 0 0 -2 3 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C52⋊C8 ρ20 8 -8 0 0 -2 3 -2 0 0 0 0 2 -3 2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ21 8 -8 0 0 3 -2 -2 0 0 0 0 -3 2 2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ22 8 -8 0 0 -2 -2 3 0 0 0 0 2 2 -3 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C52⋊C16
On 80 points
Generators in S80
```(1 34 75 60 17)(2 76 18 35 61)(3 62 36 19 77)(4 37 78 63 20)(5 21 64 79 38)(6 49 39 22 80)(7 65 23 40 50)(8 24 51 66 41)(9 42 67 52 25)(10 68 26 43 53)(11 54 44 27 69)(12 45 70 55 28)(13 29 56 71 46)(14 57 47 30 72)(15 73 31 48 58)(16 32 59 74 33)
(1 60 34 17 75)(2 61 35 18 76)(3 19 62 77 36)(4 20 63 78 37)(5 79 21 38 64)(6 80 22 39 49)(7 40 65 50 23)(8 41 66 51 24)(9 52 42 25 67)(10 53 43 26 68)(11 27 54 69 44)(12 28 55 70 45)(13 71 29 46 56)(14 72 30 47 57)(15 48 73 58 31)(16 33 74 59 32)
(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)```

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

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

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

Matrix representation of C52⋊C16 in GL8(𝔽241)

 0 240 1 0 0 0 0 0 0 240 0 1 0 0 0 0 0 240 0 0 0 0 0 0 1 240 0 0 0 0 0 0 97 34 104 240 0 0 0 1 119 54 88 24 240 240 240 240 97 34 104 240 1 0 0 0 97 34 104 240 0 1 0 0
,
 240 1 0 0 0 0 0 0 240 0 1 0 0 0 0 0 240 0 0 1 0 0 0 0 240 0 0 0 0 0 0 0 130 145 200 40 0 0 1 0 130 145 200 40 0 0 0 1 152 165 184 65 240 240 240 240 130 145 200 40 1 0 0 0
,
 0 0 0 0 240 1 0 0 0 0 0 0 240 0 1 0 0 0 0 0 240 0 0 1 22 20 225 25 239 240 240 240 125 42 53 4 33 144 240 40 2 220 192 140 33 144 240 40 141 115 103 17 33 144 240 40 52 233 40 156 33 144 240 40

`G:=sub<GL(8,GF(241))| [0,0,0,1,97,119,97,97,240,240,240,240,34,54,34,34,1,0,0,0,104,88,104,104,0,1,0,0,240,24,240,240,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,1,0,0,0,0,0,240,0,0,0,0,0,0,1,240,0,0],[240,240,240,240,130,130,152,130,1,0,0,0,145,145,165,145,0,1,0,0,200,200,184,200,0,0,1,0,40,40,65,40,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240,0],[0,0,0,22,125,2,141,52,0,0,0,20,42,220,115,233,0,0,0,225,53,192,103,40,0,0,0,25,4,140,17,156,240,240,240,239,33,33,33,33,1,0,0,240,144,144,144,144,0,1,0,240,240,240,240,240,0,0,1,240,40,40,40,40] >;`

C52⋊C16 in GAP, Magma, Sage, TeX

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

`G:=Group("C5^2:C16");`
`// GroupNames label`

`G:=SmallGroup(400,116);`
`// by ID`

`G=gap.SmallGroup(400,116);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,5,12,31,50,10564,490,496,9797,2891,2897]);`
`// Polycyclic`

`G:=Group<a,b,c|a^5=b^5=c^16=1,a*b=b*a,c*a*c^-1=a*b^2,c*b*c^-1=a*b^-1>;`
`// generators/relations`

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