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## G = C52⋊3Q16order 400 = 24·52

### 2nd semidirect product of C52 and Q16 acting via Q16/C4=C22

Aliases: C523Q16, C52Dic20, C20.15D10, C10.15D20, Dic10.1D5, C4.4D52, C52C8.1D5, C51(C5⋊Q16), (C5×C10).12D4, C10.4(C5⋊D4), (C5×C20).7C22, C2.7(C5⋊D20), C524Q8.2C2, (C5×Dic10).2C2, (C5×C52C8).2C2, SmallGroup(400,70)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C20 — C52⋊3Q16
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C5×Dic10 — C52⋊3Q16
 Lower central C52 — C5×C10 — C5×C20 — C52⋊3Q16
 Upper central C1 — C2 — C4

Generators and relations for C523Q16
G = < a,b,c,d | a5=b5=c8=1, d2=c4, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Smallest permutation representation of C523Q16
On 80 points
Generators in S80
```(1 58 53 69 46)(2 47 70 54 59)(3 60 55 71 48)(4 41 72 56 61)(5 62 49 65 42)(6 43 66 50 63)(7 64 51 67 44)(8 45 68 52 57)(9 25 73 24 40)(10 33 17 74 26)(11 27 75 18 34)(12 35 19 76 28)(13 29 77 20 36)(14 37 21 78 30)(15 31 79 22 38)(16 39 23 80 32)
(1 53 46 58 69)(2 54 47 59 70)(3 55 48 60 71)(4 56 41 61 72)(5 49 42 62 65)(6 50 43 63 66)(7 51 44 64 67)(8 52 45 57 68)(9 73 40 25 24)(10 74 33 26 17)(11 75 34 27 18)(12 76 35 28 19)(13 77 36 29 20)(14 78 37 30 21)(15 79 38 31 22)(16 80 39 32 23)
(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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 72 13 68)(10 71 14 67)(11 70 15 66)(12 69 16 65)(25 56 29 52)(26 55 30 51)(27 54 31 50)(28 53 32 49)(33 48 37 44)(34 47 38 43)(35 46 39 42)(36 45 40 41)(57 73 61 77)(58 80 62 76)(59 79 63 75)(60 78 64 74)```

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

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

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

49 conjugacy classes

 class 1 2 4A 4B 4C 5A 5B 5C 5D 5E 5F 5G 5H 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 20A 20B 20C 20D 20E ··· 20N 20O 20P 20Q 20R 40A ··· 40H order 1 2 4 4 4 5 5 5 5 5 5 5 5 8 8 10 10 10 10 10 10 10 10 20 20 20 20 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 2 20 100 2 2 2 2 4 4 4 4 10 10 2 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4 20 20 20 20 10 ··· 10

49 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + - - + + - image C1 C2 C2 C2 D4 D5 D5 Q16 D10 D20 C5⋊D4 Dic20 C5⋊Q16 D52 C5⋊D20 C52⋊3Q16 kernel C52⋊3Q16 C5×C5⋊2C8 C5×Dic10 C52⋊4Q8 C5×C10 C5⋊2C8 Dic10 C52 C20 C10 C10 C5 C5 C4 C2 C1 # reps 1 1 1 1 1 2 2 2 4 4 4 8 2 4 4 8

Matrix representation of C523Q16 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 1 34
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 35 0 0 0 0 6 35 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 13 34 0 0 0 0 22 4 0 0 0 0 0 0 2 28 0 0 0 0 13 39 0 0 0 0 0 0 13 18 0 0 0 0 27 28
,
 25 2 0 0 0 0 15 16 0 0 0 0 0 0 25 25 0 0 0 0 39 16 0 0 0 0 0 0 24 40 0 0 0 0 1 17

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,34],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,6,0,0,0,0,35,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,22,0,0,0,0,34,4,0,0,0,0,0,0,2,13,0,0,0,0,28,39,0,0,0,0,0,0,13,27,0,0,0,0,18,28],[25,15,0,0,0,0,2,16,0,0,0,0,0,0,25,39,0,0,0,0,25,16,0,0,0,0,0,0,24,1,0,0,0,0,40,17] >;`

C523Q16 in GAP, Magma, Sage, TeX

`C_5^2\rtimes_3Q_{16}`
`% in TeX`

`G:=Group("C5^2:3Q16");`
`// GroupNames label`

`G:=SmallGroup(400,70);`
`// by ID`

`G=gap.SmallGroup(400,70);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,73,79,218,50,970,11525]);`
`// 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^-1,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;`
`// generators/relations`

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