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G = C4.16D16order 128 = 27

1st central extension by C4 of D16

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C4.16D16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C4×D8 — C4.16D16
 Lower central C1 — C2 — C4 — C8 — C4.16D16
 Upper central C1 — C2×C4 — C42 — C4×C8 — C4.16D16
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C4.16D16

Generators and relations for C4.16D16
G = < a,b,c | a4=b16=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >

Smallest permutation representation of C4.16D16
On 64 points
Generators in S64
```(1 62 47 30)(2 63 48 31)(3 64 33 32)(4 49 34 17)(5 50 35 18)(6 51 36 19)(7 52 37 20)(8 53 38 21)(9 54 39 22)(10 55 40 23)(11 56 41 24)(12 57 42 25)(13 58 43 26)(14 59 44 27)(15 60 45 28)(16 61 46 29)
(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)
(1 29 62 16 47 61 30 46)(2 15 63 60 48 45 31 28)(3 59 64 44 33 27 32 14)(4 43 49 26 34 13 17 58)(5 25 50 12 35 57 18 42)(6 11 51 56 36 41 19 24)(7 55 52 40 37 23 20 10)(8 39 53 22 38 9 21 54)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A ··· 8H 8I 8J 8K 8L 16A ··· 16P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 8 ··· 8 8 8 8 8 16 ··· 16 size 1 1 1 1 8 8 1 1 1 1 2 2 2 2 8 8 2 ··· 2 8 8 8 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 D4 D4 M4(2) D8 SD16 C4≀C2 D16 SD32 D8.C4 kernel C4.16D16 C8⋊1C8 C4×C16 C4×D8 C2.D8 C2×D8 D8 C42 C2×C8 C8 C2×C4 C2×C4 C4 C4 C4 C2 # reps 1 1 1 1 2 2 8 1 1 2 2 2 4 4 4 8

Matrix representation of C4.16D16 in GL4(𝔽17) generated by

 13 0 0 0 0 13 0 0 0 0 4 0 0 0 0 4
,
 10 7 0 0 10 10 0 0 0 0 3 5 0 0 7 13
,
 7 10 0 0 10 10 0 0 0 0 4 8 0 0 7 13
`G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[10,10,0,0,7,10,0,0,0,0,3,7,0,0,5,13],[7,10,0,0,10,10,0,0,0,0,4,7,0,0,8,13] >;`

C4.16D16 in GAP, Magma, Sage, TeX

`C_4._{16}D_{16}`
`% in TeX`

`G:=Group("C4.16D16");`
`// GroupNames label`

`G:=SmallGroup(128,63);`
`// by ID`

`G=gap.SmallGroup(128,63);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,219,436,136,2804,1411,172]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^16=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;`
`// generators/relations`

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