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## G = C4.D16order 128 = 27

### 1st non-split extension by C4 of D16 acting via D16/D8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C4.D16
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×C8 — C8⋊4D4 — C4.D16
 Lower central C1 — C2 — C2×C4 — C2×C8 — C4.D16
 Upper central C1 — C22 — C42 — C4×C8 — C4.D16
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C4.D16

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

Character table of C4.D16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 16 16 2 2 2 2 4 2 2 2 2 4 4 8 8 8 8 4 4 4 4 4 4 4 4 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 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 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 -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 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 -i -i i i -i i -i -i i i i -i linear of order 4 ρ6 1 1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 i i -i -i i -i i i -i -i -i i linear of order 4 ρ7 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 i i -i -i -i i -i -i i i i -i linear of order 4 ρ8 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 -i -i i i i -i i i -i -i -i i linear of order 4 ρ9 2 2 2 2 0 0 -2 2 2 -2 -2 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 2 2 2 2 2 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 2 0 0 -2 0 √2 -√2 √2 -√2 -√2 √2 0 0 0 0 ζ167-ζ16 -ζ167+ζ16 -ζ167+ζ16 ζ165-ζ163 -ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 -ζ165+ζ163 orthogonal lifted from D16 ρ12 2 -2 2 -2 0 0 0 2 -2 0 0 2 -2 -2 2 0 0 √2 -√2 -√2 √2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ13 2 2 2 2 0 0 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 -√2 -√2 -√2 √2 √2 -√2 √2 √2 orthogonal lifted from D8 ρ14 2 -2 2 -2 0 0 0 2 -2 0 0 2 -2 -2 2 0 0 -√2 √2 √2 -√2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ15 2 -2 -2 2 0 0 2 0 0 -2 0 -√2 √2 -√2 √2 √2 -√2 0 0 0 0 ζ165-ζ163 -ζ165+ζ163 -ζ165+ζ163 -ζ167+ζ16 ζ167-ζ16 ζ165-ζ163 -ζ167+ζ16 ζ167-ζ16 orthogonal lifted from D16 ρ16 2 2 2 2 0 0 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 √2 √2 √2 -√2 -√2 √2 -√2 -√2 orthogonal lifted from D8 ρ17 2 -2 -2 2 0 0 2 0 0 -2 0 √2 -√2 √2 -√2 -√2 √2 0 0 0 0 -ζ167+ζ16 ζ167-ζ16 ζ167-ζ16 -ζ165+ζ163 ζ165-ζ163 -ζ167+ζ16 -ζ165+ζ163 ζ165-ζ163 orthogonal lifted from D16 ρ18 2 -2 -2 2 0 0 2 0 0 -2 0 -√2 √2 -√2 √2 √2 -√2 0 0 0 0 -ζ165+ζ163 ζ165-ζ163 ζ165-ζ163 ζ167-ζ16 -ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 -ζ167+ζ16 orthogonal lifted from D16 ρ19 2 -2 2 -2 0 0 0 2 -2 0 0 -2 2 2 -2 0 0 -√-2 √-2 -√-2 √-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ20 2 -2 -2 2 0 0 -2 0 0 2 0 -√2 √2 -√2 √2 -√2 √2 0 0 0 0 ζ165+ζ163 ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 ζ167+ζ16 ζ1613+ζ1611 ζ1615+ζ169 ζ1615+ζ169 complex lifted from SD32 ρ21 2 -2 2 -2 0 0 0 2 -2 0 0 -2 2 2 -2 0 0 √-2 -√-2 √-2 -√-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ22 2 2 2 2 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ23 2 2 2 2 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ24 2 -2 -2 2 0 0 -2 0 0 2 0 √2 -√2 √2 -√2 √2 -√2 0 0 0 0 ζ167+ζ16 ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 ζ1613+ζ1611 ζ1615+ζ169 ζ165+ζ163 ζ165+ζ163 complex lifted from SD32 ρ25 2 -2 -2 2 0 0 -2 0 0 2 0 -√2 √2 -√2 √2 -√2 √2 0 0 0 0 ζ1613+ζ1611 ζ1613+ζ1611 ζ165+ζ163 ζ1615+ζ169 ζ1615+ζ169 ζ165+ζ163 ζ167+ζ16 ζ167+ζ16 complex lifted from SD32 ρ26 2 -2 -2 2 0 0 -2 0 0 2 0 √2 -√2 √2 -√2 √2 -√2 0 0 0 0 ζ1615+ζ169 ζ1615+ζ169 ζ167+ζ16 ζ165+ζ163 ζ165+ζ163 ζ167+ζ16 ζ1613+ζ1611 ζ1613+ζ1611 complex lifted from SD32 ρ27 4 -4 4 -4 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ28 4 4 -4 -4 0 0 0 0 0 0 0 -2√2 -2√2 2√2 2√2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from M5(2)⋊C2 ρ29 4 4 -4 -4 0 0 0 0 0 0 0 2√2 2√2 -2√2 -2√2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from M5(2)⋊C2

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

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

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

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

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

 0 1 0 0 16 0 0 0 0 0 1 0 0 0 0 1
,
 12 5 0 0 5 5 0 0 0 0 10 8 0 0 13 2
,
 5 12 0 0 5 5 0 0 0 0 10 8 0 0 11 7
`G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,5,0,0,5,5,0,0,0,0,10,13,0,0,8,2],[5,5,0,0,12,5,0,0,0,0,10,11,0,0,8,7] >;`

C4.D16 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(128,93);`
`// by ID`

`G=gap.SmallGroup(128,93);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,520,1690,416,2804,1411,172,4037,2028,124]);`
`// Polycyclic`

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

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