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## G = C2.D16order 64 = 26

### 1st central extension by C2 of D16

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

Aliases: D81C4, C8.14D4, C2.1D16, C4.1SD16, C2.1SD32, C22.8D8, (C2×C16)⋊3C2, C8.7(C2×C4), C2.D81C2, (C2×D8).1C2, (C2×C4).60D4, C4.1(C22⋊C4), (C2×C8).69C22, C2.6(D4⋊C4), 2-Sylow(CSO-(4,7)), SmallGroup(64,38)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2.D16
G = < a,b,c | a2=b16=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >

Character table of C2.D16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 8A 8B 8C 8D 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 8 8 2 2 8 8 2 2 2 2 2 2 2 2 2 2 2 2 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 -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 linear of order 2 ρ5 1 -1 -1 1 1 -1 -1 1 i -i 1 -1 1 -1 i -i -i i i i -i -i linear of order 4 ρ6 1 -1 -1 1 -1 1 -1 1 -i i 1 -1 1 -1 i -i -i i i i -i -i linear of order 4 ρ7 1 -1 -1 1 1 -1 -1 1 -i i 1 -1 1 -1 -i i i -i -i -i i i linear of order 4 ρ8 1 -1 -1 1 -1 1 -1 1 i -i 1 -1 1 -1 -i i i -i -i -i i i linear of order 4 ρ9 2 2 2 2 0 0 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 0 0 -2 2 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 -2 -2 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ12 2 -2 2 -2 0 0 0 0 0 0 √2 √2 -√2 -√2 -ζ165+ζ163 -ζ1615+ζ169 ζ165-ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 orthogonal lifted from D16 ρ13 2 2 2 2 0 0 -2 -2 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 0 0 0 -√2 -√2 √2 √2 ζ1615-ζ169 -ζ165+ζ163 -ζ1615+ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 orthogonal lifted from D16 ρ15 2 -2 2 -2 0 0 0 0 0 0 -√2 -√2 √2 √2 -ζ1615+ζ169 ζ165-ζ163 ζ1615-ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 orthogonal lifted from D16 ρ16 2 -2 2 -2 0 0 0 0 0 0 √2 √2 -√2 -√2 ζ165-ζ163 ζ1615-ζ169 -ζ165+ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 orthogonal lifted from D16 ρ17 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 ζ165+ζ163 ζ1615+ζ169 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ1613+ζ1611 ζ167+ζ16 complex lifted from SD32 ρ18 2 -2 -2 2 0 0 2 -2 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ19 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 ζ1613+ζ1611 ζ167+ζ16 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ165+ζ163 ζ1615+ζ169 complex lifted from SD32 ρ20 2 -2 -2 2 0 0 2 -2 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ21 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 ζ167+ζ16 ζ165+ζ163 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ1615+ζ169 ζ1613+ζ1611 complex lifted from SD32 ρ22 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 ζ1615+ζ169 ζ1613+ζ1611 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ167+ζ16 ζ165+ζ163 complex lifted from SD32

Smallest permutation representation of C2.D16
On 32 points
Generators in S32
```(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)
(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)
(1 26 27 16)(2 15 28 25)(3 24 29 14)(4 13 30 23)(5 22 31 12)(6 11 32 21)(7 20 17 10)(8 9 18 19)```

`G:=sub<Sym(32)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26), (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), (1,26,27,16)(2,15,28,25)(3,24,29,14)(4,13,30,23)(5,22,31,12)(6,11,32,21)(7,20,17,10)(8,9,18,19)>;`

`G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26), (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), (1,26,27,16)(2,15,28,25)(3,24,29,14)(4,13,30,23)(5,22,31,12)(6,11,32,21)(7,20,17,10)(8,9,18,19) );`

`G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26)], [(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)], [(1,26,27,16),(2,15,28,25),(3,24,29,14),(4,13,30,23),(5,22,31,12),(6,11,32,21),(7,20,17,10),(8,9,18,19)]])`

Matrix representation of C2.D16 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 7 2 0 0 10 10 0 0 0 0 7 1 0 0 16 7
,
 7 2 0 0 9 10 0 0 0 0 7 1 0 0 1 10
`G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[7,10,0,0,2,10,0,0,0,0,7,16,0,0,1,7],[7,9,0,0,2,10,0,0,0,0,7,1,0,0,1,10] >;`

C2.D16 in GAP, Magma, Sage, TeX

`C_2.D_{16}`
`% in TeX`

`G:=Group("C2.D16");`
`// GroupNames label`

`G:=SmallGroup(64,38);`
`// by ID`

`G=gap.SmallGroup(64,38);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,362,188,230,1444,730,88]);`
`// Polycyclic`

`G:=Group<a,b,c|a^2=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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