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## G = C4○D16order 64 = 26

### Central product of C4 and D16

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

Aliases: C4D16, C4Q32, C4SD32, D163C2, Q323C2, C8.13D4, C4.20D8, SD323C2, C8.9C23, C22.1D8, C16.6C22, D8.2C22, Q16.2C22, (C2×C16)⋊6C2, C4○D81C2, C2.15(C2×D8), (C2×C4).84D4, C4.10(C2×D4), (C2×C8).89C22, SmallGroup(64,189)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C4○D16
G = < a,b,c | a4=c2=1, b8=a2, ab=ba, ac=ca, cbc=a2b7 >

Character table of C4○D16

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 2 8 8 1 1 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 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 -2 0 0 -2 -2 2 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 0 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 0 2 2 -2 0 0 0 0 0 0 -√2 √2 √2 √2 -√2 -√2 -√2 √2 orthogonal lifted from D8 ρ12 2 2 2 0 0 -2 -2 -2 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ13 2 2 2 0 0 -2 -2 -2 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ14 2 2 -2 0 0 2 2 -2 0 0 0 0 0 0 √2 -√2 -√2 -√2 √2 √2 √2 -√2 orthogonal lifted from D8 ρ15 2 -2 0 0 0 -2i 2i 0 0 0 √2 -√-2 -√2 √-2 ζ167+ζ16 ζ165+ζ163 -ζ167+ζ16 ζ1613+ζ1611 -ζ165+ζ163 ζ1615+ζ169 ζ165-ζ163 ζ167-ζ16 complex faithful ρ16 2 -2 0 0 0 -2i 2i 0 0 0 -√2 √-2 √2 -√-2 ζ165+ζ163 ζ1615+ζ169 ζ165-ζ163 ζ167+ζ16 -ζ167+ζ16 ζ1613+ζ1611 ζ167-ζ16 -ζ165+ζ163 complex faithful ρ17 2 -2 0 0 0 2i -2i 0 0 0 √2 √-2 -√2 -√-2 ζ167+ζ16 ζ165+ζ163 ζ167-ζ16 ζ1613+ζ1611 ζ165-ζ163 ζ1615+ζ169 -ζ165+ζ163 -ζ167+ζ16 complex faithful ρ18 2 -2 0 0 0 2i -2i 0 0 0 -√2 -√-2 √2 √-2 ζ165+ζ163 ζ1615+ζ169 -ζ165+ζ163 ζ167+ζ16 ζ167-ζ16 ζ1613+ζ1611 -ζ167+ζ16 ζ165-ζ163 complex faithful ρ19 2 -2 0 0 0 -2i 2i 0 0 0 √2 -√-2 -√2 √-2 ζ1615+ζ169 ζ1613+ζ1611 ζ167-ζ16 ζ165+ζ163 ζ165-ζ163 ζ167+ζ16 -ζ165+ζ163 -ζ167+ζ16 complex faithful ρ20 2 -2 0 0 0 -2i 2i 0 0 0 -√2 √-2 √2 -√-2 ζ1613+ζ1611 ζ167+ζ16 -ζ165+ζ163 ζ1615+ζ169 ζ167-ζ16 ζ165+ζ163 -ζ167+ζ16 ζ165-ζ163 complex faithful ρ21 2 -2 0 0 0 2i -2i 0 0 0 -√2 -√-2 √2 √-2 ζ1613+ζ1611 ζ167+ζ16 ζ165-ζ163 ζ1615+ζ169 -ζ167+ζ16 ζ165+ζ163 ζ167-ζ16 -ζ165+ζ163 complex faithful ρ22 2 -2 0 0 0 2i -2i 0 0 0 √2 √-2 -√2 -√-2 ζ1615+ζ169 ζ1613+ζ1611 -ζ167+ζ16 ζ165+ζ163 -ζ165+ζ163 ζ167+ζ16 ζ165-ζ163 ζ167-ζ16 complex faithful

Smallest permutation representation of C4○D16
On 32 points
Generators in S32
```(1 27 9 19)(2 28 10 20)(3 29 11 21)(4 30 12 22)(5 31 13 23)(6 32 14 24)(7 17 15 25)(8 18 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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 20)(18 19)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)```

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

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

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

C4○D16 is a maximal subgroup of
D16.C4  D163C4  D16⋊C22
D16p⋊C2: C4○D32  C32⋊C22  D487C2  D485C2  D807C2  D805C2  D1127C2  Q323D7 ...
D2p.D8: D4○D16  D4○SD32  Q8○D16  D163S3  D6.2D8  D163D5  SD323D5  D163D7 ...
C4p.D8: C8○D16  D165C4  Q64⋊C2  Q16.D6  C40.30C23  C56.30C23 ...
C4○D16 is a maximal quotient of
C23.24D8  C23.25D8  C4×D16  C4×SD32  C4×Q32  D88D4  Q16.5D4  C167D4  C168D4  D8.Q8  Q16.Q8  C23.19D8  C23.20D8  C8.22SD16  C16.5Q8
D8.D2p: D8.10D4  D8.5D4  D163S3  D6.2D8  Q16.D6  D163D5  SD323D5  C40.30C23 ...
C8.D4p: C8.21D8  D487C2  D807C2  D1127C2 ...
C16.D2p: C16.19D4  D485C2  D805C2  Q323D7 ...

Matrix representation of C4○D16 in GL2(𝔽17) generated by

 13 0 0 13
,
 13 11 6 13
,
 13 11 11 4
`G:=sub<GL(2,GF(17))| [13,0,0,13],[13,6,11,13],[13,11,11,4] >;`

C4○D16 in GAP, Magma, Sage, TeX

`C_4\circ D_{16}`
`% in TeX`

`G:=Group("C4oD16");`
`// GroupNames label`

`G:=SmallGroup(64,189);`
`// by ID`

`G=gap.SmallGroup(64,189);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,-2,-2,121,158,579,297,165,1444,730,88]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=c^2=1,b^8=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^7>;`
`// generators/relations`

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