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## G = D4○C16order 64 = 26

### Central product of D4 and C16

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — D4○C16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C8○D4 — D4○C16
 Lower central C1 — C2 — D4○C16
 Upper central C1 — C16 — D4○C16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — D4○C16

Generators and relations for D4○C16
G = < a,b,c | a4=b2=1, c8=a2, bab=a-1, ac=ca, bc=cb >

Smallest permutation representation of D4○C16
On 32 points
Generators in S32
```(1 32 9 24)(2 17 10 25)(3 18 11 26)(4 19 12 27)(5 20 13 28)(6 21 14 29)(7 22 15 30)(8 23 16 31)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 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)```

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

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

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

D4○C16 is a maximal subgroup of
D4.C16  D4.3D8  D4.4D8  D4.5D8  C16.A4  C5⋊C16.C22
D4p.C8: C16○D8  D8.C8  D12.4C8  C16.12D6  D20.6C8  D20.5C8  Dic10.C8  D28.4C8 ...
C8p.C23: D4○C32  Q8○M5(2)  D4○D16  D4○SD32  Q8○D16  C24.78C23  C40.70C23  C56.70C23 ...
D4○C16 is a maximal quotient of
C4⋊C4.7C8  C8.12M4(2)  Q8×C16  C164Q8
(Cp×D4).C8: (C2×D4).5C8  D4×C16  C169D4  C166D4  C24.78C23  C40.70C23  C5⋊C16.C22  C56.70C23 ...
C4p.(C2×C8): C162M5(2)  D12.4C8  C16.12D6  D20.6C8  D20.5C8  Dic10.C8  D28.4C8  C16.12D14 ...

40 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E ··· 8J 16A ··· 16H 16I ··· 16T order 1 2 2 2 2 4 4 4 4 4 8 8 8 8 8 ··· 8 16 ··· 16 16 ··· 16 size 1 1 2 2 2 1 1 2 2 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 type + + + + image C1 C2 C2 C2 C4 C4 C8 C8 D4○C16 kernel D4○C16 C2×C16 M5(2) C8○D4 M4(2) C4○D4 D4 Q8 C1 # reps 1 3 3 1 6 2 12 4 8

Matrix representation of D4○C16 in GL2(𝔽17) generated by

 1 15 1 16
,
 1 15 0 16
,
 7 0 0 7
`G:=sub<GL(2,GF(17))| [1,1,15,16],[1,0,15,16],[7,0,0,7] >;`

D4○C16 in GAP, Magma, Sage, TeX

`D_4\circ C_{16}`
`% in TeX`

`G:=Group("D4oC16");`
`// GroupNames label`

`G:=SmallGroup(64,185);`
`// by ID`

`G=gap.SmallGroup(64,185);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,-2,-2,48,332,69,88]);`
`// Polycyclic`

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

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