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## G = D4.C16order 128 = 27

### The non-split extension by D4 of C16 acting via C16/C8=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for D4.C16
G = < a,b,c | a4=b2=1, c16=a2, bab=a-1, ac=ca, cbc-1=ab >

Smallest permutation representation of D4.C16
On 64 points
Generators in S64
```(1 54 17 38)(2 55 18 39)(3 56 19 40)(4 57 20 41)(5 58 21 42)(6 59 22 43)(7 60 23 44)(8 61 24 45)(9 62 25 46)(10 63 26 47)(11 64 27 48)(12 33 28 49)(13 34 29 50)(14 35 30 51)(15 36 31 52)(16 37 32 53)
(1 38)(2 18)(3 56)(5 42)(6 22)(7 60)(9 46)(10 26)(11 64)(13 50)(14 30)(15 36)(17 54)(19 40)(21 58)(23 44)(25 62)(27 48)(29 34)(31 52)(33 49)(37 53)(41 57)(45 61)
(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)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G 8H 16A ··· 16H 16I 16J 16K 16L 16M 16N 16O 16P 32A ··· 32P 32Q ··· 32X order 1 2 2 2 4 4 4 4 8 8 8 8 8 8 8 8 16 ··· 16 16 16 16 16 16 16 16 16 32 ··· 32 32 ··· 32 size 1 1 2 4 1 1 2 4 1 1 1 1 2 2 4 4 1 ··· 1 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 C16 C16 D4 M4(2) M5(2) D4.C16 kernel D4.C16 C2×C32 M6(2) D4○C16 M5(2) C8○D4 M4(2) C4○D4 D4 Q8 C16 C8 C22 C1 # reps 1 1 1 1 2 2 4 4 8 8 2 2 4 16

Matrix representation of D4.C16 in GL2(𝔽97) generated by

 79 43 15 18
,
 79 43 94 18
,
 46 32 45 57
`G:=sub<GL(2,GF(97))| [79,15,43,18],[79,94,43,18],[46,45,32,57] >;`

D4.C16 in GAP, Magma, Sage, TeX

`D_4.C_{16}`
`% in TeX`

`G:=Group("D4.C16");`
`// GroupNames label`

`G:=SmallGroup(128,133);`
`// by ID`

`G=gap.SmallGroup(128,133);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,723,352,1018,80,102,124]);`
`// Polycyclic`

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

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