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## G = C16.D4order 128 = 27

### 1st non-split extension by C16 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C16.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×M5(2) — C16.D4
 Lower central C1 — C2 — C4 — C2×C8 — C16.D4
 Upper central C1 — C22 — C22×C4 — C22×C8 — C16.D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C16.D4

Generators and relations for C16.D4
G = < a,b,c | a16=b4=1, c2=a8, bab-1=a7, cac-1=a-1, cbc-1=a8b-1 >

Subgroups: 168 in 75 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C8 [×2], C8, C2×C4 [×2], C2×C4 [×6], Q8 [×4], C23, C16 [×2], C16, C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], Q16 [×4], C22×C4, C2×Q8 [×2], Q8⋊C4 [×2], C2.D8 [×2], C2×C16 [×2], M5(2) [×2], Q32 [×2], C22⋊Q8 [×2], C22×C8, C2×Q16 [×2], C2.Q32 [×2], C164C4, C8.18D4 [×2], C2×M5(2), C2×Q32, C16.D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C4○D8, C87D4, Q32⋊C2 [×2], C16.D4

Character table of C16.D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 4 2 2 4 16 16 16 16 2 2 2 2 4 4 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 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 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 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 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 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 -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 -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 1 1 1 1 linear of order 2 ρ9 2 2 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 2 2 -2 0 0 0 0 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 -2 2 0 0 0 0 0 -2 2 2 -2 0 0 2 0 -2 0 -2 0 0 2 orthogonal lifted from D4 ρ12 2 -2 2 -2 0 -2 2 0 0 0 0 0 -2 2 2 -2 0 0 -2 0 2 0 2 0 0 -2 orthogonal lifted from D4 ρ13 2 2 2 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 -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 ρ15 2 2 2 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 ρ16 2 2 2 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 2 -2 0 0 0 0 0 0 0 0 0 -2i 2i √2 -√-2 -√2 √-2 √2 -√-2 √-2 -√2 complex lifted from C4○D8 ρ18 2 -2 2 -2 0 -2 2 0 0 0 0 0 2 -2 -2 2 0 0 0 2i 0 -2i 0 -2i 2i 0 complex lifted from C4○D4 ρ19 2 -2 2 -2 0 -2 2 0 0 0 0 0 2 -2 -2 2 0 0 0 -2i 0 2i 0 2i -2i 0 complex lifted from C4○D4 ρ20 2 -2 2 -2 0 2 -2 0 0 0 0 0 0 0 0 0 2i -2i √2 √-2 -√2 -√-2 √2 √-2 -√-2 -√2 complex lifted from C4○D8 ρ21 2 -2 2 -2 0 2 -2 0 0 0 0 0 0 0 0 0 2i -2i -√2 -√-2 √2 √-2 -√2 -√-2 √-2 √2 complex lifted from C4○D8 ρ22 2 -2 2 -2 0 2 -2 0 0 0 0 0 0 0 0 0 -2i 2i -√2 √-2 √2 -√-2 -√2 √-2 -√-2 √2 complex lifted from C4○D8 ρ23 4 -4 -4 4 0 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 symplectic lifted from Q32⋊C2, Schur index 2 ρ24 4 4 -4 -4 0 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 symplectic lifted from Q32⋊C2, Schur index 2 ρ25 4 -4 -4 4 0 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 symplectic lifted from Q32⋊C2, Schur index 2 ρ26 4 4 -4 -4 0 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 symplectic lifted from Q32⋊C2, Schur index 2

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

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

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

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

Matrix representation of C16.D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 12 15 0 0 0 0 3 0 15 0 0 2 5 1 5 0 0 7 6 0 14
,
 7 2 0 0 0 0 9 10 0 0 0 0 0 0 5 6 10 7 0 0 12 7 7 7 0 0 16 16 3 13 0 0 12 13 2 2
,
 7 2 0 0 0 0 10 10 0 0 0 0 0 0 5 6 10 7 0 0 12 7 7 7 0 0 4 11 3 13 0 0 7 8 2 2

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,2,7,0,0,12,3,5,6,0,0,15,0,1,0,0,0,0,15,5,14],[7,9,0,0,0,0,2,10,0,0,0,0,0,0,5,12,16,12,0,0,6,7,16,13,0,0,10,7,3,2,0,0,7,7,13,2],[7,10,0,0,0,0,2,10,0,0,0,0,0,0,5,12,4,7,0,0,6,7,11,8,0,0,10,7,3,2,0,0,7,7,13,2] >;`

C16.D4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(128,951);`
`// by ID`

`G=gap.SmallGroup(128,951);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,288,422,723,1123,360,4037,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^16=b^4=1,c^2=a^8,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=a^8*b^-1>;`
`// generators/relations`

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