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## G = C42.16D4order 128 = 27

### 16th non-split extension by C42 of D4 acting faithfully

p-group, non-abelian, nilpotent (class 4), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×Q8 — C42.16D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — C4.4D4 — C22.57C24 — C42.16D4
 Lower central C1 — C2 — C22 — C2×Q8 — C42.16D4
 Upper central C1 — C2 — C22 — C2×Q8 — C42.16D4
 Jennings C1 — C2 — C22 — C2×Q8 — C42.16D4

Generators and relations for C42.16D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=a-1b-1, dad=a-1b, cbc-1=a2b, bd=db, dcd=c3 >

Subgroups: 264 in 108 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2 [×3], C4 [×9], C22, C22 [×4], C8 [×3], C2×C4, C2×C4 [×2], C2×C4 [×9], D4 [×5], Q8 [×6], C23, C42, C42 [×2], C22⋊C4 [×5], C4⋊C4 [×8], M4(2) [×3], D8, SD16 [×3], Q16 [×2], C22×C4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8, C4○D4 [×4], C4.10D4, C4.10D4 [×2], C4≀C2 [×3], C22⋊Q8 [×2], C22.D4, C4.4D4, C42.C2, C422C2 [×2], C4⋊Q8 [×2], C8⋊C22, C8.C22 [×2], 2- 1+4, C42.C4, C42.3C4 [×2], D4.8D4, D4.10D4 [×2], C22.57C24, C42.16D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C2≀C22, C42.16D4

Character table of C42.16D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C size 1 1 2 8 8 4 4 4 8 8 8 8 8 8 16 16 16 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 2 0 -2 -2 2 -2 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 0 -2 -2 2 2 0 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 0 0 2 -2 -2 0 2 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 0 0 2 -2 -2 0 -2 0 2 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 0 0 -2 -2 2 -2 0 0 0 2 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 0 2 -2 2 -2 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 4 4 -4 2 0 0 0 0 0 0 0 0 0 -2 0 0 0 orthogonal lifted from C2≀C22 ρ16 4 4 -4 -2 0 0 0 0 0 0 0 0 0 2 0 0 0 orthogonal lifted from C2≀C22 ρ17 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C42.16D4
On 32 points
Generators in S32
```(1 23 5 19)(2 13)(3 21 7 17)(4 15)(6 9)(8 11)(10 27 14 31)(12 25 16 29)(18 28)(20 30)(22 32)(24 26)
(1 29 5 25)(2 26 6 30)(3 27 7 31)(4 32 8 28)(9 20 13 24)(10 21 14 17)(11 18 15 22)(12 19 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)
(1 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)(17 32)(18 27)(19 30)(20 25)(21 28)(22 31)(23 26)(24 29)```

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

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

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

Matrix representation of C42.16D4 in GL8(𝔽17)

 5 5 0 0 0 0 0 0 5 12 0 0 0 0 0 0 5 0 12 12 0 0 0 0 0 5 12 5 0 0 0 0 12 12 0 0 7 0 7 10 12 0 0 0 5 0 0 7 0 5 0 0 10 5 5 12 0 0 0 0 12 5 12 5
,
 1 0 0 15 0 0 0 0 0 16 2 0 0 0 0 0 0 16 1 0 0 0 0 0 1 0 0 16 0 0 0 0 0 0 0 0 1 0 0 15 16 0 16 1 16 0 16 1 0 1 16 1 0 1 0 1 0 0 0 0 1 0 0 16
,
 12 12 0 0 7 7 0 0 12 12 0 0 0 7 0 0 0 0 0 0 5 12 5 12 12 12 0 0 12 12 12 12 5 12 10 7 0 0 0 0 0 5 0 10 5 10 0 0 5 5 12 5 5 10 0 0 0 12 5 12 0 0 0 0
,
 0 0 0 0 1 0 0 0 16 16 0 0 16 15 0 0 0 0 0 0 0 16 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0

`G:=sub<GL(8,GF(17))| [5,5,5,0,12,12,0,0,5,12,0,5,12,0,5,0,0,0,12,12,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,7,5,10,12,0,0,0,0,0,0,5,5,0,0,0,0,7,0,5,12,0,0,0,0,10,7,12,5],[1,0,0,1,0,16,0,0,0,16,16,0,0,0,1,0,0,2,1,0,0,16,16,0,15,0,0,16,0,1,1,0,0,0,0,0,1,16,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,15,1,1,16],[12,12,0,12,5,0,5,0,12,12,0,12,12,5,5,12,0,0,0,0,10,0,12,5,0,0,0,0,7,10,5,12,7,0,5,12,0,5,5,0,7,7,12,12,0,10,10,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0],[0,16,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,16,0,0,0,0,0,0,0,15,16,0,0,1,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;`

C42.16D4 in GAP, Magma, Sage, TeX

`C_4^2._{16}D_4`
`% in TeX`

`G:=Group("C4^2.16D4");`
`// GroupNames label`

`G:=SmallGroup(128,935);`
`// by ID`

`G=gap.SmallGroup(128,935);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,456,422,2019,1018,297,248,2804,1971,718,375,172,4037]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=a^-1*b^-1,d*a*d=a^-1*b,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^3>;`
`// generators/relations`

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