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## G = C42.14D4order 128 = 27

### 14th 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×D4 — C42.14D4
 Chief series C1 — C2 — C22 — C23 — C2×D4 — C22.D4 — C22.57C24 — C42.14D4
 Lower central C1 — C2 — C22 — C2×D4 — C42.14D4
 Upper central C1 — C2 — C22 — C2×D4 — C42.14D4
 Jennings C1 — C2 — C22 — C2×D4 — C42.14D4

Generators and relations for C42.14D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b, dad=a-1b-1, cbc-1=a2b-1, bd=db, dcd=b2c-1 >

Subgroups: 296 in 113 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2 [×4], C4 [×10], C22, C22 [×7], C8, C2×C4, C2×C4 [×13], D4 [×5], Q8 [×3], C23 [×2], C23 [×2], C42, C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×10], M4(2), SD16, Q16, C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×2], C23⋊C4 [×2], C23⋊C4 [×2], C4.D4, C4≀C2, C22⋊Q8 [×2], C22.D4 [×2], C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], C4⋊Q8, C8.C22, 2+ 1+4, C23.D4 [×2], C423C4, D4.9D4, C23.7D4 [×2], C22.57C24, C42.14D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C2≀C22, C42.14D4

Character table of C42.14D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8 size 1 1 2 4 4 8 4 8 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 -2 -2 0 2 0 0 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 0 -2 0 2 0 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 -2 -2 0 2 0 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 -2 2 0 -2 2 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 0 -2 0 -2 0 0 0 0 2 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 -2 2 0 -2 -2 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 4 4 -4 0 0 2 0 0 0 0 0 0 -2 0 0 0 0 orthogonal lifted from C2≀C22 ρ16 4 4 -4 0 0 -2 0 0 0 0 0 0 2 0 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 C42.14D4
On 32 points
Generators in S32
```(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 9 31 26)(2 10 32 27)(3 11 29 28)(4 12 30 25)(5 16 17 24)(6 13 18 21)(7 14 19 22)(8 15 20 23)
(1 5 9 14)(2 13 25 18)(3 19 11 24)(4 23 27 8)(6 32 21 12)(7 28 16 29)(10 20 30 15)(17 26 22 31)
(2 25)(3 29)(4 10)(5 22)(6 18)(7 16)(11 28)(12 32)(13 21)(14 17)(19 24)(27 30)```

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

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

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

Matrix representation of C42.14D4 in GL8(𝔽17)

 11 6 11 11 6 11 6 6 6 11 11 11 11 6 6 6 6 6 11 6 11 11 11 6 6 6 6 11 11 11 6 11 6 11 6 6 6 11 6 6 11 6 6 6 11 6 6 6 11 11 11 6 11 11 6 11 11 11 6 11 11 11 11 6
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0
,
 11 6 11 11 11 6 11 11 11 6 6 6 11 6 6 6 11 11 11 6 11 11 6 11 6 6 11 6 6 6 6 11 11 6 11 11 6 11 6 6 11 6 6 6 6 11 11 11 6 6 11 6 11 11 11 6 11 11 11 6 6 6 11 6
,
 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1

`G:=sub<GL(8,GF(17))| [11,6,6,6,6,11,11,11,6,11,6,6,11,6,11,11,11,11,11,6,6,6,11,6,11,11,6,11,6,6,6,11,6,11,11,11,6,11,11,11,11,6,11,11,11,6,11,11,6,6,11,6,6,6,6,11,6,6,6,11,6,6,11,6],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[11,11,11,6,11,11,6,11,6,6,11,6,6,6,6,11,11,6,11,11,11,6,11,11,11,6,6,6,11,6,6,6,11,11,11,6,6,6,11,6,6,6,11,6,11,11,11,6,11,6,6,6,6,11,11,11,11,6,11,11,6,11,6,6],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1] >;`

C42.14D4 in GAP, Magma, Sage, TeX

`C_4^2._{14}D_4`
`% in TeX`

`G:=Group("C4^2.14D4");`
`// GroupNames label`

`G:=SmallGroup(128,933);`
`// by ID`

`G=gap.SmallGroup(128,933);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,456,422,723,352,297,1971,375,4037]);`
`// Polycyclic`

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

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