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## G = C42.131D4order 128 = 27

### 113rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C42.131D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — M4(2).8C22 — C42.131D4
 Lower central C1 — C2 — C22×C4 — C42.131D4
 Upper central C1 — C4 — C22×C4 — C42.131D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.131D4

Generators and relations for C42.131D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=a-1b2, dad=a-1b, cbc-1=b-1, bd=db, dcd=b2c3 >

Subgroups: 288 in 133 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×5], C8 [×5], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×2], C42, C2×C8 [×2], C2×C8 [×3], M4(2) [×8], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4.D4 [×2], C4.10D4 [×2], C4≀C2 [×4], C4⋊C8 [×2], C2×C42, C2×M4(2) [×2], C2×M4(2) [×2], C4○D8 [×2], C8⋊C22 [×2], C8.C22 [×2], C2×C4○D4 [×2], C4.10C42, M4(2).8C22 [×2], C2×C4≀C2 [×2], C4⋊M4(2), D8⋊C22, C42.131D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4, C42.131D4

Character table of C42.131D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 8 8 1 1 2 2 2 4 4 4 4 8 8 8 8 8 8 8 8 8 8 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 0 -2 -2 2 2 -2 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 2 -2 0 2 -2 -2 -2 2 2 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 2 0 0 -2 -2 2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 2 0 0 2 2 -2 2 -2 0 0 0 0 0 0 0 0 -2 0 0 0 2 0 orthogonal lifted from D4 ρ13 2 2 -2 -2 2 0 0 -2 -2 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 -2 -2 -2 0 -2 -2 2 2 -2 0 0 0 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 2 0 0 2 2 -2 2 -2 0 0 0 0 0 0 0 0 2 0 0 0 -2 0 orthogonal lifted from D4 ρ16 2 2 -2 2 -2 0 -2 -2 -2 -2 2 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 -2 2 -2 0 0 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 -2i 0 0 0 2i complex lifted from C4○D4 ρ18 2 2 2 -2 -2 0 0 2 2 -2 -2 2 0 0 0 0 0 0 0 -2i 0 0 2i 0 0 0 complex lifted from C4○D4 ρ19 2 2 2 -2 -2 0 0 2 2 -2 -2 2 0 0 0 0 0 0 0 2i 0 0 -2i 0 0 0 complex lifted from C4○D4 ρ20 2 2 2 2 2 0 0 -2 -2 -2 -2 -2 0 0 0 0 0 0 -2i 0 0 0 0 2i 0 0 complex lifted from C4○D4 ρ21 2 2 2 2 2 0 0 -2 -2 -2 -2 -2 0 0 0 0 0 0 2i 0 0 0 0 -2i 0 0 complex lifted from C4○D4 ρ22 2 2 -2 2 -2 0 0 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 2i 0 0 0 -2i complex lifted from C4○D4 ρ23 4 -4 0 0 0 0 0 -4i 4i 0 0 0 -2 -2i 2 2i 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 0 0 0 -4i 4i 0 0 0 2 2i -2 -2i 0 0 0 0 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 0 0 0 4i -4i 0 0 0 -2 2i 2 -2i 0 0 0 0 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 0 0 0 4i -4i 0 0 0 2 -2i -2 2i 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C42.131D4
On 16 points - transitive group 16T360
Generators in S16
```(1 5)(3 7)(9 15 13 11)(10 16 14 12)
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 10)(2 9)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)```

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

`G:=Group( (1,5)(3,7)(9,15,13,11)(10,16,14,12), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,10)(2,9)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11) );`

`G=PermutationGroup([(1,5),(3,7),(9,15,13,11),(10,16,14,12)], [(1,7,5,3),(2,4,6,8),(9,15,13,11),(10,12,14,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,10),(2,9),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11)])`

`G:=TransitiveGroup(16,360);`

Matrix representation of C42.131D4 in GL4(𝔽5) generated by

 4 0 4 0 0 1 0 1 3 0 0 0 0 4 0 1
,
 4 0 3 0 0 0 0 4 1 0 1 0 0 1 0 0
,
 0 4 0 4 0 0 4 0 0 0 0 1 4 0 4 0
,
 0 4 0 1 4 0 4 0 0 0 0 4 0 0 4 0
`G:=sub<GL(4,GF(5))| [4,0,3,0,0,1,0,4,4,0,0,0,0,1,0,1],[4,0,1,0,0,0,0,1,3,0,1,0,0,4,0,0],[0,0,0,4,4,0,0,0,0,4,0,4,4,0,1,0],[0,4,0,0,4,0,0,0,0,4,0,4,1,0,4,0] >;`

C42.131D4 in GAP, Magma, Sage, TeX

`C_4^2._{131}D_4`
`% in TeX`

`G:=Group("C4^2.131D4");`
`// GroupNames label`

`G:=SmallGroup(128,782);`
`// by ID`

`G=gap.SmallGroup(128,782);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,1411,718,4037,1027]);`
`// 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^2,d*a*d=a^-1*b,c*b*c^-1=b^-1,b*d=d*b,d*c*d=b^2*c^3>;`
`// generators/relations`

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