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## G = C42.3D4order 128 = 27

### 3rd non-split extension by C42 of D4 acting faithfully

p-group, non-abelian, nilpotent (class 5), monomial

Aliases: C42.3D4, 2- 1+4⋊C4, C2.10C2≀C4, C4.10D4⋊C4, (C2×D4).2D4, C423C41C2, D4.8D4.1C2, C22.3(C23⋊C4), C4.4D4.2C22, C42.C226C2, (C2×Q8).1(C2×C4), (C2×C4).7(C22⋊C4), SmallGroup(128,136)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×Q8 — C42.3D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×D4 — C4.4D4 — D4.8D4 — C42.3D4
 Lower central C1 — C2 — C22 — C2×C4 — C2×Q8 — C42.3D4
 Upper central C1 — C2 — C22 — C2×C4 — C4.4D4 — C42.3D4
 Jennings C1 — C2 — C2 — C22 — C2×C4 — C4.4D4 — C42.3D4

Generators and relations for C42.3D4
G = < a,b,c,d | a4=b4=c4=1, d2=a-1, ab=ba, cac-1=a-1b-1, ad=da, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a-1c-1 >

Character table of C42.3D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E size 1 1 2 8 8 4 4 4 8 8 16 16 8 8 8 8 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 i -i -i i i -i 1 linear of order 4 ρ6 1 1 1 1 -1 -1 -1 1 1 1 -i i -i i i -i -1 linear of order 4 ρ7 1 1 1 1 -1 -1 -1 1 1 1 i -i i -i -i i -1 linear of order 4 ρ8 1 1 1 -1 -1 -1 -1 1 -1 1 -i i i -i -i i 1 linear of order 4 ρ9 2 2 2 0 -2 2 2 2 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 2 -2 -2 2 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 -2 0 0 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C4 ρ12 4 4 4 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ13 4 4 -4 2 0 0 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C4 ρ14 4 -4 0 0 0 -2i 2i 0 0 0 0 0 1-i -1-i 1+i -1+i 0 complex faithful ρ15 4 -4 0 0 0 -2i 2i 0 0 0 0 0 -1+i 1+i -1-i 1-i 0 complex faithful ρ16 4 -4 0 0 0 2i -2i 0 0 0 0 0 -1-i 1-i -1+i 1+i 0 complex faithful ρ17 4 -4 0 0 0 2i -2i 0 0 0 0 0 1+i -1+i 1-i -1-i 0 complex faithful

Permutation representations of C42.3D4
On 16 points - transitive group 16T397
Generators in S16
```(5 7)(6 8)(9 15 13 11)(10 16 14 12)
(1 4 2 3)(5 8 7 6)(9 11 13 15)(10 16 14 12)
(1 12 8 15)(2 16 6 11)(3 10 7 13)(4 14 5 9)
(1 2)(3 4)(5 6 7 8)(9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,397);`

Matrix representation of C42.3D4 in GL4(𝔽5) generated by

 1 3 1 3 3 3 2 1 1 4 0 2 4 1 2 0
,
 1 0 4 0 1 3 3 0 2 0 4 0 3 0 3 2
,
 0 0 3 2 1 0 1 1 0 1 2 2 0 0 0 3
,
 0 4 1 0 0 3 1 4 0 4 3 3 1 3 2 2
`G:=sub<GL(4,GF(5))| [1,3,1,4,3,3,4,1,1,2,0,2,3,1,2,0],[1,1,2,3,0,3,0,0,4,3,4,3,0,0,0,2],[0,1,0,0,0,0,1,0,3,1,2,0,2,1,2,3],[0,0,0,1,4,3,4,3,1,1,3,2,0,4,3,2] >;`

C42.3D4 in GAP, Magma, Sage, TeX

`C_4^2._3D_4`
`% in TeX`

`G:=Group("C4^2.3D4");`
`// GroupNames label`

`G:=SmallGroup(128,136);`
`// by ID`

`G=gap.SmallGroup(128,136);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,723,352,346,745,1684,1411,718,375,172,4037,2028]);`
`// Polycyclic`

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

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