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## G = C42.8D4order 128 = 27

### 8th non-split extension by C42 of D4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C42.8D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C4≀C2 — C42.8D4
 Lower central C1 — C2 — C22×C4 — C42.8D4
 Upper central C1 — C4 — C22×C4 — C42.8D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.8D4

Generators and relations for C42.8D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=a-1b-1, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2b2c3 >

Subgroups: 264 in 122 conjugacy classes, 40 normal (36 characteristic)
C1, C2, C2 [×4], C4 [×4], C4 [×9], C22 [×3], C22 [×3], C8 [×2], C2×C4 [×6], C2×C4 [×12], D4 [×4], Q8 [×6], C23, C23, C42 [×2], C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C4○D4 [×4], C23⋊C4 [×2], C4≀C2 [×4], C2×C42, C42⋊C2 [×3], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2, C4⋊Q8, C2×M4(2) [×2], C2×C4○D4, C4.9C42, C426C4, M4(2)⋊4C4, C23.C23, C2×C4≀C2, C42⋊C22, C23.37C23, C42.8D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], C23⋊Q8, C42.8D4

Character table of C42.8D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 8A 8B 8C 8D size 1 1 2 2 2 8 1 1 2 2 2 4 4 4 4 8 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 0 -2 -2 2 -2 2 2 2 -2 -2 0 0 0 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 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 0 -2 -2 -2 -2 -2 0 0 0 0 0 0 2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 2 0 2 2 -2 2 -2 0 0 0 0 0 -2 0 0 0 2 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 -2 2 0 2 2 -2 2 -2 0 0 0 0 0 2 0 0 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 2 0 -2 -2 -2 -2 -2 0 0 0 0 0 0 -2 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 2 -2 2 -2 -2 2 2 -2 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 2 -2 2 -2 -2 -2 -2 2 2 -2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 2 2 -2 -2 0 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i 0 complex lifted from C4○D4 ρ18 2 2 2 -2 -2 0 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 2i 0 0 -2i complex lifted from C4○D4 ρ19 2 2 2 -2 -2 0 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i 0 complex lifted from C4○D4 ρ20 2 2 -2 2 -2 0 2 2 -2 -2 2 0 0 0 0 0 0 0 2i 0 0 -2i 0 0 0 0 complex lifted from C4○D4 ρ21 2 2 -2 2 -2 0 2 2 -2 -2 2 0 0 0 0 0 0 0 -2i 0 0 2i 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 -2 -2 0 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 -2i 0 0 2i complex lifted from C4○D4 ρ23 4 -4 0 0 0 0 4i -4i 0 0 0 -2i 2i -2 2 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 0 0 -4i 4i 0 0 0 2i -2i -2 2 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 0 0 -4i 4i 0 0 0 -2i 2i 2 -2 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 0 0 4i -4i 0 0 0 2i -2i 2 -2 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C42.8D4
On 16 points - transitive group 16T338
Generators in S16
```(1 3)(2 10 6 14)(4 16 8 12)(5 7)(9 15)(11 13)
(1 11 5 15)(2 16 6 12)(3 13 7 9)(4 10 8 14)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 5)(2 8 6 4)(3 7)(9 13)(10 16 14 12)(11 15)```

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

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

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

`G:=TransitiveGroup(16,338);`

On 16 points - transitive group 16T353
Generators in S16
```(1 15)(2 10 6 14)(3 13)(4 16 8 12)(5 11)(7 9)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 4 6 8)(10 12 14 16)```

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

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

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

`G:=TransitiveGroup(16,353);`

Matrix representation of C42.8D4 in GL4(𝔽5) generated by

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

C42.8D4 in GAP, Magma, Sage, TeX

`C_4^2._8D_4`
`% in TeX`

`G:=Group("C4^2.8D4");`
`// GroupNames label`

`G:=SmallGroup(128,763);`
`// by ID`

`G=gap.SmallGroup(128,763);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,521,718,172,4037,1027,124]);`
`// Polycyclic`

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

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