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## G = C42.5D4order 128 = 27

### 5th 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 — C2×C4 — C42.5D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C22.11C24 — C42.5D4
 Lower central C1 — C2 — C2×C4 — C42.5D4
 Upper central C1 — C2 — C22×C4 — C42.5D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.5D4

Generators and relations for C42.5D4
G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=dad-1=a-1b-1, cbc-1=b-1, bd=db, dcd-1=bc-1 >

Subgroups: 412 in 173 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2 [×8], C4 [×4], C4 [×6], C22 [×3], C22 [×15], C8 [×3], C2×C4 [×6], C2×C4 [×12], D4 [×4], D4 [×10], Q8 [×2], C23, C23 [×9], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×3], C2×C8 [×2], C2×C8, M4(2) [×4], D8 [×4], SD16 [×4], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, C23⋊C4 [×2], D4⋊C4 [×2], C4≀C2 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×3], C4×D4 [×4], C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×4], C22×D4, C2×C4○D4, C4.9C42, M4(2)⋊4C4, C23.C23, C23.37D4, C42⋊C22, C22.11C24, C2×C8⋊C22, C42.5D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C42.5D4

Character table of C42.5D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 8A 8B 8C 8D size 1 1 2 2 2 4 4 4 4 8 2 2 2 2 4 4 4 4 4 4 4 4 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 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 -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 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 -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 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 -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 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 -1 1 1 linear of order 2 ρ9 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 -i i -i i -i i -i i -i i -1 i -i 1 -1 linear of order 4 ρ10 1 1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 i -i i -i -i i -i i -i i -1 -i i -1 1 linear of order 4 ρ11 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 i -i i -i i -i i -i i -i -1 -i i 1 -1 linear of order 4 ρ12 1 1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 -i i -i i i -i i -i i -i -1 i -i -1 1 linear of order 4 ρ13 1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 -i i -i i -i i -i i i -i 1 -i i -1 1 linear of order 4 ρ14 1 1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 i -i i -i -i i -i i i -i 1 i -i 1 -1 linear of order 4 ρ15 1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 i -i i -i i -i i -i -i i 1 i -i -1 1 linear of order 4 ρ16 1 1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 -i i -i i i -i i -i -i i 1 -i i 1 -1 linear of order 4 ρ17 2 2 2 2 2 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 2 -2 2 2 -2 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 2 -2 -2 -2 2 2 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 -2 -2 2 0 0 0 0 0 2 -2 -2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 2 -2 2 2 -2 -2 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 -2 -2 2 2 -2 -2 2 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 -2 2 0 0 0 0 0 2 -2 -2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 -2 2 -2 0 0 0 0 0 2 2 -2 -2 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 2 -2 -2 0 0 0 0 0 -2 2 -2 2 0 0 0 0 2i -2i -2i 2i 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 2 -2 2 -2 0 0 0 0 0 2 2 -2 -2 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 2 -2 -2 0 0 0 0 0 -2 2 -2 2 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C42.5D4
On 16 points - transitive group 16T221
Generators in S16
```(1 9 5 13)(2 12)(3 11 7 15)(4 14)(6 16)(8 10)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)
(1 10 13 6)(2 5 14 9)(3 16 15 4)(7 12 11 8)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,221);`

On 16 points - transitive group 16T307
Generators in S16
```(1 11)(2 10 6 14)(3 13)(4 12 8 16)(5 15)(7 9)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)
(1 4 5 8)(2 7 6 3)(9 12)(10 11)(13 16)(14 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,307);`

Matrix representation of C42.5D4 in GL8(ℤ)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0
,
 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0
,
 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0

`G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0],[0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;`

C42.5D4 in GAP, Magma, Sage, TeX

`C_4^2._5D_4`
`% in TeX`

`G:=Group("C4^2.5D4");`
`// GroupNames label`

`G:=SmallGroup(128,636);`
`// by ID`

`G=gap.SmallGroup(128,636);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,521,248,1411,718,172,1027,124]);`
`// Polycyclic`

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

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