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## G = C4○C2≀C4order 128 = 27

### Central product of C4 and C2≀C4

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C4○C2≀C4
 Chief series C1 — C2 — C22 — C23 — C2×D4 — C2×C4○D4 — C22.19C24 — C4○C2≀C4
 Lower central C1 — C2 — C22 — C23 — C4○C2≀C4
 Upper central C1 — C4 — C2×C4 — C2×C4○D4 — C4○C2≀C4
 Jennings C1 — C2 — C22 — C2×D4 — C4○C2≀C4

Generators and relations for C4○C2≀C4
G = < a,b,c,d,e,f | a4=b2=c2=d2=f4=1, e1=a2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=a2bcd, cd=dc, ce=ec, fcf-1=a2cd, de=ed, fdf-1=a2d, ef=fe >

Subgroups: 324 in 132 conjugacy classes, 42 normal (34 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×8], C22, C22 [×14], C8 [×2], C2×C4 [×4], C2×C4 [×16], D4 [×7], Q8, C23 [×3], C23 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C2×C8, M4(2) [×3], C22×C4 [×4], C22×C4 [×4], C2×D4 [×3], C2×D4 [×2], C2×Q8, C4○D4 [×2], C24, C23⋊C4 [×2], C23⋊C4, C4.D4 [×2], C4.10D4, C42⋊C2, C42⋊C2, C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C22⋊Q8, C22.D4 [×2], C2×M4(2), C23×C4, C2×C4○D4, C2≀C4 [×2], C23.D4 [×2], C23.C23, M4(2).8C22, C22.19C24, C4○C2≀C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C2×C23⋊C4, C4○C2≀C4

Character table of C4○C2≀C4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 8A 8B 8C 8D size 1 1 2 4 4 4 4 4 1 1 2 4 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 1 1 1 -1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -i -i i i -1 -1 i -i i -i linear of order 4 ρ10 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 1 i -i i -i -1 1 -i i i -i linear of order 4 ρ11 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 -i i -i i 1 -1 -i i i -i linear of order 4 ρ12 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 -1 -1 i i -i -i 1 1 i -i i -i linear of order 4 ρ13 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 i -i i -i 1 -1 i -i -i i linear of order 4 ρ14 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 -1 -1 -i -i i i 1 1 -i i -i i linear of order 4 ρ15 1 1 1 -1 1 -1 1 1 1 1 1 1 1 1 -1 -1 i i -i -i -1 -1 -i i -i i linear of order 4 ρ16 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 1 -i i -i i -1 1 i -i -i i linear of order 4 ρ17 2 2 2 -2 0 2 -2 0 2 2 2 0 -2 0 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 0 -2 -2 0 -2 -2 -2 0 2 0 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 0 -2 -2 0 2 2 2 0 -2 0 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 0 2 -2 0 -2 -2 -2 0 2 0 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 -4 0 0 0 0 0 4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 -4 0 0 0 0 0 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 4 -4 0 0 -2 0 0 2 4i -4i 0 2i 0 -2i 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 2 0 0 -2 4i -4i 0 -2i 0 2i 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 -2 0 0 2 -4i 4i 0 -2i 0 2i 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 2 0 0 -2 -4i 4i 0 2i 0 -2i 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C4○C2≀C4
On 16 points - transitive group 16T243
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 11)(10 12)
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)
(5 7)(6 8)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 14 12 8)(2 15 9 5)(3 16 10 6)(4 13 11 7)```

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

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

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

`G:=TransitiveGroup(16,243);`

On 16 points - transitive group 16T280
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(5 14)(6 15)(7 16)(8 13)
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)
(5 7)(6 8)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 14)(2 15)(3 16)(4 13)(5 11 7 9)(6 12 8 10)```

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

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

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

`G:=TransitiveGroup(16,280);`

On 16 points - transitive group 16T292
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(5 7)(6 8)
(5 7)(6 8)(9 11)(10 12)
(5 7)(6 8)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 5 11 16)(2 6 12 13)(3 7 9 14)(4 8 10 15)```

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

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

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

`G:=TransitiveGroup(16,292);`

Matrix representation of C4○C2≀C4 in GL4(𝔽5) generated by

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

C4○C2≀C4 in GAP, Magma, Sage, TeX

`C_4\circ C_2\wr C_4`
`% in TeX`

`G:=Group("C4oC2wrC4");`
`// GroupNames label`

`G:=SmallGroup(128,852);`
`// by ID`

`G=gap.SmallGroup(128,852);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,352,1123,851,375,4037]);`
`// Polycyclic`

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

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