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## G = C43⋊C7order 448 = 26·7

### The semidirect product of C43 and C7 acting faithfully

Aliases: C43⋊C7, C23.1F8, SmallGroup(448,178)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C43 — C43⋊C7
 Chief series C1 — C23 — C43 — C43⋊C7
 Lower central C43 — C43⋊C7
 Upper central C1

Generators and relations for C43⋊C7
G = < a,b,c,d | a4=b4=c4=d7=1, ab=ba, ac=ca, dad-1=bc-1, bc=cb, dbd-1=a-1, dcd-1=b-1c2 >

Character table of C43⋊C7

 class 1 2 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 7D 7E 7F size 1 7 7 7 7 7 7 7 7 7 64 64 64 64 64 64 ρ1 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 ζ73 ζ76 ζ72 ζ75 ζ7 ζ74 linear of order 7 ρ3 1 1 1 1 1 1 1 1 1 1 ζ75 ζ73 ζ7 ζ76 ζ74 ζ72 linear of order 7 ρ4 1 1 1 1 1 1 1 1 1 1 ζ72 ζ74 ζ76 ζ7 ζ73 ζ75 linear of order 7 ρ5 1 1 1 1 1 1 1 1 1 1 ζ74 ζ7 ζ75 ζ72 ζ76 ζ73 linear of order 7 ρ6 1 1 1 1 1 1 1 1 1 1 ζ7 ζ72 ζ73 ζ74 ζ75 ζ76 linear of order 7 ρ7 1 1 1 1 1 1 1 1 1 1 ζ76 ζ75 ζ74 ζ73 ζ72 ζ7 linear of order 7 ρ8 7 7 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from F8 ρ9 7 -1 1-2i 1+2i 1+2i -3-2i -3+2i 1-2i 1+2i 1-2i 0 0 0 0 0 0 complex faithful ρ10 7 -1 1+2i 1-2i -3+2i 1-2i 1+2i -3-2i 1-2i 1+2i 0 0 0 0 0 0 complex faithful ρ11 7 -1 1-2i 1+2i -3-2i 1+2i 1-2i -3+2i 1+2i 1-2i 0 0 0 0 0 0 complex faithful ρ12 7 -1 1+2i 1-2i 1-2i -3+2i -3-2i 1+2i 1-2i 1+2i 0 0 0 0 0 0 complex faithful ρ13 7 -1 -3-2i -3+2i 1-2i 1-2i 1+2i 1+2i 1-2i 1+2i 0 0 0 0 0 0 complex faithful ρ14 7 -1 1+2i 1-2i 1-2i 1-2i 1+2i 1+2i -3+2i -3-2i 0 0 0 0 0 0 complex faithful ρ15 7 -1 -3+2i -3-2i 1+2i 1+2i 1-2i 1-2i 1+2i 1-2i 0 0 0 0 0 0 complex faithful ρ16 7 -1 1-2i 1+2i 1+2i 1+2i 1-2i 1-2i -3-2i -3+2i 0 0 0 0 0 0 complex faithful

Permutation representations of C43⋊C7
On 28 points - transitive group 28T61
Generators in S28
```(1 20)(2 25 21 9)(3 15)(4 27 16 11)(5 28 17 12)(6 13 18 22)(7 19)(8 24)(10 26)(14 23)
(1 20)(2 21)(3 10 15 26)(4 16)(5 12 17 28)(6 13 18 22)(7 23 19 14)(8 24)(9 25)(11 27)
(1 24 20 8)(3 15)(4 27 16 11)(6 22 18 13)(7 14 19 23)(10 26)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)```

`G:=sub<Sym(28)| (1,20)(2,25,21,9)(3,15)(4,27,16,11)(5,28,17,12)(6,13,18,22)(7,19)(8,24)(10,26)(14,23), (1,20)(2,21)(3,10,15,26)(4,16)(5,12,17,28)(6,13,18,22)(7,23,19,14)(8,24)(9,25)(11,27), (1,24,20,8)(3,15)(4,27,16,11)(6,22,18,13)(7,14,19,23)(10,26), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)>;`

`G:=Group( (1,20)(2,25,21,9)(3,15)(4,27,16,11)(5,28,17,12)(6,13,18,22)(7,19)(8,24)(10,26)(14,23), (1,20)(2,21)(3,10,15,26)(4,16)(5,12,17,28)(6,13,18,22)(7,23,19,14)(8,24)(9,25)(11,27), (1,24,20,8)(3,15)(4,27,16,11)(6,22,18,13)(7,14,19,23)(10,26), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28) );`

`G=PermutationGroup([[(1,20),(2,25,21,9),(3,15),(4,27,16,11),(5,28,17,12),(6,13,18,22),(7,19),(8,24),(10,26),(14,23)], [(1,20),(2,21),(3,10,15,26),(4,16),(5,12,17,28),(6,13,18,22),(7,23,19,14),(8,24),(9,25),(11,27)], [(1,24,20,8),(3,15),(4,27,16,11),(6,22,18,13),(7,14,19,23),(10,26)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28)]])`

`G:=TransitiveGroup(28,61);`

Matrix representation of C43⋊C7 in GL7(𝔽29)

 28 0 0 0 0 0 0 0 28 0 0 0 0 0 23 0 17 0 0 0 0 18 0 0 12 0 0 0 10 0 0 0 12 0 0 0 0 0 0 0 28 0 26 0 0 0 0 0 12
,
 28 0 0 0 0 0 0 27 12 0 0 0 0 0 23 0 17 0 0 0 0 16 0 0 17 0 0 0 0 0 0 0 28 0 0 1 0 0 0 0 17 0 0 0 0 0 0 0 28
,
 12 0 0 0 0 0 0 26 17 0 0 0 0 0 0 0 12 0 0 0 0 16 0 0 1 0 0 0 0 0 0 0 12 0 0 17 0 0 0 0 28 0 7 0 0 0 0 0 1
,
 7 27 0 0 0 0 0 0 22 1 0 0 0 0 0 9 0 1 0 0 0 0 5 0 0 1 0 0 0 6 0 0 0 1 0 0 13 0 0 0 0 1 0 4 0 0 0 0 0

`G:=sub<GL(7,GF(29))| [28,0,23,18,10,0,26,0,28,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,12],[28,27,23,16,0,1,0,0,12,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,28],[12,26,0,16,0,17,7,0,17,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,27,22,9,5,6,13,4,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;`

C43⋊C7 in GAP, Magma, Sage, TeX

`C_4^3\rtimes C_7`
`% in TeX`

`G:=Group("C4^3:C7");`
`// GroupNames label`

`G:=SmallGroup(448,178);`
`// by ID`

`G=gap.SmallGroup(448,178);`
`# by ID`

`G:=PCGroup([7,-7,-2,2,2,-2,2,2,197,792,590,352,7255,360,9804,16469,16470]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^4=d^7=1,a*b=b*a,a*c=c*a,d*a*d^-1=b*c^-1,b*c=c*b,d*b*d^-1=a^-1,d*c*d^-1=b^-1*c^2>;`
`// generators/relations`

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