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## G = C7⋊A4order 84 = 22·3·7

### The semidirect product of C7 and A4 acting via A4/C22=C3

Aliases: C7⋊A4, C22⋊(C7⋊C3), (C2×C14)⋊2C3, SmallGroup(84,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C7⋊A4
 Chief series C1 — C7 — C2×C14 — C7⋊A4
 Lower central C2×C14 — C7⋊A4
 Upper central C1

Generators and relations for C7⋊A4
G = < a,b,c,d | a7=b2=c2=d3=1, ab=ba, ac=ca, dad-1=a4, dbd-1=bc=cb, dcd-1=b >

Character table of C7⋊A4

 class 1 2 3A 3B 7A 7B 14A 14B 14C 14D 14E 14F size 1 3 28 28 3 3 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ3 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ4 3 -1 0 0 3 3 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ5 3 -1 0 0 -1+√-7/2 -1-√-7/2 -ζ76+ζ75-ζ73 ζ74-ζ72-ζ7 -ζ74-ζ72+ζ7 ζ76-ζ75-ζ73 -ζ74+ζ72-ζ7 -ζ76-ζ75+ζ73 complex faithful ρ6 3 -1 0 0 -1-√-7/2 -1+√-7/2 ζ74-ζ72-ζ7 ζ76-ζ75-ζ73 -ζ76+ζ75-ζ73 -ζ74+ζ72-ζ7 -ζ76-ζ75+ζ73 -ζ74-ζ72+ζ7 complex faithful ρ7 3 -1 0 0 -1-√-7/2 -1+√-7/2 -ζ74-ζ72+ζ7 -ζ76+ζ75-ζ73 -ζ76-ζ75+ζ73 ζ74-ζ72-ζ7 ζ76-ζ75-ζ73 -ζ74+ζ72-ζ7 complex faithful ρ8 3 -1 0 0 -1-√-7/2 -1+√-7/2 -ζ74+ζ72-ζ7 -ζ76-ζ75+ζ73 ζ76-ζ75-ζ73 -ζ74-ζ72+ζ7 -ζ76+ζ75-ζ73 ζ74-ζ72-ζ7 complex faithful ρ9 3 3 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ10 3 -1 0 0 -1+√-7/2 -1-√-7/2 -ζ76-ζ75+ζ73 -ζ74-ζ72+ζ7 -ζ74+ζ72-ζ7 -ζ76+ζ75-ζ73 ζ74-ζ72-ζ7 ζ76-ζ75-ζ73 complex faithful ρ11 3 3 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ12 3 -1 0 0 -1+√-7/2 -1-√-7/2 ζ76-ζ75-ζ73 -ζ74+ζ72-ζ7 ζ74-ζ72-ζ7 -ζ76-ζ75+ζ73 -ζ74-ζ72+ζ7 -ζ76+ζ75-ζ73 complex faithful

Permutation representations of C7⋊A4
On 28 points - transitive group 28T16
Generators in S28
```(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)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)
(2 3 5)(4 7 6)(8 24 21)(9 26 18)(10 28 15)(11 23 19)(12 25 16)(13 27 20)(14 22 17)```

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

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

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

`G:=TransitiveGroup(28,16);`

C7⋊A4 is a maximal subgroup of   D7⋊A4  A4×C7⋊C3  C42⋊(C7⋊C3)  C7⋊(C22⋊A4)
C7⋊A4 is a maximal quotient of   C14.A4  C21.A4  C42⋊(C7⋊C3)  C7⋊(C22⋊A4)

Matrix representation of C7⋊A4 in GL3(𝔽43) generated by

 18 1 0 19 0 1 1 0 0
,
 15 35 4 24 30 6 35 4 40
,
 38 32 2 8 21 39 32 2 26
,
 24 18 0 42 19 1 42 1 0
`G:=sub<GL(3,GF(43))| [18,19,1,1,0,0,0,1,0],[15,24,35,35,30,4,4,6,40],[38,8,32,32,21,2,2,39,26],[24,42,42,18,19,1,0,1,0] >;`

C7⋊A4 in GAP, Magma, Sage, TeX

`C_7\rtimes A_4`
`% in TeX`

`G:=Group("C7:A4");`
`// GroupNames label`

`G:=SmallGroup(84,11);`
`// by ID`

`G=gap.SmallGroup(84,11);`
`# by ID`

`G:=PCGroup([4,-3,-2,2,-7,49,110,387]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^7=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^4,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;`
`// generators/relations`

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