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## G = C7⋊S4order 168 = 23·3·7

### The semidirect product of C7 and S4 acting via S4/A4=C2

Aliases: C7⋊S4, A4⋊D7, C22⋊D21, (C7×A4)⋊1C2, (C2×C14)⋊2S3, SmallGroup(168,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×A4 — C7⋊S4
 Chief series C1 — C22 — C2×C14 — C7×A4 — C7⋊S4
 Lower central C7×A4 — C7⋊S4
 Upper central C1

Generators and relations for C7⋊S4
G = < a,b,c,d,e | a7=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

3C2
42C2
4C3
21C22
21C4
28S3
3C14
6D7
4C21
21D4
3D14
3Dic7
4D21
7S4

Character table of C7⋊S4

 class 1 2A 2B 3 4 7A 7B 7C 14A 14B 14C 21A 21B 21C 21D 21E 21F size 1 3 42 8 42 2 2 2 6 6 6 8 8 8 8 8 8 ρ1 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 linear of order 2 ρ3 2 2 0 -1 0 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 2 0 2 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ5 2 2 0 2 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ6 2 2 0 2 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ7 2 2 0 -1 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ3ζ74+ζ3ζ73-ζ74 -ζ32ζ74+ζ32ζ73-ζ74 ζ3ζ75-ζ3ζ72-ζ72 -ζ32ζ76+ζ32ζ7-ζ76 ζ32ζ76-ζ32ζ7-ζ7 -ζ3ζ75+ζ3ζ72-ζ75 orthogonal lifted from D21 ρ8 2 2 0 -1 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ3ζ75-ζ3ζ72-ζ72 -ζ3ζ75+ζ3ζ72-ζ75 -ζ32ζ76+ζ32ζ7-ζ76 -ζ3ζ74+ζ3ζ73-ζ74 -ζ32ζ74+ζ32ζ73-ζ74 ζ32ζ76-ζ32ζ7-ζ7 orthogonal lifted from D21 ρ9 2 2 0 -1 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ32ζ76+ζ32ζ7-ζ76 ζ32ζ76-ζ32ζ7-ζ7 -ζ3ζ74+ζ3ζ73-ζ74 ζ3ζ75-ζ3ζ72-ζ72 -ζ3ζ75+ζ3ζ72-ζ75 -ζ32ζ74+ζ32ζ73-ζ74 orthogonal lifted from D21 ρ10 2 2 0 -1 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ74+ζ3ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ76+ζ32ζ7-ζ76 ζ3ζ75-ζ3ζ72-ζ72 orthogonal lifted from D21 ρ11 2 2 0 -1 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ76+ζ32ζ7-ζ76 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ75+ζ3ζ72-ζ75 ζ3ζ75-ζ3ζ72-ζ72 -ζ3ζ74+ζ3ζ73-ζ74 orthogonal lifted from D21 ρ12 2 2 0 -1 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ3ζ75+ζ3ζ72-ζ75 ζ3ζ75-ζ3ζ72-ζ72 ζ32ζ76-ζ32ζ7-ζ7 -ζ32ζ74+ζ32ζ73-ζ74 -ζ3ζ74+ζ3ζ73-ζ74 -ζ32ζ76+ζ32ζ7-ζ76 orthogonal lifted from D21 ρ13 3 -1 1 0 -1 3 3 3 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ14 3 -1 -1 0 1 3 3 3 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ15 6 -2 0 0 0 3ζ74+3ζ73 3ζ75+3ζ72 3ζ76+3ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 0 0 0 0 0 0 orthogonal faithful ρ16 6 -2 0 0 0 3ζ76+3ζ7 3ζ74+3ζ73 3ζ75+3ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 0 0 0 0 0 0 orthogonal faithful ρ17 6 -2 0 0 0 3ζ75+3ζ72 3ζ76+3ζ7 3ζ74+3ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C7⋊S4
On 28 points - transitive group 28T30
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 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 26)(16 27)(17 28)(18 22)(19 23)(20 24)(21 25)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 20)(9 21)(10 15)(11 16)(12 17)(13 18)(14 19)
(8 24 20)(9 25 21)(10 26 15)(11 27 16)(12 28 17)(13 22 18)(14 23 19)
(2 7)(3 6)(4 5)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(14 21)(22 26)(23 25)(27 28)```

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

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

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

`G:=TransitiveGroup(28,30);`

C7⋊S4 is a maximal subgroup of   D7×S4
C7⋊S4 is a maximal quotient of   Q8.D21  Q8⋊D21  A4⋊Dic7

Matrix representation of C7⋊S4 in GL5(𝔽337)

 103 274 0 0 0 63 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 336 0 0 0 1 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 336 336 0 0 336 0 1 0 0 0 0 336
,
 336 1 0 0 0 336 0 0 0 0 0 0 336 336 0 0 0 1 0 0 0 0 336 0 1
,
 336 0 0 0 0 336 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 336

`G:=sub<GL(5,GF(337))| [103,63,0,0,0,274,40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,336,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,0,336,0,0,0,0,336,1,336],[336,336,0,0,0,1,0,0,0,0,0,0,336,1,336,0,0,336,0,0,0,0,0,0,1],[336,336,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,336] >;`

C7⋊S4 in GAP, Magma, Sage, TeX

`C_7\rtimes S_4`
`% in TeX`

`G:=Group("C7:S4");`
`// GroupNames label`

`G:=SmallGroup(168,46);`
`// by ID`

`G=gap.SmallGroup(168,46);`
`# by ID`

`G:=PCGroup([5,-2,-3,-7,-2,2,41,542,1683,848,1054,1584]);`
`// Polycyclic`

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

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