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## G = C6.7S4order 144 = 24·32

### 7th non-split extension by C6 of S4 acting via S4/A4=C2

Aliases: C6.7S4, A4⋊Dic3, C3⋊(A4⋊C4), (C2×A4).S3, (C3×A4)⋊2C4, C2.1(C3⋊S4), (C6×A4).2C2, C23.(C3⋊S3), (C2×C6)⋊2Dic3, C22⋊(C3⋊Dic3), (C22×C6).4S3, SmallGroup(144,126)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — C6.7S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — C6.7S4
 Lower central C3×A4 — C6.7S4
 Upper central C1 — C2

Generators and relations for C6.7S4
G = < a,b,c,d,e | a6=b2=c2=d3=1, e2=a3, ab=ba, ac=ca, ad=da, eae-1=a-1, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

Character table of C6.7S4

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F size 1 1 3 3 2 8 8 8 18 18 18 18 2 6 6 8 8 8 ρ1 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 linear of order 2 ρ3 1 -1 -1 1 1 1 1 1 -i i -i i -1 1 -1 -1 -1 -1 linear of order 4 ρ4 1 -1 -1 1 1 1 1 1 i -i i -i -1 1 -1 -1 -1 -1 linear of order 4 ρ5 2 2 2 2 2 -1 -1 -1 0 0 0 0 2 2 2 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 2 -1 2 -1 -1 0 0 0 0 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ7 2 2 2 2 -1 -1 -1 2 0 0 0 0 -1 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ8 2 2 2 2 -1 -1 2 -1 0 0 0 0 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ9 2 -2 -2 2 -1 -1 -1 2 0 0 0 0 1 -1 1 1 1 -2 symplectic lifted from Dic3, Schur index 2 ρ10 2 -2 -2 2 -1 2 -1 -1 0 0 0 0 1 -1 1 1 -2 1 symplectic lifted from Dic3, Schur index 2 ρ11 2 -2 -2 2 2 -1 -1 -1 0 0 0 0 -2 2 -2 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ12 2 -2 -2 2 -1 -1 2 -1 0 0 0 0 1 -1 1 -2 1 1 symplectic lifted from Dic3, Schur index 2 ρ13 3 3 -1 -1 3 0 0 0 1 1 -1 -1 3 -1 -1 0 0 0 orthogonal lifted from S4 ρ14 3 3 -1 -1 3 0 0 0 -1 -1 1 1 3 -1 -1 0 0 0 orthogonal lifted from S4 ρ15 3 -3 1 -1 3 0 0 0 i -i -i i -3 -1 1 0 0 0 complex lifted from A4⋊C4 ρ16 3 -3 1 -1 3 0 0 0 -i i i -i -3 -1 1 0 0 0 complex lifted from A4⋊C4 ρ17 6 6 -2 -2 -3 0 0 0 0 0 0 0 -3 1 1 0 0 0 orthogonal lifted from C3⋊S4 ρ18 6 -6 2 -2 -3 0 0 0 0 0 0 0 3 1 -1 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C6.7S4
On 36 points
Generators in S36
```(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 29 30)(31 32 33 34 35 36)
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 10)(8 11)(9 12)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(31 34)(32 35)(33 36)
(1 4)(2 5)(3 6)(7 36)(8 31)(9 32)(10 33)(11 34)(12 35)(13 16)(14 17)(15 18)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)
(1 26 9)(2 27 10)(3 28 11)(4 29 12)(5 30 7)(6 25 8)(13 21 33)(14 22 34)(15 23 35)(16 24 36)(17 19 31)(18 20 32)
(1 15 4 18)(2 14 5 17)(3 13 6 16)(7 19 10 22)(8 24 11 21)(9 23 12 20)(25 36 28 33)(26 35 29 32)(27 34 30 31)```

`G:=sub<Sym(36)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,10)(8,11)(9,12)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,34)(32,35)(33,36), (1,4)(2,5)(3,6)(7,36)(8,31)(9,32)(10,33)(11,34)(12,35)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27), (1,26,9)(2,27,10)(3,28,11)(4,29,12)(5,30,7)(6,25,8)(13,21,33)(14,22,34)(15,23,35)(16,24,36)(17,19,31)(18,20,32), (1,15,4,18)(2,14,5,17)(3,13,6,16)(7,19,10,22)(8,24,11,21)(9,23,12,20)(25,36,28,33)(26,35,29,32)(27,34,30,31)>;`

`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)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,10)(8,11)(9,12)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,34)(32,35)(33,36), (1,4)(2,5)(3,6)(7,36)(8,31)(9,32)(10,33)(11,34)(12,35)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27), (1,26,9)(2,27,10)(3,28,11)(4,29,12)(5,30,7)(6,25,8)(13,21,33)(14,22,34)(15,23,35)(16,24,36)(17,19,31)(18,20,32), (1,15,4,18)(2,14,5,17)(3,13,6,16)(7,19,10,22)(8,24,11,21)(9,23,12,20)(25,36,28,33)(26,35,29,32)(27,34,30,31) );`

`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),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,15),(2,16),(3,17),(4,18),(5,13),(6,14),(7,10),(8,11),(9,12),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(31,34),(32,35),(33,36)], [(1,4),(2,5),(3,6),(7,36),(8,31),(9,32),(10,33),(11,34),(12,35),(13,16),(14,17),(15,18),(19,28),(20,29),(21,30),(22,25),(23,26),(24,27)], [(1,26,9),(2,27,10),(3,28,11),(4,29,12),(5,30,7),(6,25,8),(13,21,33),(14,22,34),(15,23,35),(16,24,36),(17,19,31),(18,20,32)], [(1,15,4,18),(2,14,5,17),(3,13,6,16),(7,19,10,22),(8,24,11,21),(9,23,12,20),(25,36,28,33),(26,35,29,32),(27,34,30,31)]])`

C6.7S4 is a maximal subgroup of
Dic3.S4  Dic3×S4  S3×A4⋊C4  D6⋊S4  A4⋊Dic6  C4×C3⋊S4  (C2×C6)⋊4S4  C625Dic3  A4⋊Dic9  C6210Dic3
C6.7S4 is a maximal quotient of
C12.12S4  C6.GL2(𝔽3)  C3⋊U2(𝔽3)  A4⋊Dic9  C62.10Dic3  C626Dic3  C6210Dic3

Matrix representation of C6.7S4 in GL5(𝔽13)

 1 1 0 0 0 12 0 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 12 0 0 0 0 0 12 0 0 0 12 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 8 12 0 0 0 1 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 0 4 0 0 11 0 3 0 0 1 6 2
,
 8 0 0 0 0 5 5 0 0 0 0 0 12 0 0 0 0 8 0 2 0 0 4 7 0

`G:=sub<GL(5,GF(13))| [1,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,12,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,8,1,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,11,11,1,0,0,0,0,6,0,0,4,3,2],[8,5,0,0,0,0,5,0,0,0,0,0,12,8,4,0,0,0,0,7,0,0,0,2,0] >;`

C6.7S4 in GAP, Magma, Sage, TeX

`C_6._7S_4`
`% in TeX`

`G:=Group("C6.7S4");`
`// GroupNames label`

`G:=SmallGroup(144,126);`
`// by ID`

`G=gap.SmallGroup(144,126);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-3,-2,2,12,146,579,2164,556,1301,989]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^6=b^2=c^2=d^3=1,e^2=a^3,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=d^-1>;`
`// generators/relations`

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