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G = C6.S4order 144 = 24·32

3rd non-split extension by C6 of S4 acting via S4/A4=C2

Aliases: C6.3S4, C23.D9, C22⋊Dic9, C3.A4⋊C4, C3.(A4⋊C4), (C2×C6).Dic3, C2.1(C3.S4), (C22×C6).2S3, (C2×C3.A4).C2, SmallGroup(144,33)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6.S4
G = < a,b,c,d,e | a6=b2=c2=1, d3=a2, 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=a4d2 >

Character table of C6.S4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 9A 9B 9C 18A 18B 18C size 1 1 3 3 2 18 18 18 18 2 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 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 -i i -i i -1 -1 1 1 1 1 -1 -1 -1 linear of order 4 ρ4 1 -1 1 -1 1 i -i i -i -1 -1 1 1 1 1 -1 -1 -1 linear of order 4 ρ5 2 2 2 2 -1 0 0 0 0 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ6 2 2 2 2 2 0 0 0 0 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 2 -1 0 0 0 0 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ8 2 2 2 2 -1 0 0 0 0 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ9 2 -2 2 -2 -1 0 0 0 0 1 1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 symplectic lifted from Dic9, Schur index 2 ρ10 2 -2 2 -2 2 0 0 0 0 -2 -2 2 -1 -1 -1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ11 2 -2 2 -2 -1 0 0 0 0 1 1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 symplectic lifted from Dic9, Schur index 2 ρ12 2 -2 2 -2 -1 0 0 0 0 1 1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 symplectic lifted from Dic9, Schur index 2 ρ13 3 3 -1 -1 3 -1 -1 1 1 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ14 3 3 -1 -1 3 1 1 -1 -1 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ15 3 -3 -1 1 3 -i i i -i -3 1 -1 0 0 0 0 0 0 complex lifted from A4⋊C4 ρ16 3 -3 -1 1 3 i -i -i i -3 1 -1 0 0 0 0 0 0 complex lifted from A4⋊C4 ρ17 6 6 -2 -2 -3 0 0 0 0 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C3.S4 ρ18 6 -6 -2 2 -3 0 0 0 0 3 -1 1 0 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

C6.S4 is a maximal subgroup of   C12.1S4  C4×C3.S4  C23.D18  C62.Dic3  A4⋊Dic9  C62.10Dic3
C6.S4 is a maximal quotient of   C12.S4  Q8⋊Dic9  C12.9S4  C18.S4  C62.10Dic3

Matrix representation of C6.S4 in GL5(𝔽37)

 0 1 0 0 0 36 1 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 36 0 0 0 0 0 36 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 0 36
,
 31 17 0 0 0 20 11 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 0 31 0 0 0 31 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 36 0

`G:=sub<GL(5,GF(37))| [0,36,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36],[31,20,0,0,0,17,11,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[0,31,0,0,0,31,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,36,0] >;`

C6.S4 in GAP, Magma, Sage, TeX

`C_6.S_4`
`% in TeX`

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

`G:=SmallGroup(144,33);`
`// by ID`

`G=gap.SmallGroup(144,33);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-3,-2,2,12,362,122,579,2164,556,1301,989]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^6=b^2=c^2=1,d^3=a^2,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=a^4*d^2>;`
`// generators/relations`

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