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## G = C6.6S4order 144 = 24·32

### 6th non-split extension by C6 of S4 acting via S4/A4=C2

Aliases: C6.6S4, C3⋊GL2(𝔽3), SL2(𝔽3)⋊S3, Q8⋊(C3⋊S3), (C3×Q8)⋊1S3, C2.3(C3⋊S4), (C3×SL2(𝔽3))⋊1C2, SmallGroup(144,125)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — C6.6S4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C6.6S4
 Lower central C3×SL2(𝔽3) — C6.6S4
 Upper central C1 — C2

Generators and relations for C6.6S4
G = < a,b,c,d,e | a6=d3=e2=1, b2=c2=a3, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a3b, dbd-1=a3bc, ebe=bc, dcd-1=b, ece=a3c, ede=d-1 >

36C2
4C3
4C3
4C3
3C4
18C22
4C6
4C6
4C6
12S3
12S3
12S3
12S3
12S3
12S3
12S3
4C32
9C8
9D4
3C12
6D6
12D6
12D6
12D6
9SD16
3D12

Character table of C6.6S4

 class 1 2A 2B 3A 3B 3C 3D 4 6A 6B 6C 6D 8A 8B 12 size 1 1 36 2 8 8 8 6 2 8 8 8 18 18 12 ρ1 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 linear of order 2 ρ3 2 2 0 -1 -1 2 -1 2 -1 -1 -1 2 0 0 -1 orthogonal lifted from S3 ρ4 2 2 0 -1 -1 -1 2 2 -1 2 -1 -1 0 0 -1 orthogonal lifted from S3 ρ5 2 2 0 -1 2 -1 -1 2 -1 -1 2 -1 0 0 -1 orthogonal lifted from S3 ρ6 2 2 0 2 -1 -1 -1 2 2 -1 -1 -1 0 0 2 orthogonal lifted from S3 ρ7 2 -2 0 2 -1 -1 -1 0 -2 1 1 1 -√-2 √-2 0 complex lifted from GL2(𝔽3) ρ8 2 -2 0 2 -1 -1 -1 0 -2 1 1 1 √-2 -√-2 0 complex lifted from GL2(𝔽3) ρ9 3 3 1 3 0 0 0 -1 3 0 0 0 -1 -1 -1 orthogonal lifted from S4 ρ10 3 3 -1 3 0 0 0 -1 3 0 0 0 1 1 -1 orthogonal lifted from S4 ρ11 4 -4 0 -2 -2 1 1 0 2 -1 2 -1 0 0 0 orthogonal faithful ρ12 4 -4 0 4 1 1 1 0 -4 -1 -1 -1 0 0 0 orthogonal lifted from GL2(𝔽3) ρ13 4 -4 0 -2 1 1 -2 0 2 2 -1 -1 0 0 0 orthogonal faithful ρ14 4 -4 0 -2 1 -2 1 0 2 -1 -1 2 0 0 0 orthogonal faithful ρ15 6 6 0 -3 0 0 0 -2 -3 0 0 0 0 0 1 orthogonal lifted from C3⋊S4

Permutation representations of C6.6S4
On 24 points - transitive group 24T252
Generators in S24
```(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 16 4 13)(2 17 5 14)(3 18 6 15)(7 19 10 22)(8 20 11 23)(9 21 12 24)
(1 11 4 8)(2 12 5 9)(3 7 6 10)(13 23 16 20)(14 24 17 21)(15 19 18 22)
(7 19 18)(8 20 13)(9 21 14)(10 22 15)(11 23 16)(12 24 17)
(1 4)(2 3)(5 6)(7 9)(10 12)(13 20)(14 19)(15 24)(16 23)(17 22)(18 21)```

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

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

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

`G:=TransitiveGroup(24,252);`

C6.6S4 is a maximal subgroup of
Dic3.5S4  GL2(𝔽3)⋊S3  D6.2S4  S3×GL2(𝔽3)  SL2(𝔽3).D6  C12.14S4  C12.7S4  C322GL2(𝔽3)  C18.6S4  C325GL2(𝔽3)
C6.6S4 is a maximal quotient of
C6.GL2(𝔽3)  C18.6S4  C32.3GL2(𝔽3)  C323GL2(𝔽3)  C325GL2(𝔽3)

Matrix representation of C6.6S4 in GL4(ℚ) generated by

 1/2 1/2 -1/2 1/2 -1/2 1/2 1/2 1/2 1/2 -1/2 1/2 1/2 -1/2 -1/2 -1/2 1/2
,
 0 0 1 0 0 0 0 1 -1 0 0 0 0 -1 0 0
,
 0 0 0 -1 0 0 1 0 0 -1 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 1 0 -1 0 0 0 0 -1 0
,
 -1/2 -1/2 1/2 -1/2 -1/2 1/2 -1/2 -1/2 1/2 -1/2 -1/2 -1/2 -1/2 -1/2 -1/2 1/2
`G:=sub<GL(4,Rationals())| [1/2,-1/2,1/2,-1/2,1/2,1/2,-1/2,-1/2,-1/2,1/2,1/2,-1/2,1/2,1/2,1/2,1/2],[0,0,-1,0,0,0,0,-1,1,0,0,0,0,1,0,0],[0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[1,0,0,0,0,0,-1,0,0,0,0,-1,0,1,0,0],[-1/2,-1/2,1/2,-1/2,-1/2,1/2,-1/2,-1/2,1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,1/2] >;`

C6.6S4 in GAP, Magma, Sage, TeX

`C_6._6S_4`
`% in TeX`

`G:=Group("C6.6S4");`
`// GroupNames label`

`G:=SmallGroup(144,125);`
`// by ID`

`G=gap.SmallGroup(144,125);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-2,2,-2,49,218,867,1305,117,544,820,202,88]);`
`// Polycyclic`

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

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