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## G = C4.6S4order 96 = 25·3

### 3rd central extension by C4 of S4

Aliases: C4.6S4, Q8.4D6, C4GL2(𝔽3), C4CSU2(𝔽3), GL2(𝔽3)⋊3C2, CSU2(𝔽3)⋊3C2, SL2(𝔽3).4C22, C4.A42C2, C4○D41S3, C2.9(C2×S4), SmallGroup(96,192)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C4.6S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3) — C4.6S4
 Lower central SL2(𝔽3) — C4.6S4
 Upper central C1 — C4

Generators and relations for C4.6S4
G = < a,b,c,d,e | a4=d3=e2=1, b2=c2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=a2b, dbd-1=a2bc, ebe=bc, dcd-1=b, ece=a2c, ede=d-1 >

Character table of C4.6S4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 8A 8B 8C 8D 12A 12B size 1 1 6 12 8 1 1 6 12 8 6 6 6 6 8 8 ρ1 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 linear of order 2 ρ3 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ5 2 2 2 0 -1 2 2 2 0 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ6 2 2 -2 0 -1 -2 -2 2 0 -1 0 0 0 0 1 1 orthogonal lifted from D6 ρ7 2 -2 0 0 -1 2i -2i 0 0 1 -√2 -√-2 √2 √-2 -i i complex faithful ρ8 2 -2 0 0 -1 -2i 2i 0 0 1 -√2 √-2 √2 -√-2 i -i complex faithful ρ9 2 -2 0 0 -1 2i -2i 0 0 1 √2 √-2 -√2 -√-2 -i i complex faithful ρ10 2 -2 0 0 -1 -2i 2i 0 0 1 √2 -√-2 -√2 √-2 i -i complex faithful ρ11 3 3 1 -1 0 -3 -3 -1 1 0 -1 1 -1 1 0 0 orthogonal lifted from C2×S4 ρ12 3 3 -1 1 0 3 3 -1 1 0 -1 -1 -1 -1 0 0 orthogonal lifted from S4 ρ13 3 3 -1 -1 0 3 3 -1 -1 0 1 1 1 1 0 0 orthogonal lifted from S4 ρ14 3 3 1 1 0 -3 -3 -1 -1 0 1 -1 1 -1 0 0 orthogonal lifted from C2×S4 ρ15 4 -4 0 0 1 -4i 4i 0 0 -1 0 0 0 0 -i i complex faithful ρ16 4 -4 0 0 1 4i -4i 0 0 -1 0 0 0 0 i -i complex faithful

Permutation representations of C4.6S4
On 16 points - transitive group 16T189
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 8 3 6)(2 5 4 7)(9 15 11 13)(10 16 12 14)
(1 9 3 11)(2 10 4 12)(5 14 7 16)(6 15 8 13)
(5 10 16)(6 11 13)(7 12 14)(8 9 15)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)```

`G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8,3,6)(2,5,4,7)(9,15,11,13)(10,16,12,14), (1,9,3,11)(2,10,4,12)(5,14,7,16)(6,15,8,13), (5,10,16)(6,11,13)(7,12,14)(8,9,15), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)>;`

`G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8,3,6)(2,5,4,7)(9,15,11,13)(10,16,12,14), (1,9,3,11)(2,10,4,12)(5,14,7,16)(6,15,8,13), (5,10,16)(6,11,13)(7,12,14)(8,9,15), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15) );`

`G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,8,3,6),(2,5,4,7),(9,15,11,13),(10,16,12,14)], [(1,9,3,11),(2,10,4,12),(5,14,7,16),(6,15,8,13)], [(5,10,16),(6,11,13),(7,12,14),(8,9,15)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15)]])`

`G:=TransitiveGroup(16,189);`

C4.6S4 is a maximal subgroup of
CU2(𝔽3)  C8.5S4  GL2(𝔽3)⋊C22  Q8.6S4  Q8.7S4  D4.4S4  D4.5S4  Dic3.4S4  Dic3.5S4  C12.14S4  C4.6S5  Dic5.6S4  Dic5.7S4  C20.6S4
C4.6S4 is a maximal quotient of
C4×CSU2(𝔽3)  Q8.Dic6  C4×GL2(𝔽3)  Q8.2D12  C4.A4⋊C4  SL2(𝔽3).D4  SL2(𝔽3)⋊D4  C12.11S4  Dic3.4S4  Dic3.5S4  C12.14S4  Dic5.6S4  Dic5.7S4  C20.6S4

Matrix representation of C4.6S4 in GL2(𝔽17) generated by

 4 0 0 4
,
 9 9 6 8
,
 6 11 9 11
,
 1 10 15 15
,
 1 10 0 16
`G:=sub<GL(2,GF(17))| [4,0,0,4],[9,6,9,8],[6,9,11,11],[1,15,10,15],[1,0,10,16] >;`

C4.6S4 in GAP, Magma, Sage, TeX

`C_4._6S_4`
`% in TeX`

`G:=Group("C4.6S4");`
`// GroupNames label`

`G:=SmallGroup(96,192);`
`// by ID`

`G=gap.SmallGroup(96,192);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-2,2,-2,295,146,579,447,117,364,286,202,88]);`
`// Polycyclic`

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

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