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## G = C4.3S4order 96 = 25·3

### 3rd non-split extension by C4 of S4 acting via S4/A4=C2

Aliases: C4.3S4, Q8.5D6, GL2(𝔽3)⋊2C2, SL2(𝔽3)⋊2C22, C4.A41C2, C4○D42S3, C2.10(C2×S4), SmallGroup(96,193)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C4.3S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3) — C4.3S4
 Lower central SL2(𝔽3) — C4.3S4
 Upper central C1 — C2 — C4

Generators and relations for C4.3S4
G = < a,b,c,d,e | a4=d3=e2=1, b2=c2=a2, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a2b, dbd-1=a2bc, ebe=bc, dcd-1=b, ece=a2c, ede=d-1 >

6C2
12C2
12C2
4C3
3C22
3C4
6C22
6C22
12C22
12C22
4C6
8S3
8S3
3D4
3C8
3D4
3D4
3C8
6C23
6D4
4D6
4D6
4C12
3D8
3SD16
3D8
3SD16
4D12

Character table of C4.3S4

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

Permutation representations of C4.3S4
On 16 points - transitive group 16T190
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6 3 8)(2 7 4 5)(9 15 11 13)(10 16 12 14)
(1 9 3 11)(2 10 4 12)(5 16 7 14)(6 13 8 15)
(5 12 14)(6 9 15)(7 10 16)(8 11 13)
(1 4)(2 3)(5 13)(6 16)(7 15)(8 14)(9 10)(11 12)```

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

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

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

`G:=TransitiveGroup(16,190);`

On 24 points - transitive group 24T127
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 12 3 10)(2 9 4 11)(5 8 7 6)(13 18 15 20)(14 19 16 17)(21 22 23 24)
(1 4 3 2)(5 24 7 22)(6 21 8 23)(9 10 11 12)(13 17 15 19)(14 18 16 20)
(1 17 7)(2 18 8)(3 19 5)(4 20 6)(9 16 24)(10 13 21)(11 14 22)(12 15 23)
(2 4)(5 19)(6 18)(7 17)(8 20)(9 12)(10 11)(13 22)(14 21)(15 24)(16 23)```

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

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

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

`G:=TransitiveGroup(24,127);`

C4.3S4 is a maximal subgroup of
C8.4S4  C8.3S4  Q8.5S4  D4.3S4  GL2(𝔽3)⋊C22  Q8.7S4  D4.4S4  GL2(𝔽3)⋊S3  C12.7S4  C4.3S5  GL2(𝔽3)⋊D5  C20.3S4
C4.3S4 is a maximal quotient of
SL2(𝔽3)⋊Q8  Q8⋊D12  GL2(𝔽3)⋊C4  (C2×C4).S4  SL2(𝔽3)⋊D4  C12.4S4  GL2(𝔽3)⋊S3  C12.7S4  GL2(𝔽3)⋊D5  C20.3S4

Matrix representation of C4.3S4 in GL4(ℚ) generated by

 0 0 0 -1 0 0 -1 0 0 1 0 0 1 0 0 0
,
 0 0 0 -1 0 0 1 0 0 -1 0 0 1 0 0 0
,
 0 -1 0 0 1 0 0 0 0 0 0 1 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
,
 -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())| [0,0,0,1,0,0,1,0,0,-1,0,0,-1,0,0,0],[0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[0,1,0,0,-1,0,0,0,0,0,0,-1,0,0,1,0],[-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],[-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] >;`

C4.3S4 in GAP, Magma, Sage, TeX

`C_4._3S_4`
`% in TeX`

`G:=Group("C4.3S4");`
`// GroupNames label`

`G:=SmallGroup(96,193);`
`// by ID`

`G=gap.SmallGroup(96,193);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-2,2,-2,601,295,146,579,447,117,364,286,202,88]);`
`// 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,e*a*e=a^-1,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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