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## G = C23.9S4order 192 = 26·3

### 3rd non-split extension by C23 of S4 acting via S4/C22=S3

Aliases: C23.9S4, C422Dic3, C42⋊C33C4, (C2×C42).S3, C2.1(C42⋊S3), C22.1(A4⋊C4), (C2×C42⋊C3).3C2, SmallGroup(192,182)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C42⋊C3 — C23.9S4
 Chief series C1 — C22 — C42 — C42⋊C3 — C2×C42⋊C3 — C23.9S4
 Lower central C42⋊C3 — C23.9S4
 Upper central C1 — C2

Generators and relations for C23.9S4
G = < a,b,c,d,e,f,g | a2=b2=c2=f3=1, d2=cb=gbg-1=fcf-1=bc, e2=fbf-1=c, g2=a, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bd=db, be=eb, fef-1=cd=dc, geg-1=ce=ec, cg=gc, fdf-1=gdg-1=de=ed, gfg-1=f-1 >

Character table of C23.9S4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6 8A 8B 8C 8D size 1 1 3 3 32 3 3 3 3 6 6 12 12 12 12 32 12 12 12 12 ρ1 1 1 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 -1 -1 linear of order 2 ρ3 1 -1 1 -1 1 -1 1 -1 1 -1 1 i -i i -i -1 i -i i -i linear of order 4 ρ4 1 -1 1 -1 1 -1 1 -1 1 -1 1 -i i -i i -1 -i i -i i linear of order 4 ρ5 2 2 2 2 -1 2 2 2 2 2 2 0 0 0 0 -1 0 0 0 0 orthogonal lifted from S3 ρ6 2 -2 2 -2 -1 -2 2 -2 2 -2 2 0 0 0 0 1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ7 3 3 3 3 0 -1 -1 -1 -1 -1 -1 1 1 1 1 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ8 3 3 3 3 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 1 1 1 1 orthogonal lifted from S4 ρ9 3 -3 -1 1 0 1+2i -1+2i 1-2i -1-2i -1 1 i -i -i i 0 -1 1 1 -1 complex faithful ρ10 3 3 -1 -1 0 -1-2i -1+2i -1+2i -1-2i 1 1 1 1 -1 -1 0 i i -i -i complex lifted from C42⋊S3 ρ11 3 -3 -1 1 0 1+2i -1+2i 1-2i -1-2i -1 1 -i i i -i 0 1 -1 -1 1 complex faithful ρ12 3 3 -1 -1 0 -1+2i -1-2i -1-2i -1+2i 1 1 1 1 -1 -1 0 -i -i i i complex lifted from C42⋊S3 ρ13 3 -3 3 -3 0 1 -1 1 -1 1 -1 -i i -i i 0 i -i i -i complex lifted from A4⋊C4 ρ14 3 -3 -1 1 0 1-2i -1-2i 1+2i -1+2i -1 1 i -i -i i 0 1 -1 -1 1 complex faithful ρ15 3 3 -1 -1 0 -1+2i -1-2i -1-2i -1+2i 1 1 -1 -1 1 1 0 i i -i -i complex lifted from C42⋊S3 ρ16 3 -3 3 -3 0 1 -1 1 -1 1 -1 i -i i -i 0 -i i -i i complex lifted from A4⋊C4 ρ17 3 -3 -1 1 0 1-2i -1-2i 1+2i -1+2i -1 1 -i i i -i 0 -1 1 1 -1 complex faithful ρ18 3 3 -1 -1 0 -1-2i -1+2i -1+2i -1-2i 1 1 -1 -1 1 1 0 -i -i i i complex lifted from C42⋊S3 ρ19 6 6 -2 -2 0 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from C42⋊S3 ρ20 6 -6 -2 2 0 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of C23.9S4
On 12 points - transitive group 12T98
Generators in S12
```(1 2)(3 4)(5 7)(6 8)(9 11)(10 12)
(1 2)(3 4)(9 11)(10 12)
(1 2)(3 4)(5 7)(6 8)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 3 2 4)(5 6 7 8)
(1 11 7)(2 9 5)(3 10 6)(4 12 8)
(1 8 2 6)(3 7 4 5)(9 10 11 12)```

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

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

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

`G:=TransitiveGroup(12,98);`

On 24 points - transitive group 24T317
Generators in S24
```(1 2)(3 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 7)(6 8)(17 19)(18 20)(21 23)(22 24)
(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 3 2 4)(5 8 7 6)(9 11)(10 12)(13 15)(14 16)(17 20 19 18)(21 24 23 22)
(1 11 20)(2 9 18)(3 12 19)(4 10 17)(5 13 23)(6 16 24)(7 15 21)(8 14 22)
(1 23 2 21)(3 24 4 22)(5 18 7 20)(6 17 8 19)(9 15 11 13)(10 14 12 16)```

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

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

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

`G:=TransitiveGroup(24,317);`

On 24 points - transitive group 24T481
Generators in S24
```(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)
(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 4 2 3)(5 7 6 8)(9 10 11 12)(13 14 15 16)
(1 23 9)(2 21 11)(3 24 10)(4 22 12)(5 17 14)(6 19 16)(7 20 13)(8 18 15)
(1 16 2 14)(3 13 4 15)(5 9 6 11)(7 12 8 10)(17 23 19 21)(18 24 20 22)```

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

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

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

`G:=TransitiveGroup(24,481);`

On 24 points - transitive group 24T482
Generators in S24
```(1 2)(3 4)(5 6)(7 8)(9 15)(10 16)(11 13)(12 14)(17 22)(18 23)(19 24)(20 21)
(1 6)(2 5)(3 7)(4 8)(17 19)(18 20)(21 23)(22 24)
(1 6)(2 5)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)
(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 8 6 4)(2 7 5 3)(9 14 11 16)(10 15 12 13)(17 24)(18 21)(19 22)(20 23)
(1 23 9)(2 18 15)(3 22 16)(4 17 10)(5 20 13)(6 21 11)(7 24 14)(8 19 12)
(1 12 2 14)(3 11 4 13)(5 16 6 10)(7 9 8 15)(17 20 22 21)(18 24 23 19)```

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

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

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

`G:=TransitiveGroup(24,482);`

Polynomial with Galois group C23.9S4 over ℚ
actionf(x)Disc(f)
12T98x12-14210x10-5541749x8+46928284x6-72912385x4-52237562x2+8560357268·520·1314·3713·140837291602386138174

Matrix representation of C23.9S4 in GL3(𝔽5) generated by

 4 0 0 0 4 0 0 0 4
,
 2 2 4 0 4 0 3 2 3
,
 2 4 2 3 3 2 0 0 4
,
 1 2 2 1 2 4 1 4 2
,
 2 4 1 1 0 4 4 1 4
,
 1 0 1 0 0 4 0 1 4
,
 2 0 2 0 2 3 0 0 3
`G:=sub<GL(3,GF(5))| [4,0,0,0,4,0,0,0,4],[2,0,3,2,4,2,4,0,3],[2,3,0,4,3,0,2,2,4],[1,1,1,2,2,4,2,4,2],[2,1,4,4,0,1,1,4,4],[1,0,0,0,0,1,1,4,4],[2,0,0,0,2,0,2,3,3] >;`

C23.9S4 in GAP, Magma, Sage, TeX

`C_2^3._9S_4`
`% in TeX`

`G:=Group("C2^3.9S4");`
`// GroupNames label`

`G:=SmallGroup(192,182);`
`// by ID`

`G=gap.SmallGroup(192,182);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,14,170,675,185,360,424,1173,102,6053,1027,1784]);`
`// Polycyclic`

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

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