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## G = C24.3A4order 192 = 26·3

### 3rd non-split extension by C24 of A4 acting faithfully

Aliases: C24.3A4, C23.3SL2(𝔽3), C23.4Q8⋊C3, C23.18(C2×A4), C2.C422C6, C23.3A43C2, C2.3(C23.A4), C22.4(C2×SL2(𝔽3)), SmallGroup(192,198)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C2.C42 — C24.3A4
 Chief series C1 — C2 — C23 — C2.C42 — C23.3A4 — C24.3A4
 Lower central C2.C42 — C24.3A4
 Upper central C1 — C2

Generators and relations for C24.3A4
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g3=1, e2=gbg-1=bcd, f2=gcg-1=b, ab=ba, ac=ca, ad=da, eae-1=abc, faf-1=abd, ag=ga, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fef-1=de=ed, df=fd, dg=gd, geg-1=bef, gfg-1=cde >

Character table of C24.3A4

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F size 1 1 3 3 4 4 16 16 12 12 12 12 16 16 16 16 16 16 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 1 ζ3 ζ32 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 1 1 1 -1 -1 ζ32 ζ3 1 1 -1 -1 ζ65 ζ3 ζ6 ζ6 ζ65 ζ32 linear of order 6 ρ5 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 linear of order 3 ρ6 1 1 1 1 -1 -1 ζ3 ζ32 1 1 -1 -1 ζ6 ζ32 ζ65 ζ65 ζ6 ζ3 linear of order 6 ρ7 2 -2 2 -2 2 -2 -1 -1 0 0 0 0 -1 1 1 -1 1 1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ8 2 -2 2 -2 -2 2 -1 -1 0 0 0 0 1 1 -1 1 -1 1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ9 2 -2 2 -2 -2 2 ζ6 ζ65 0 0 0 0 ζ3 ζ3 ζ6 ζ32 ζ65 ζ32 complex lifted from SL2(𝔽3) ρ10 2 -2 2 -2 -2 2 ζ65 ζ6 0 0 0 0 ζ32 ζ32 ζ65 ζ3 ζ6 ζ3 complex lifted from SL2(𝔽3) ρ11 2 -2 2 -2 2 -2 ζ6 ζ65 0 0 0 0 ζ65 ζ3 ζ32 ζ6 ζ3 ζ32 complex lifted from SL2(𝔽3) ρ12 2 -2 2 -2 2 -2 ζ65 ζ6 0 0 0 0 ζ6 ζ32 ζ3 ζ65 ζ32 ζ3 complex lifted from SL2(𝔽3) ρ13 3 3 3 3 -3 -3 0 0 -1 -1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 3 3 3 3 3 0 0 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ15 6 6 -2 -2 0 0 0 0 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from C23.A4 ρ16 6 6 -2 -2 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C23.A4 ρ17 6 -6 -2 2 0 0 0 0 0 0 2i -2i 0 0 0 0 0 0 complex faithful ρ18 6 -6 -2 2 0 0 0 0 0 0 -2i 2i 0 0 0 0 0 0 complex faithful

Permutation representations of C24.3A4
On 24 points - transitive group 24T293
Generators in S24
```(2 14)(4 7)(5 12)(10 15)(17 19)(18 20)
(3 8)(4 7)(5 12)(6 11)
(1 13)(2 14)(9 16)(10 15)
(1 13)(2 14)(3 8)(4 7)(5 12)(6 11)(9 16)(10 15)(17 19)(18 20)(21 23)(22 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 10)(2 16)(3 11 8 6)(4 5 7 12)(9 14)(13 15)(17 23)(18 22)(19 21)(20 24)
(1 24 11)(2 20 7)(3 16 21)(4 14 18)(5 10 17)(6 13 22)(8 9 23)(12 15 19)```

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

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

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

`G:=TransitiveGroup(24,293);`

On 24 points - transitive group 24T307
Generators in S24
```(1 13)(2 12)(3 14)(4 11)(5 9)(6 15)(7 16)(8 10)(17 22)(18 23)(19 24)(20 21)
(1 3)(2 4)(11 12)(13 14)
(5 7)(6 8)(9 16)(10 15)
(1 3)(2 4)(5 7)(6 8)(9 16)(10 15)(11 12)(13 14)(17 19)(18 20)(21 23)(22 24)
(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 12 3 11)(2 14 4 13)(5 7)(10 15)(17 20)(18 19)(21 24)(22 23)
(1 7 18)(2 15 22)(3 5 20)(4 10 24)(6 17 12)(8 19 11)(9 21 14)(13 16 23)```

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

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

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

`G:=TransitiveGroup(24,307);`

Matrix representation of C24.3A4 in GL6(𝔽13)

 0 5 0 0 0 0 8 0 0 0 0 0 0 0 0 5 0 0 0 0 8 0 0 0 11 11 7 7 8 3 2 0 6 0 5 5
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 9 9 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 3 3 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 10 10 4 4 12 11 0 3 0 0 1 1
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 4 0 12 12
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 10 10 4 4 12 11 1 0 0 0 0 0 0 0 0 0 0 9

`G:=sub<GL(6,GF(13))| [0,8,0,0,11,2,5,0,0,0,11,0,0,0,0,8,7,6,0,0,5,0,7,0,0,0,0,0,8,5,0,0,0,0,3,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,9,0,0,0,12,0,9,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,3,0,12,0,0,0,3,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,10,0,0,12,0,0,10,3,0,0,0,1,4,0,0,0,1,0,4,0,0,0,0,0,12,1,0,0,0,0,11,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,4,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[0,0,0,10,1,0,0,0,0,10,0,0,1,0,0,4,0,0,0,1,0,4,0,0,0,0,1,12,0,0,0,0,0,11,0,9] >;`

C24.3A4 in GAP, Magma, Sage, TeX

`C_2^4._3A_4`
`% in TeX`

`G:=Group("C2^4.3A4");`
`// GroupNames label`

`G:=SmallGroup(192,198);`
`// by ID`

`G=gap.SmallGroup(192,198);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,1640,135,604,1011,934,521,304,851,1524]);`
`// Polycyclic`

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

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