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## G = C24.A4order 192 = 26·3

### 1st non-split extension by C24 of A4 acting faithfully

Aliases: C24.1A4, C231SL2(𝔽3), C23⋊Q8⋊C3, Q8⋊A41C2, (C22×Q8)⋊1C6, C23.15(C2×A4), C2.2(C24⋊C6), C22.2(C2×SL2(𝔽3)), SmallGroup(192,195)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22×Q8 — C24.A4
 Chief series C1 — C2 — C23 — C22×Q8 — Q8⋊A4 — C24.A4
 Lower central C22×Q8 — C24.A4
 Upper central C1 — C2

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

Subgroups: 299 in 57 conjugacy classes, 11 normal (9 characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22, C22 [×6], C6 [×3], C2×C4 [×7], Q8 [×4], C23, C23 [×2], C23 [×2], A4, C2×C6, C22⋊C4 [×2], C22×C4 [×2], C2×Q8 [×2], C24, SL2(𝔽3) [×2], C2×A4 [×3], C2.C42, C2×C22⋊C4, C22×Q8, C22×A4, C23⋊Q8, Q8⋊A4, C24.A4
Quotients: C1, C2, C3, C6, A4, SL2(𝔽3) [×2], C2×A4, C2×SL2(𝔽3), C24⋊C6, C24.A4

Character table of C24.A4

 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 C24⋊C6 ρ16 6 6 -2 -2 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C6 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.A4
On 24 points - transitive group 24T291
Generators in S24
```(2 4)(5 7)(10 12)(17 19)(18 20)(21 23)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 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 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 9 3 11)(2 12 4 10)(5 22 7 24)(6 21 8 23)(13 19 15 17)(14 18 16 20)
(1 6 13)(2 21 20)(3 8 15)(4 23 18)(5 17 12)(7 19 10)(9 22 14)(11 24 16)```

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

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

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

`G:=TransitiveGroup(24,291);`

On 24 points - transitive group 24T303
Generators in S24
```(1 4)(2 3)(5 24)(6 23)(7 22)(8 21)(9 10)(11 12)(13 18)(14 19)(15 20)(16 17)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 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 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 9 3 11)(2 12 4 10)(5 22 7 24)(6 21 8 23)(13 19 15 17)(14 18 16 20)
(1 6 13)(2 21 20)(3 8 15)(4 23 18)(5 17 12)(7 19 10)(9 22 14)(11 24 16)```

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

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

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

`G:=TransitiveGroup(24,303);`

Matrix representation of C24.A4 in GL6(𝔽13)

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

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

C24.A4 in GAP, Magma, Sage, TeX

`C_2^4.A_4`
`% in TeX`

`G:=Group("C2^4.A4");`
`// GroupNames label`

`G:=SmallGroup(192,195);`
`// by ID`

`G=gap.SmallGroup(192,195);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,632,135,352,1683,262,521,248,851,1524]);`
`// Polycyclic`

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

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