Copied to
clipboard

## G = C24.7A4order 192 = 26·3

### 7th non-split extension by C24 of A4 acting faithfully

Aliases: C24.7A4, C232SL2(𝔽3), (C2×Q8)⋊1A4, C232Q8⋊C3, C2.1(C23⋊A4), C2.1(Q8⋊A4), C22.6(C22⋊A4), SmallGroup(192,1021)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C23⋊2Q8 — C24.7A4
 Chief series C1 — C2 — C22 — C2×Q8 — C23⋊2Q8 — C24.7A4
 Lower central C23⋊2Q8 — C24.7A4
 Upper central C1 — C22

Generators and relations for C24.7A4
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g3=1, e2=f2=d, gag-1=ab=ba, faf-1=ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, bf=fb, gbg-1=a, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, geg-1=def, gfg-1=e >

Subgroups: 326 in 72 conjugacy classes, 13 normal (7 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C22, C22 [×6], C6 [×3], C2×C4 [×6], Q8 [×4], C23, C23 [×4], A4, C2×C6, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C2×Q8 [×4], C24, SL2(𝔽3) [×4], C2×A4 [×3], C2×C22⋊C4, C22⋊Q8 [×4], C2×SL2(𝔽3) [×4], C22×A4, C232Q8, C24.7A4
Quotients: C1, C3, A4 [×5], SL2(𝔽3), C22⋊A4, Q8⋊A4, C23⋊A4 [×2], C24.7A4

Character table of C24.7A4

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F size 1 1 1 1 6 6 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 ζ3 ζ32 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ3 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ4 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 ρ5 2 -2 2 -2 2 -2 ζ6 ζ65 0 0 0 0 ζ65 ζ3 ζ3 ζ32 ζ6 ζ32 complex lifted from SL2(𝔽3) ρ6 2 -2 2 -2 2 -2 ζ65 ζ6 0 0 0 0 ζ6 ζ32 ζ32 ζ3 ζ65 ζ3 complex lifted from SL2(𝔽3) ρ7 3 3 3 3 -1 -1 0 0 -1 -1 -1 3 0 0 0 0 0 0 orthogonal lifted from A4 ρ8 3 3 3 3 -1 -1 0 0 -1 -1 3 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ9 3 3 3 3 3 3 0 0 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ10 3 3 3 3 -1 -1 0 0 3 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ11 3 3 3 3 -1 -1 0 0 -1 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ12 4 4 -4 -4 0 0 1 1 0 0 0 0 -1 1 -1 -1 -1 1 orthogonal lifted from C23⋊A4 ρ13 4 -4 -4 4 0 0 1 1 0 0 0 0 -1 -1 1 1 -1 -1 orthogonal lifted from C23⋊A4 ρ14 4 -4 -4 4 0 0 ζ32 ζ3 0 0 0 0 ζ65 ζ65 ζ3 ζ32 ζ6 ζ6 complex lifted from C23⋊A4 ρ15 4 -4 -4 4 0 0 ζ3 ζ32 0 0 0 0 ζ6 ζ6 ζ32 ζ3 ζ65 ζ65 complex lifted from C23⋊A4 ρ16 4 4 -4 -4 0 0 ζ3 ζ32 0 0 0 0 ζ6 ζ32 ζ6 ζ65 ζ65 ζ3 complex lifted from C23⋊A4 ρ17 4 4 -4 -4 0 0 ζ32 ζ3 0 0 0 0 ζ65 ζ3 ζ65 ζ6 ζ6 ζ32 complex lifted from C23⋊A4 ρ18 6 -6 6 -6 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8⋊A4, Schur index 2

Permutation representations of C24.7A4
On 16 points - transitive group 16T438
Generators in S16
```(5 9)(6 10)(7 11)(8 12)
(2 14)(4 16)(5 9)(7 11)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(1 3)(2 4)(5 7)(6 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)
(1 6 3 8)(2 5 4 7)(9 16 11 14)(10 15 12 13)
(2 6 7)(4 8 5)(9 16 12)(10 11 14)```

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

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

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

`G:=TransitiveGroup(16,438);`

Matrix representation of C24.7A4 in GL6(𝔽13)

 1 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 1 0 0 0 12 0 12 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 12 12 0 12
,
 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 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
,
 3 4 0 0 0 0 4 10 0 0 0 0 0 0 0 0 12 0 0 0 1 1 1 2 0 0 1 0 0 0 0 0 12 12 0 12
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 12 12 12 11 0 0 0 1 1 1
,
 1 0 0 0 0 0 10 9 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 12 12 12 11 0 0 0 0 0 1

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

C24.7A4 in GAP, Magma, Sage, TeX

`C_2^4._7A_4`
`% in TeX`

`G:=Group("C2^4.7A4");`
`// GroupNames label`

`G:=SmallGroup(192,1021);`
`// by ID`

`G=gap.SmallGroup(192,1021);`
`# by ID`

`G:=PCGroup([7,-3,-2,2,-2,2,-2,-2,85,191,675,297,248,1264,851,375,172]);`
`// 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,g*a*g^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,g*b*g^-1=a,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,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`

Export

׿
×
𝔽