Copied to
clipboard

## G = C23⋊A4order 96 = 25·3

### 2nd semidirect product of C23 and A4 acting faithfully

Aliases: Q82A4, C232A4, 2+ 1+42C3, C2.2(C22⋊A4), SmallGroup(96,204)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — 2+ 1+4 — C23⋊A4
 Chief series C1 — C2 — C23 — 2+ 1+4 — C23⋊A4
 Lower central 2+ 1+4 — C23⋊A4
 Upper central C1 — C2

Generators and relations for C23⋊A4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f3=1, faf-1=ab=ba, eae=ac=ca, ad=da, dbd=bc=cb, be=eb, fbf-1=a, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

6C2
6C2
6C2
16C3
3C22
3C4
3C22
3C4
3C22
4C22
4C22
4C22
12C22
16C6
3D4
3D4
3C23
3D4
3D4
3D4
3D4
4A4
4A4
4A4

Character table of C23⋊A4

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 6A 6B size 1 1 6 6 6 16 16 6 6 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 ζ32 ζ3 1 1 ζ3 ζ32 linear of order 3 ρ3 1 1 1 1 1 ζ3 ζ32 1 1 ζ32 ζ3 linear of order 3 ρ4 3 3 -1 -1 3 0 0 -1 -1 0 0 orthogonal lifted from A4 ρ5 3 3 -1 -1 -1 0 0 -1 3 0 0 orthogonal lifted from A4 ρ6 3 3 -1 3 -1 0 0 -1 -1 0 0 orthogonal lifted from A4 ρ7 3 3 -1 -1 -1 0 0 3 -1 0 0 orthogonal lifted from A4 ρ8 3 3 3 -1 -1 0 0 -1 -1 0 0 orthogonal lifted from A4 ρ9 4 -4 0 0 0 1 1 0 0 -1 -1 orthogonal faithful ρ10 4 -4 0 0 0 ζ32 ζ3 0 0 ζ65 ζ6 complex faithful ρ11 4 -4 0 0 0 ζ3 ζ32 0 0 ζ6 ζ65 complex faithful

Permutation representations of C23⋊A4
On 8 points - transitive group 8T32
Generators in S8
```(1 5)(2 8)(3 4)(6 7)
(1 3)(2 6)(4 5)(7 8)
(1 2)(3 6)(4 7)(5 8)
(3 6)(4 7)
(4 7)(5 8)
(3 4 5)(6 7 8)```

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

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

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

`G:=TransitiveGroup(8,32);`

On 24 points - transitive group 24T97
Generators in S24
```(5 9)(6 7)(10 15)(12 14)(16 19)(18 21)
(4 8)(6 7)(10 15)(11 13)(16 19)(17 20)
(1 23)(2 24)(3 22)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(1 8)(2 20)(3 11)(4 23)(5 15)(6 18)(7 21)(9 10)(12 19)(13 22)(14 16)(17 24)
(1 12)(2 9)(3 21)(4 16)(5 24)(6 13)(7 11)(8 19)(10 20)(14 23)(15 17)(18 22)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)```

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

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

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

`G:=TransitiveGroup(24,97);`

On 24 points - transitive group 24T149
Generators in S24
```(2 10)(3 11)(4 8)(5 20)(6 18)(7 21)(9 17)(13 22)(15 24)(16 19)
(1 12)(3 11)(4 16)(5 9)(6 21)(7 18)(8 19)(13 22)(14 23)(17 20)
(1 23)(2 24)(3 22)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(2 9)(3 7)(5 24)(6 22)(10 17)(11 21)(12 14)(13 18)(15 20)(16 19)
(1 8)(3 7)(4 23)(6 22)(10 15)(11 18)(12 19)(13 21)(14 16)(17 20)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)```

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

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

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

`G:=TransitiveGroup(24,149);`

C23⋊A4 is a maximal subgroup of
C2≀A4  2+ 1+4.C6  C23.S4  Q8.S4  C23⋊S4  Q82S4  2+ 1+4.3C6  Ω4+ (𝔽3)
C23⋊A4 is a maximal quotient of
C24.7A4  Q8⋊SL2(𝔽3)  C245A4  2+ 1+42C9

Polynomial with Galois group C23⋊A4 over ℚ
actionf(x)Disc(f)
8T32x8+2x7-27x6-93x5-3x4+272x3+263x2+35x-2212·34·532·614·3892

Matrix representation of C23⋊A4 in GL4(ℤ) generated by

 1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1
,
 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1
,
 -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
`G:=sub<GL(4,Integers())| [1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;`

C23⋊A4 in GAP, Magma, Sage, TeX

`C_2^3\rtimes A_4`
`% in TeX`

`G:=Group("C2^3:A4");`
`// GroupNames label`

`G:=SmallGroup(96,204);`
`// by ID`

`G=gap.SmallGroup(96,204);`
`# by ID`

`G:=PCGroup([6,-3,-2,2,-2,2,-2,73,164,579,255,1084,730]);`
`// Polycyclic`

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

Export

׿
×
𝔽