Copied to
clipboard

## G = C22⋊A4order 48 = 24·3

### The semidirect product of C22 and A4 acting via A4/C22=C3

Aliases: C22⋊A4, C242C3, SmallGroup(48,50)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C22⋊A4
 Chief series C1 — C22 — C24 — C22⋊A4
 Lower central C24 — C22⋊A4
 Upper central C1

Generators and relations for C22⋊A4
G = < a,b,c,d,e | a2=b2=c2=d2=e3=1, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, ebe-1=a, ece-1=cd=dc, ede-1=c >

3C2
3C2
3C2
3C2
3C2
16C3
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C23
3C23
3C23
3C23
3C23
4A4
4A4
4A4
4A4
4A4

Character table of C22⋊A4

 class 1 2A 2B 2C 2D 2E 3A 3B size 1 3 3 3 3 3 16 16 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 ζ3 ζ32 linear of order 3 ρ3 1 1 1 1 1 1 ζ32 ζ3 linear of order 3 ρ4 3 3 -1 -1 -1 -1 0 0 orthogonal lifted from A4 ρ5 3 -1 -1 -1 3 -1 0 0 orthogonal lifted from A4 ρ6 3 -1 3 -1 -1 -1 0 0 orthogonal lifted from A4 ρ7 3 -1 -1 3 -1 -1 0 0 orthogonal lifted from A4 ρ8 3 -1 -1 -1 -1 3 0 0 orthogonal lifted from A4

Permutation representations of C22⋊A4
On 12 points - transitive group 12T32
Generators in S12
```(1 10)(3 12)(4 8)(6 7)
(1 10)(2 11)(4 8)(5 9)
(1 10)(2 9)(3 6)(4 8)(5 11)(7 12)
(1 4)(2 11)(3 7)(5 9)(6 12)(8 10)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)```

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

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

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

`G:=TransitiveGroup(12,32);`

On 16 points - transitive group 16T64
Generators in S16
```(1 4)(2 3)(5 13)(6 15)(7 10)(8 14)(9 11)(12 16)
(1 2)(3 4)(5 8)(6 11)(7 16)(9 15)(10 12)(13 14)
(1 12)(2 10)(3 7)(4 16)(5 9)(6 14)(8 15)(11 13)
(1 13)(2 14)(3 8)(4 5)(6 10)(7 15)(9 16)(11 12)
(2 3 4)(5 6 7)(8 9 10)(11 12 13)(14 15 16)```

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

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

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

`G:=TransitiveGroup(16,64);`

On 24 points - transitive group 24T59
Generators in S24
```(1 12)(2 15)(3 22)(4 16)(5 20)(6 7)(8 19)(9 17)(10 24)(11 13)(14 23)(18 21)
(1 23)(2 10)(3 13)(4 8)(5 17)(6 21)(7 18)(9 20)(11 22)(12 14)(15 24)(16 19)
(1 14)(2 17)(3 7)(4 19)(5 10)(6 22)(8 16)(9 15)(11 21)(12 23)(13 18)(20 24)
(1 8)(2 15)(3 18)(4 23)(5 20)(6 11)(7 13)(9 17)(10 24)(12 19)(14 16)(21 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)| (1,12)(2,15)(3,22)(4,16)(5,20)(6,7)(8,19)(9,17)(10,24)(11,13)(14,23)(18,21), (1,23)(2,10)(3,13)(4,8)(5,17)(6,21)(7,18)(9,20)(11,22)(12,14)(15,24)(16,19), (1,14)(2,17)(3,7)(4,19)(5,10)(6,22)(8,16)(9,15)(11,21)(12,23)(13,18)(20,24), (1,8)(2,15)(3,18)(4,23)(5,20)(6,11)(7,13)(9,17)(10,24)(12,19)(14,16)(21,22), (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( (1,12)(2,15)(3,22)(4,16)(5,20)(6,7)(8,19)(9,17)(10,24)(11,13)(14,23)(18,21), (1,23)(2,10)(3,13)(4,8)(5,17)(6,21)(7,18)(9,20)(11,22)(12,14)(15,24)(16,19), (1,14)(2,17)(3,7)(4,19)(5,10)(6,22)(8,16)(9,15)(11,21)(12,23)(13,18)(20,24), (1,8)(2,15)(3,18)(4,23)(5,20)(6,11)(7,13)(9,17)(10,24)(12,19)(14,16)(21,22), (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([[(1,12),(2,15),(3,22),(4,16),(5,20),(6,7),(8,19),(9,17),(10,24),(11,13),(14,23),(18,21)], [(1,23),(2,10),(3,13),(4,8),(5,17),(6,21),(7,18),(9,20),(11,22),(12,14),(15,24),(16,19)], [(1,14),(2,17),(3,7),(4,19),(5,10),(6,22),(8,16),(9,15),(11,21),(12,23),(13,18),(20,24)], [(1,8),(2,15),(3,18),(4,23),(5,20),(6,11),(7,13),(9,17),(10,24),(12,19),(14,16),(21,22)], [(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(24,59);`

C22⋊A4 is a maximal subgroup of
C24⋊C6  C22⋊S4  A42  C422A4  C42⋊A4  C26⋊C3  F16  C7⋊(C22⋊A4)
C22⋊A4 is a maximal quotient of
Q8⋊A4  C23⋊A4  C24⋊C9  C422A4  C42⋊A4  C42.A4  C26⋊C3  C7⋊(C22⋊A4)

Polynomial with Galois group C22⋊A4 over ℚ
actionf(x)Disc(f)
12T32x12-47x10+646x8-2800x6+2584x4-752x2+64270·312·194·3078

Matrix representation of C22⋊A4 in GL6(ℤ)

 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 -1 0 0 0 0 0 0 1
,
 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 1 0 0 0 0 0 0 -1
,
 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 1 0 0 0 0 0 0 1
,
 -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 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

`G:=sub<GL(6,Integers())| [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,-1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C22⋊A4 in GAP, Magma, Sage, TeX

`C_2^2\rtimes A_4`
`% in TeX`

`G:=Group("C2^2:A4");`
`// GroupNames label`

`G:=SmallGroup(48,50);`
`// by ID`

`G=gap.SmallGroup(48,50);`
`# by ID`

`G:=PCGroup([5,-3,-2,2,-2,2,61,137,483,904]);`
`// Polycyclic`

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

Export

׿
×
𝔽