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G = C42⋊C3order 48 = 24·3

The semidirect product of C42 and C3 acting faithfully

Aliases: C42⋊C3, C22.A4, SmallGroup(48,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C42⋊C3
 Chief series C1 — C22 — C42 — C42⋊C3
 Lower central C42 — C42⋊C3
 Upper central C1

Generators and relations for C42⋊C3
G = < a,b,c | a4=b4=c3=1, ab=ba, cac-1=ab-1, cbc-1=a-1b2 >

Character table of C42⋊C3

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

Permutation representations of C42⋊C3
On 12 points - transitive group 12T31
Generators in S12
```(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 4 2 3)(5 8 7 6)
(1 11 5)(2 9 7)(3 10 8)(4 12 6)```

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

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

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

`G:=TransitiveGroup(12,31);`

On 16 points - transitive group 16T63
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 12 13 5)(2 9 14 6)(3 10 15 7)(4 11 16 8)
(2 10 6)(3 13 15)(4 7 11)(5 8 14)(9 16 12)```

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

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

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

`G:=TransitiveGroup(16,63);`

On 24 points - transitive group 24T58
Generators in S24
```(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 7 6 3)(2 8 5 4)(9 23)(10 24)(11 21)(12 22)(13 20 15 18)(14 17 16 19)
(1 24 16)(2 12 18)(3 9 15)(4 21 17)(5 10 20)(6 22 14)(7 11 13)(8 23 19)```

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

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

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

`G:=TransitiveGroup(24,58);`

C42⋊C3 is a maximal subgroup of
C42⋊S3  C42⋊C6  C23.A4  C82⋊C3  C422A4  C42⋊A4  C42.A4  C42⋊(C7⋊C3)
C42⋊C3 is a maximal quotient of
C23.3A4  C42⋊C9  C82⋊C3  C422A4  C42⋊(C7⋊C3)

Polynomial with Galois group C42⋊C3 over ℚ
actionf(x)Disc(f)
12T31x12-38x10+450x8-2106x6+3679x4-2028x2+169224·720·1314·7974

Matrix representation of C42⋊C3 in GL3(𝔽5) generated by

 2 0 0 0 2 0 0 0 4
,
 1 0 0 0 3 0 0 0 2
,
 0 0 1 1 0 0 0 1 0
`G:=sub<GL(3,GF(5))| [2,0,0,0,2,0,0,0,4],[1,0,0,0,3,0,0,0,2],[0,1,0,0,0,1,1,0,0] >;`

C42⋊C3 in GAP, Magma, Sage, TeX

`C_4^2\rtimes C_3`
`% in TeX`

`G:=Group("C4^2:C3");`
`// GroupNames label`

`G:=SmallGroup(48,3);`
`// by ID`

`G=gap.SmallGroup(48,3);`
`# by ID`

`G:=PCGroup([5,-3,-2,2,-2,2,61,126,497,42,483,904]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^4=c^3=1,a*b=b*a,c*a*c^-1=a*b^-1,c*b*c^-1=a^-1*b^2>;`
`// generators/relations`

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