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G = C3⋊D4order 24 = 23·3

The semidirect product of C3 and D4 acting via D4/C22=C2

Aliases: C32D4, D62C2, Dic3⋊C2, C2.5D6, C222S3, C6.5C22, (C2×C6)⋊2C2, SmallGroup(24,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3⋊D4
 Chief series C1 — C3 — C6 — D6 — C3⋊D4
 Lower central C3 — C6 — C3⋊D4
 Upper central C1 — C2 — C22

Generators and relations for C3⋊D4
G = < a,b,c | a3=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Character table of C3⋊D4

 class 1 2A 2B 2C 3 4 6A 6B 6C size 1 1 2 6 2 6 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 -1 1 1 1 linear of order 2 ρ3 1 1 -1 1 1 -1 -1 -1 1 linear of order 2 ρ4 1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ5 2 2 -2 0 -1 0 1 1 -1 orthogonal lifted from D6 ρ6 2 2 2 0 -1 0 -1 -1 -1 orthogonal lifted from S3 ρ7 2 -2 0 0 2 0 0 0 -2 orthogonal lifted from D4 ρ8 2 -2 0 0 -1 0 -√-3 √-3 1 complex faithful ρ9 2 -2 0 0 -1 0 √-3 -√-3 1 complex faithful

Permutation representations of C3⋊D4
On 12 points - transitive group 12T13
Generators in S12
```(1 9 8)(2 5 10)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 12)(6 11)(7 10)(8 9)```

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

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

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

`G:=TransitiveGroup(12,13);`

On 12 points - transitive group 12T15
Generators in S12
```(1 9 8)(2 5 10)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)```

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

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

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

`G:=TransitiveGroup(12,15);`

Regular action on 24 points - transitive group 24T14
Generators in S24
```(1 21 20)(2 17 22)(3 23 18)(4 19 24)(5 14 10)(6 11 15)(7 16 12)(8 9 13)
(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 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)```

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

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

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

`G:=TransitiveGroup(24,14);`

C3⋊D4 is a maximal subgroup of
C3.S4  C327D4  C3⋊S4  Q8.D6  A4⋊D4  C33⋊D4  C2.S5
D6p⋊C2: C4○D12  S3×D4  C9⋊D4  C3⋊D20  C157D4  C3⋊D28  C217D4  C3⋊D44 ...
D2p⋊S3: D42S3  D6⋊S3  C3⋊D12  C15⋊D4  C21⋊D4  C33⋊D4  C39⋊D4  C51⋊D4 ...
C3⋊D4 is a maximal quotient of
A4⋊D4  C33⋊D4
C3⋊D4p: D4⋊S3  C3⋊D12  C3⋊D20  C3⋊D28  C3⋊D44  C3⋊D52  C3⋊D68  C3⋊D76 ...
C6.D2p: Dic3⋊C4  D6⋊C4  D4.S3  Q82S3  C3⋊Q16  C6.D4  C9⋊D4  D6⋊S3 ...

Polynomial with Galois group C3⋊D4 over ℚ
actionf(x)Disc(f)
12T13x12-9x8+3x4-3-256·315
12T15x12-24x10+216x8-896x6+1680x4-1152x2+48268·317·136

Matrix representation of C3⋊D4 in GL2(𝔽7) generated by

 4 0 0 2
,
 0 6 1 0
,
 0 1 1 0
`G:=sub<GL(2,GF(7))| [4,0,0,2],[0,1,6,0],[0,1,1,0] >;`

C3⋊D4 in GAP, Magma, Sage, TeX

`C_3\rtimes D_4`
`% in TeX`

`G:=Group("C3:D4");`
`// GroupNames label`

`G:=SmallGroup(24,8);`
`// by ID`

`G=gap.SmallGroup(24,8);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-3,49,259]);`
`// Polycyclic`

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

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