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## G = C3⋊D24order 144 = 24·32

### The semidirect product of C3 and D24 acting via D24/D12=C2

Aliases: C32D24, C323D8, D121S3, C12.10D6, C6.12D12, C3⋊C81S3, C4.1S32, (C3×C6).7D4, C31(D4⋊S3), (C3×D12)⋊2C2, C12⋊S32C2, C6.1(C3⋊D4), (C3×C12).2C22, C2.4(C3⋊D12), (C3×C3⋊C8)⋊1C2, SmallGroup(144,57)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3⋊D24
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — C3⋊D24
 Lower central C32 — C3×C6 — C3×C12 — C3⋊D24
 Upper central C1 — C2 — C4

Generators and relations for C3⋊D24
G = < a,b,c | a3=b24=c2=1, bab-1=cac=a-1, cbc=b-1 >

12C2
36C2
2C3
6C22
18C22
2C6
4S3
12S3
12S3
12S3
12C6
12S3
3C8
3D4
9D4
2C12
2D6
6D6
6D6
6D6
6D6
9D8
3C24
3D12
3D12
6D12
3D24

Character table of C3⋊D24

 class 1 2A 2B 2C 3A 3B 3C 4 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 24A 24B 24C 24D size 1 1 12 36 2 2 4 2 2 2 4 12 12 6 6 2 2 4 4 4 6 6 6 6 ρ1 1 1 1 1 1 1 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 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 2 2 0 0 -1 2 -1 2 2 -1 -1 0 0 2 2 -1 -1 -1 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 0 2 -1 -1 2 -1 2 -1 -1 -1 0 0 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ7 2 2 -2 0 2 -1 -1 2 -1 2 -1 1 1 0 0 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from D6 ρ8 2 2 0 0 -1 2 -1 2 2 -1 -1 0 0 -2 -2 -1 -1 -1 -1 2 1 1 1 1 orthogonal lifted from D6 ρ9 2 2 0 0 2 2 2 -2 2 2 2 0 0 0 0 -2 -2 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 0 -1 2 -1 -2 2 -1 -1 0 0 0 0 1 1 1 1 -2 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ11 2 2 0 0 -1 2 -1 -2 2 -1 -1 0 0 0 0 1 1 1 1 -2 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ12 2 -2 0 0 2 2 2 0 -2 -2 -2 0 0 √2 -√2 0 0 0 0 0 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ13 2 -2 0 0 2 2 2 0 -2 -2 -2 0 0 -√2 √2 0 0 0 0 0 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ14 2 -2 0 0 -1 2 -1 0 -2 1 1 0 0 √2 -√2 √3 -√3 √3 -√3 0 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ3+ζ83+ζ8ζ3 orthogonal lifted from D24 ρ15 2 -2 0 0 -1 2 -1 0 -2 1 1 0 0 -√2 √2 -√3 √3 -√3 √3 0 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ32+ζ87+ζ85ζ32 orthogonal lifted from D24 ρ16 2 -2 0 0 -1 2 -1 0 -2 1 1 0 0 √2 -√2 -√3 √3 -√3 √3 0 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ3+ζ85ζ3+ζ85 orthogonal lifted from D24 ρ17 2 -2 0 0 -1 2 -1 0 -2 1 1 0 0 -√2 √2 √3 -√3 √3 -√3 0 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ32+ζ8ζ32+ζ8 orthogonal lifted from D24 ρ18 2 2 0 0 2 -1 -1 -2 -1 2 -1 √-3 -√-3 0 0 -2 -2 1 1 1 0 0 0 0 complex lifted from C3⋊D4 ρ19 2 2 0 0 2 -1 -1 -2 -1 2 -1 -√-3 √-3 0 0 -2 -2 1 1 1 0 0 0 0 complex lifted from C3⋊D4 ρ20 4 -4 0 0 4 -2 -2 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ21 4 4 0 0 -2 -2 1 4 -2 -2 1 0 0 0 0 -2 -2 1 1 -2 0 0 0 0 orthogonal lifted from S32 ρ22 4 4 0 0 -2 -2 1 -4 -2 -2 1 0 0 0 0 2 2 -1 -1 2 0 0 0 0 orthogonal lifted from C3⋊D12 ρ23 4 -4 0 0 -2 -2 1 0 2 2 -1 0 0 0 0 -2√3 2√3 √3 -√3 0 0 0 0 0 orthogonal faithful ρ24 4 -4 0 0 -2 -2 1 0 2 2 -1 0 0 0 0 2√3 -2√3 -√3 √3 0 0 0 0 0 orthogonal faithful

Permutation representations of C3⋊D24
On 24 points - transitive group 24T234
Generators in S24
```(1 17 9)(2 10 18)(3 19 11)(4 12 20)(5 21 13)(6 14 22)(7 23 15)(8 16 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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)```

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

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

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

`G:=TransitiveGroup(24,234);`

Matrix representation of C3⋊D24 in GL6(𝔽73)

 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 0 72 0 0 0 0 1 72
,
 13 48 0 0 0 0 3 28 0 0 0 0 0 0 1 1 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 14 72 0 0 0 0 0 0 72 72 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[13,3,0,0,0,0,48,28,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,14,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C3⋊D24 in GAP, Magma, Sage, TeX

`C_3\rtimes D_{24}`
`% in TeX`

`G:=Group("C3:D24");`
`// GroupNames label`

`G:=SmallGroup(144,57);`
`// by ID`

`G=gap.SmallGroup(144,57);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,79,218,50,490,3461]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=b^24=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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