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## G = C6.D6order 72 = 23·32

### 2nd non-split extension by C6 of D6 acting via D6/S3=C2

Aliases: C6.2D6, Dic3Dic3, Dic32S3, C2.2S32, C3⋊S31C4, C31(C4×S3), C323(C2×C4), (C3×Dic3)⋊3C2, (C3×C6).2C22, (C2×C3⋊S3).1C2, SmallGroup(72,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C6.D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×Dic3 — C6.D6
 Lower central C32 — C6.D6
 Upper central C1 — C2

Generators and relations for C6.D6
G = < a,b,c | a6=c2=1, b6=a3, bab-1=cac=a-1, cbc=b5 >

Character table of C6.D6

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 12A 12B 12C 12D size 1 1 9 9 2 2 4 3 3 3 3 2 2 4 6 6 6 6 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 1 -1 1 1 1 -i i -i i -1 -1 -1 -i -i i i linear of order 4 ρ6 1 -1 -1 1 1 1 1 -i -i i i -1 -1 -1 i -i -i i linear of order 4 ρ7 1 -1 1 -1 1 1 1 i -i i -i -1 -1 -1 i i -i -i linear of order 4 ρ8 1 -1 -1 1 1 1 1 i i -i -i -1 -1 -1 -i i i -i linear of order 4 ρ9 2 2 0 0 -1 2 -1 0 -2 -2 0 2 -1 -1 1 0 1 0 orthogonal lifted from D6 ρ10 2 2 0 0 2 -1 -1 -2 0 0 -2 -1 2 -1 0 1 0 1 orthogonal lifted from D6 ρ11 2 2 0 0 2 -1 -1 2 0 0 2 -1 2 -1 0 -1 0 -1 orthogonal lifted from S3 ρ12 2 2 0 0 -1 2 -1 0 2 2 0 2 -1 -1 -1 0 -1 0 orthogonal lifted from S3 ρ13 2 -2 0 0 2 -1 -1 2i 0 0 -2i 1 -2 1 0 -i 0 i complex lifted from C4×S3 ρ14 2 -2 0 0 2 -1 -1 -2i 0 0 2i 1 -2 1 0 i 0 -i complex lifted from C4×S3 ρ15 2 -2 0 0 -1 2 -1 0 2i -2i 0 -2 1 1 i 0 -i 0 complex lifted from C4×S3 ρ16 2 -2 0 0 -1 2 -1 0 -2i 2i 0 -2 1 1 -i 0 i 0 complex lifted from C4×S3 ρ17 4 4 0 0 -2 -2 1 0 0 0 0 -2 -2 1 0 0 0 0 orthogonal lifted from S32 ρ18 4 -4 0 0 -2 -2 1 0 0 0 0 2 2 -1 0 0 0 0 orthogonal faithful

Permutation representations of C6.D6
On 12 points - transitive group 12T39
Generators in S12
```(1 3 5 7 9 11)(2 12 10 8 6 4)
(1 2 3 4 5 6 7 8 9 10 11 12)
(1 9)(3 7)(4 12)(6 10)```

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

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

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

`G:=TransitiveGroup(12,39);`

On 24 points - transitive group 24T75
Generators in S24
```(1 3 5 7 9 11)(2 12 10 8 6 4)(13 23 21 19 17 15)(14 16 18 20 22 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 24)(2 17)(3 22)(4 15)(5 20)(6 13)(7 18)(8 23)(9 16)(10 21)(11 14)(12 19)```

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

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

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

`G:=TransitiveGroup(24,75);`

Polynomial with Galois group C6.D6 over ℚ
actionf(x)Disc(f)
12T39x12-8x10+24x8-32x6+17x4-8x2+8241·74·234

Matrix representation of C6.D6 in GL4(ℤ) generated by

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

C6.D6 in GAP, Magma, Sage, TeX

`C_6.D_6`
`% in TeX`

`G:=Group("C6.D6");`
`// GroupNames label`

`G:=SmallGroup(72,21);`
`// by ID`

`G=gap.SmallGroup(72,21);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-3,-3,20,26,168,1204]);`
`// Polycyclic`

`G:=Group<a,b,c|a^6=c^2=1,b^6=a^3,b*a*b^-1=c*a*c=a^-1,c*b*c=b^5>;`
`// generators/relations`

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