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## G = D6.3D6order 144 = 24·32

### 3rd non-split extension by D6 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D6.3D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — D6.3D6
 Lower central C32 — C3×C6 — D6.3D6
 Upper central C1 — C2 — C22

Generators and relations for D6.3D6
G = < a,b,c,d | a6=b2=c6=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >

Subgroups: 284 in 88 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C4○D12, D42S3, S3×Dic3, C6.D6, C3⋊D12, C322Q8, C6×Dic3, C3×C3⋊D4, C327D4, D6.3D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D42S3, C2×S32, D6.3D6

Character table of D6.3D6

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B 12C 12D 12E size 1 1 2 6 18 2 2 4 3 3 6 6 18 2 2 2 2 4 4 4 4 12 6 6 6 6 12 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 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 -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 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 -1 1 1 linear of order 2 ρ5 1 1 1 -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 ρ6 1 1 -1 -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 ρ7 1 1 1 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 ρ8 1 1 -1 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 ρ9 2 2 -2 0 0 -1 2 -1 2 2 -2 0 0 1 -1 2 1 -2 1 1 -1 0 -1 1 -1 1 0 orthogonal lifted from D6 ρ10 2 2 2 0 0 -1 2 -1 2 2 2 0 0 -1 -1 2 -1 2 -1 -1 -1 0 -1 -1 -1 -1 0 orthogonal lifted from S3 ρ11 2 2 -2 -2 0 2 -1 -1 0 0 0 2 0 -2 2 -1 -2 1 1 1 -1 1 0 0 0 0 -1 orthogonal lifted from D6 ρ12 2 2 2 2 0 2 -1 -1 0 0 0 2 0 2 2 -1 2 -1 -1 -1 -1 -1 0 0 0 0 -1 orthogonal lifted from S3 ρ13 2 2 -2 0 0 -1 2 -1 -2 -2 2 0 0 1 -1 2 1 -2 1 1 -1 0 1 -1 1 -1 0 orthogonal lifted from D6 ρ14 2 2 2 -2 0 2 -1 -1 0 0 0 -2 0 2 2 -1 2 -1 -1 -1 -1 1 0 0 0 0 1 orthogonal lifted from D6 ρ15 2 2 -2 2 0 2 -1 -1 0 0 0 -2 0 -2 2 -1 -2 1 1 1 -1 -1 0 0 0 0 1 orthogonal lifted from D6 ρ16 2 2 2 0 0 -1 2 -1 -2 -2 -2 0 0 -1 -1 2 -1 2 -1 -1 -1 0 1 1 1 1 0 orthogonal lifted from D6 ρ17 2 -2 0 0 0 2 2 2 2i -2i 0 0 0 0 -2 -2 0 0 0 0 -2 0 2i 0 -2i 0 0 complex lifted from C4○D4 ρ18 2 -2 0 0 0 2 2 2 -2i 2i 0 0 0 0 -2 -2 0 0 0 0 -2 0 -2i 0 2i 0 0 complex lifted from C4○D4 ρ19 2 -2 0 0 0 -1 2 -1 2i -2i 0 0 0 -√-3 1 -2 √-3 0 √-3 -√-3 1 0 -i -√3 i √3 0 complex lifted from C4○D12 ρ20 2 -2 0 0 0 -1 2 -1 -2i 2i 0 0 0 √-3 1 -2 -√-3 0 -√-3 √-3 1 0 i -√3 -i √3 0 complex lifted from C4○D12 ρ21 2 -2 0 0 0 -1 2 -1 -2i 2i 0 0 0 -√-3 1 -2 √-3 0 √-3 -√-3 1 0 i √3 -i -√3 0 complex lifted from C4○D12 ρ22 2 -2 0 0 0 -1 2 -1 2i -2i 0 0 0 √-3 1 -2 -√-3 0 -√-3 √-3 1 0 -i √3 i -√3 0 complex lifted from C4○D12 ρ23 4 4 -4 0 0 -2 -2 1 0 0 0 0 0 2 -2 -2 2 2 -1 -1 1 0 0 0 0 0 0 orthogonal lifted from C2×S32 ρ24 4 4 4 0 0 -2 -2 1 0 0 0 0 0 -2 -2 -2 -2 -2 1 1 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ25 4 -4 0 0 0 4 -2 -2 0 0 0 0 0 0 -4 2 0 0 0 0 2 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ26 4 -4 0 0 0 -2 -2 1 0 0 0 0 0 -2√-3 2 2 2√-3 0 -√-3 √-3 -1 0 0 0 0 0 0 complex faithful ρ27 4 -4 0 0 0 -2 -2 1 0 0 0 0 0 2√-3 2 2 -2√-3 0 √-3 -√-3 -1 0 0 0 0 0 0 complex faithful

Permutation representations of D6.3D6
On 24 points - transitive group 24T205
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 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 14 15 16 17 18)(19 24 23 22 21 20)
(1 12 4 9)(2 7 5 10)(3 8 6 11)(13 23 16 20)(14 24 17 21)(15 19 18 22)```

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

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

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

`G:=TransitiveGroup(24,205);`

On 24 points - transitive group 24T221
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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 15)(8 14)(9 13)(10 18)(11 17)(12 16)
(1 9 5 7 3 11)(2 10 6 8 4 12)(13 23 15 19 17 21)(14 24 16 20 18 22)
(1 24 4 21)(2 19 5 22)(3 20 6 23)(7 18 10 15)(8 13 11 16)(9 14 12 17)```

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

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

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

`G:=TransitiveGroup(24,221);`

Matrix representation of D6.3D6 in GL4(𝔽7) generated by

 0 0 3 1 5 4 6 0 1 6 5 5 3 3 2 0
,
 3 3 1 5 2 5 2 0 6 6 1 2 3 1 5 5
,
 1 1 1 3 0 6 6 3 2 2 6 4 2 6 2 1
,
 0 0 1 3 4 0 1 4 3 3 5 1 1 6 3 2
`G:=sub<GL(4,GF(7))| [0,5,1,3,0,4,6,3,3,6,5,2,1,0,5,0],[3,2,6,3,3,5,6,1,1,2,1,5,5,0,2,5],[1,0,2,2,1,6,2,6,1,6,6,2,3,3,4,1],[0,4,3,1,0,0,3,6,1,1,5,3,3,4,1,2] >;`

D6.3D6 in GAP, Magma, Sage, TeX

`D_6._3D_6`
`% in TeX`

`G:=Group("D6.3D6");`
`// GroupNames label`

`G:=SmallGroup(144,147);`
`// by ID`

`G=gap.SmallGroup(144,147);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,490,3461]);`
`// Polycyclic`

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

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