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## G = D6.6D6order 144 = 24·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D6.6D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C3⋊D12 — D6.6D6
 Lower central C32 — C3×C6 — D6.6D6
 Upper central C1 — C2 — C4

Generators and relations for D6.6D6
G = < a,b,c,d | a6=b2=1, c6=d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a3c5 >

Subgroups: 316 in 88 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×7], C6 [×2], C6 [×2], C2×C4 [×3], D4 [×3], Q8, C32, Dic3, Dic3 [×2], C12 [×2], C12 [×4], D6, D6 [×6], C2×C6, C4○D4, C3×S3, C3⋊S3 [×2], C3×C6, Dic6, C4×S3, C4×S3 [×4], D12 [×5], C3⋊D4 [×2], C2×C12, C3×Q8, C3×Dic3, C3×Dic3 [×2], C3×C12, S3×C6, C2×C3⋊S3 [×2], C4○D12, Q83S3, C6.D6 [×2], C3⋊D12 [×2], C3×Dic6, S3×C12, C12⋊S3, D6.6D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4, C22×S3 [×2], S32, C4○D12, Q83S3, C2×S32, D6.6D6

Character table of D6.6D6

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F 12G 12H 12I size 1 1 6 18 18 2 2 4 2 3 3 6 6 2 2 4 6 6 2 2 4 4 4 6 6 12 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 0 0 0 -1 2 -1 2 0 0 -2 -2 -1 2 -1 0 0 2 2 -1 -1 -1 0 0 1 1 orthogonal lifted from D6 ρ10 2 2 0 0 0 -1 2 -1 -2 0 0 2 -2 -1 2 -1 0 0 -2 -2 1 1 1 0 0 -1 1 orthogonal lifted from D6 ρ11 2 2 -2 0 0 2 -1 -1 2 -2 -2 0 0 2 -1 -1 1 1 -1 -1 -1 2 -1 1 1 0 0 orthogonal lifted from D6 ρ12 2 2 2 0 0 2 -1 -1 2 2 2 0 0 2 -1 -1 -1 -1 -1 -1 -1 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ13 2 2 0 0 0 -1 2 -1 -2 0 0 -2 2 -1 2 -1 0 0 -2 -2 1 1 1 0 0 1 -1 orthogonal lifted from D6 ρ14 2 2 -2 0 0 2 -1 -1 -2 2 2 0 0 2 -1 -1 1 1 1 1 1 -2 1 -1 -1 0 0 orthogonal lifted from D6 ρ15 2 2 2 0 0 2 -1 -1 -2 -2 -2 0 0 2 -1 -1 -1 -1 1 1 1 -2 1 1 1 0 0 orthogonal lifted from D6 ρ16 2 2 0 0 0 -1 2 -1 2 0 0 2 2 -1 2 -1 0 0 2 2 -1 -1 -1 0 0 -1 -1 orthogonal lifted from S3 ρ17 2 -2 0 0 0 2 2 2 0 -2i 2i 0 0 -2 -2 -2 0 0 0 0 0 0 0 -2i 2i 0 0 complex lifted from C4○D4 ρ18 2 -2 0 0 0 2 2 2 0 2i -2i 0 0 -2 -2 -2 0 0 0 0 0 0 0 2i -2i 0 0 complex lifted from C4○D4 ρ19 2 -2 0 0 0 2 -1 -1 0 2i -2i 0 0 -2 1 1 -√-3 √-3 -√3 √3 -√3 0 √3 -i i 0 0 complex lifted from C4○D12 ρ20 2 -2 0 0 0 2 -1 -1 0 -2i 2i 0 0 -2 1 1 √-3 -√-3 -√3 √3 -√3 0 √3 i -i 0 0 complex lifted from C4○D12 ρ21 2 -2 0 0 0 2 -1 -1 0 2i -2i 0 0 -2 1 1 √-3 -√-3 √3 -√3 √3 0 -√3 -i i 0 0 complex lifted from C4○D12 ρ22 2 -2 0 0 0 2 -1 -1 0 -2i 2i 0 0 -2 1 1 -√-3 √-3 √3 -√3 √3 0 -√3 i -i 0 0 complex lifted from C4○D12 ρ23 4 4 0 0 0 -2 -2 1 -4 0 0 0 0 -2 -2 1 0 0 2 2 -1 2 -1 0 0 0 0 orthogonal lifted from C2×S32 ρ24 4 -4 0 0 0 -2 4 -2 0 0 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ25 4 4 0 0 0 -2 -2 1 4 0 0 0 0 -2 -2 1 0 0 -2 -2 1 -2 1 0 0 0 0 orthogonal lifted from S32 ρ26 4 -4 0 0 0 -2 -2 1 0 0 0 0 0 2 2 -1 0 0 -2√3 2√3 √3 0 -√3 0 0 0 0 orthogonal faithful ρ27 4 -4 0 0 0 -2 -2 1 0 0 0 0 0 2 2 -1 0 0 2√3 -2√3 -√3 0 √3 0 0 0 0 orthogonal faithful

Permutation representations of D6.6D6
On 24 points - transitive group 24T222
Generators in S24
```(1 3 5 7 9 11)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 24 22 20 18 16)
(1 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(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 21 7 15)(2 20 8 14)(3 19 9 13)(4 18 10 24)(5 17 11 23)(6 16 12 22)```

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

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

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

`G:=TransitiveGroup(24,222);`

Matrix representation of D6.6D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 8 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12 1

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;`

D6.6D6 in GAP, Magma, Sage, TeX

`D_6._6D_6`
`% in TeX`

`G:=Group("D6.6D6");`
`// GroupNames label`

`G:=SmallGroup(144,142);`
`// by ID`

`G=gap.SmallGroup(144,142);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,55,116,50,490,3461]);`
`// Polycyclic`

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

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