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## G = D6.D6order 144 = 24·32

### 1st 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.D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — D6⋊S3 — D6.D6
 Lower central C32 — C3×C6 — D6.D6
 Upper central C1 — C4

Generators and relations for D6.D6
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=c5 >

Subgroups: 288 in 88 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, D6⋊S3, C3⋊D12, C322Q8, S3×C12, C4×C3⋊S3, D6.D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, C2×S32, D6.D6

Character table of D6.D6

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J size 1 1 6 6 18 2 2 4 1 1 6 6 18 2 2 4 6 6 6 6 2 2 2 2 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 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 -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 -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 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 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 -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 -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 1 -1 -1 linear of order 2 ρ9 2 2 -2 0 0 2 -1 -1 -2 -2 2 0 0 -1 2 -1 1 1 0 0 -2 -2 1 1 1 1 0 0 -1 -1 orthogonal lifted from D6 ρ10 2 2 -2 0 0 2 -1 -1 2 2 -2 0 0 -1 2 -1 1 1 0 0 2 2 -1 -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 -1 2 -1 -1 -1 0 0 -2 -2 1 1 1 1 0 0 1 1 orthogonal lifted from D6 ρ12 2 2 0 -2 0 -1 2 -1 2 2 0 -2 0 2 -1 -1 0 0 1 1 -1 -1 2 2 -1 -1 1 1 0 0 orthogonal lifted from D6 ρ13 2 2 0 -2 0 -1 2 -1 -2 -2 0 2 0 2 -1 -1 0 0 1 1 1 1 -2 -2 1 1 -1 -1 0 0 orthogonal lifted from D6 ρ14 2 2 0 2 0 -1 2 -1 -2 -2 0 -2 0 2 -1 -1 0 0 -1 -1 1 1 -2 -2 1 1 1 1 0 0 orthogonal lifted from D6 ρ15 2 2 0 2 0 -1 2 -1 2 2 0 2 0 2 -1 -1 0 0 -1 -1 -1 -1 2 2 -1 -1 -1 -1 0 0 orthogonal lifted from S3 ρ16 2 2 2 0 0 2 -1 -1 2 2 2 0 0 -1 2 -1 -1 -1 0 0 2 2 -1 -1 -1 -1 0 0 -1 -1 orthogonal lifted from S3 ρ17 2 -2 0 0 0 2 2 2 2i -2i 0 0 0 -2 -2 -2 0 0 0 0 -2i 2i 2i -2i 2i -2i 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 0 0 0 2 2 2 -2i 2i 0 0 0 -2 -2 -2 0 0 0 0 2i -2i -2i 2i -2i 2i 0 0 0 0 complex lifted from C4○D4 ρ19 2 -2 0 0 0 -1 2 -1 2i -2i 0 0 0 -2 1 1 0 0 -√-3 √-3 i -i 2i -2i -i i √3 -√3 0 0 complex lifted from C4○D12 ρ20 2 -2 0 0 0 -1 2 -1 -2i 2i 0 0 0 -2 1 1 0 0 √-3 -√-3 -i i -2i 2i i -i √3 -√3 0 0 complex lifted from C4○D12 ρ21 2 -2 0 0 0 2 -1 -1 2i -2i 0 0 0 1 -2 1 √-3 -√-3 0 0 -2i 2i -i i -i i 0 0 -√3 √3 complex lifted from C4○D12 ρ22 2 -2 0 0 0 -1 2 -1 2i -2i 0 0 0 -2 1 1 0 0 √-3 -√-3 i -i 2i -2i -i i -√3 √3 0 0 complex lifted from C4○D12 ρ23 2 -2 0 0 0 -1 2 -1 -2i 2i 0 0 0 -2 1 1 0 0 -√-3 √-3 -i i -2i 2i i -i -√3 √3 0 0 complex lifted from C4○D12 ρ24 2 -2 0 0 0 2 -1 -1 -2i 2i 0 0 0 1 -2 1 √-3 -√-3 0 0 2i -2i i -i i -i 0 0 √3 -√3 complex lifted from C4○D12 ρ25 2 -2 0 0 0 2 -1 -1 2i -2i 0 0 0 1 -2 1 -√-3 √-3 0 0 -2i 2i -i i -i i 0 0 √3 -√3 complex lifted from C4○D12 ρ26 2 -2 0 0 0 2 -1 -1 -2i 2i 0 0 0 1 -2 1 -√-3 √-3 0 0 2i -2i i -i i -i 0 0 -√3 √3 complex lifted from C4○D12 ρ27 4 4 0 0 0 -2 -2 1 -4 -4 0 0 0 -2 -2 1 0 0 0 0 2 2 2 2 -1 -1 0 0 0 0 orthogonal lifted from C2×S32 ρ28 4 4 0 0 0 -2 -2 1 4 4 0 0 0 -2 -2 1 0 0 0 0 -2 -2 -2 -2 1 1 0 0 0 0 orthogonal lifted from S32 ρ29 4 -4 0 0 0 -2 -2 1 4i -4i 0 0 0 2 2 -1 0 0 0 0 2i -2i -2i 2i i -i 0 0 0 0 complex faithful ρ30 4 -4 0 0 0 -2 -2 1 -4i 4i 0 0 0 2 2 -1 0 0 0 0 -2i 2i 2i -2i -i i 0 0 0 0 complex faithful

Permutation representations of D6.D6
On 24 points - transitive group 24T228
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 24)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)
(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 18 7 24)(2 23 8 17)(3 16 9 22)(4 21 10 15)(5 14 11 20)(6 19 12 13)```

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

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

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

`G:=TransitiveGroup(24,228);`

Matrix representation of D6.D6 in GL4(𝔽5) generated by

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

D6.D6 in GAP, Magma, Sage, TeX

`D_6.D_6`
`% in TeX`

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

`G:=SmallGroup(144,141);`
`// by ID`

`G=gap.SmallGroup(144,141);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,50,490,3461]);`
`// 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=c^5>;`
`// generators/relations`

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