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## G = D6.4D6order 144 = 24·32

### 4th non-split extension by D6 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D6.4D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — D6.4D6
 Lower central C32 — C3×C6 — D6.4D6
 Upper central C1 — C2 — C22

Generators and relations for D6.4D6
G = < a,b,c,d | a6=b2=1, c6=d2=a3, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a4b, dcd-1=c5 >

Subgroups: 256 in 88 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3⋊D4, C3×D4, C3×Dic3, C3⋊Dic3, S3×C6, C62, D42S3, S3×Dic3, D6⋊S3, C322Q8, C3×C3⋊D4, C2×C3⋊Dic3, D6.4D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, D42S3, C2×S32, D6.4D6

Character table of D6.4D6

 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 size 1 1 2 6 6 2 2 4 6 6 9 9 18 2 2 4 4 4 4 4 12 12 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 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 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 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 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 linear of order 2 ρ9 2 2 2 0 2 2 -1 -1 2 0 0 0 0 2 -1 -1 -1 2 -1 -1 0 -1 0 -1 orthogonal lifted from S3 ρ10 2 2 2 2 0 -1 2 -1 0 2 0 0 0 -1 2 -1 2 -1 -1 -1 -1 0 -1 0 orthogonal lifted from S3 ρ11 2 2 -2 0 2 2 -1 -1 -2 0 0 0 0 2 -1 1 1 -2 1 -1 0 -1 0 1 orthogonal lifted from D6 ρ12 2 2 -2 2 0 -1 2 -1 0 -2 0 0 0 -1 2 1 -2 1 1 -1 -1 0 1 0 orthogonal lifted from D6 ρ13 2 2 2 0 -2 2 -1 -1 -2 0 0 0 0 2 -1 -1 -1 2 -1 -1 0 1 0 1 orthogonal lifted from D6 ρ14 2 2 -2 0 -2 2 -1 -1 2 0 0 0 0 2 -1 1 1 -2 1 -1 0 1 0 -1 orthogonal lifted from D6 ρ15 2 2 -2 -2 0 -1 2 -1 0 2 0 0 0 -1 2 1 -2 1 1 -1 1 0 -1 0 orthogonal lifted from D6 ρ16 2 2 2 -2 0 -1 2 -1 0 -2 0 0 0 -1 2 -1 2 -1 -1 -1 1 0 1 0 orthogonal lifted from D6 ρ17 2 -2 0 0 0 2 2 2 0 0 2i -2i 0 -2 -2 0 0 0 0 -2 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 0 0 0 2 2 2 0 0 -2i 2i 0 -2 -2 0 0 0 0 -2 0 0 0 0 complex lifted from C4○D4 ρ19 4 4 -4 0 0 -2 -2 1 0 0 0 0 0 -2 -2 -1 2 2 -1 1 0 0 0 0 orthogonal lifted from C2×S32 ρ20 4 4 4 0 0 -2 -2 1 0 0 0 0 0 -2 -2 1 -2 -2 1 1 0 0 0 0 orthogonal lifted from S32 ρ21 4 -4 0 0 0 -2 4 -2 0 0 0 0 0 2 -4 0 0 0 0 2 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ22 4 -4 0 0 0 -2 -2 1 0 0 0 0 0 2 2 -3 0 0 3 -1 0 0 0 0 symplectic faithful, Schur index 2 ρ23 4 -4 0 0 0 4 -2 -2 0 0 0 0 0 -4 2 0 0 0 0 2 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ24 4 -4 0 0 0 -2 -2 1 0 0 0 0 0 2 2 3 0 0 -3 -1 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of D6.4D6
On 24 points - transitive group 24T206
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 6)(2 9)(3 8)(4 11)(5 10)(7 12)(13 20)(14 19)(15 22)(16 21)(17 24)(18 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 16 7 22)(2 21 8 15)(3 14 9 20)(4 19 10 13)(5 24 11 18)(6 17 12 23)```

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

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

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

`G:=TransitiveGroup(24,206);`

Matrix representation of D6.4D6 in GL4(𝔽5) generated by

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

D6.4D6 in GAP, Magma, Sage, TeX

`D_6._4D_6`
`% in TeX`

`G:=Group("D6.4D6");`
`// GroupNames label`

`G:=SmallGroup(144,148);`
`// by ID`

`G=gap.SmallGroup(144,148);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,490,3461]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^2=1,c^6=d^2=a^3,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^4*b,d*c*d^-1=c^5>;`
`// generators/relations`

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