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## G = D6.D4order 96 = 25·3

### 2nd non-split extension by D6 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D6.D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — D6.D4
 Lower central C3 — C2×C6 — D6.D4
 Upper central C1 — C22 — C4⋊C4

Generators and relations for D6.D4
G = < a,b,c,d | a6=b2=c4=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >

Subgroups: 202 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], S3 [×3], C6 [×3], C2×C4 [×3], C2×C4 [×4], D4 [×2], C23 [×2], Dic3 [×2], C12 [×3], D6 [×2], D6 [×5], C2×C6, C22⋊C4 [×3], C4⋊C4, C4⋊C4, C22×C4, C2×D4, C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C2×C12 [×3], C22×S3 [×2], C22.D4, Dic3⋊C4, D6⋊C4 [×3], C3×C4⋊C4, S3×C2×C4, C2×D12, D6.D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C22×S3, C22.D4, C4○D12, S3×D4, Q83S3, D6.D4

Character table of D6.D4

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

Smallest permutation representation of D6.D4
On 48 points
Generators in S48
```(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)
(1 8)(2 7)(3 12)(4 11)(5 10)(6 9)(13 45)(14 44)(15 43)(16 48)(17 47)(18 46)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)
(1 29 17 24)(2 30 18 19)(3 25 13 20)(4 26 14 21)(5 27 15 22)(6 28 16 23)(7 34 46 39)(8 35 47 40)(9 36 48 41)(10 31 43 42)(11 32 44 37)(12 33 45 38)
(1 12 4 9)(2 7 5 10)(3 8 6 11)(13 47 16 44)(14 48 17 45)(15 43 18 46)(19 34 22 31)(20 35 23 32)(21 36 24 33)(25 40 28 37)(26 41 29 38)(27 42 30 39)```

`G:=sub<Sym(48)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,8)(2,7)(3,12)(4,11)(5,10)(6,9)(13,45)(14,44)(15,43)(16,48)(17,47)(18,46)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31), (1,29,17,24)(2,30,18,19)(3,25,13,20)(4,26,14,21)(5,27,15,22)(6,28,16,23)(7,34,46,39)(8,35,47,40)(9,36,48,41)(10,31,43,42)(11,32,44,37)(12,33,45,38), (1,12,4,9)(2,7,5,10)(3,8,6,11)(13,47,16,44)(14,48,17,45)(15,43,18,46)(19,34,22,31)(20,35,23,32)(21,36,24,33)(25,40,28,37)(26,41,29,38)(27,42,30,39)>;`

`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)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,8)(2,7)(3,12)(4,11)(5,10)(6,9)(13,45)(14,44)(15,43)(16,48)(17,47)(18,46)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31), (1,29,17,24)(2,30,18,19)(3,25,13,20)(4,26,14,21)(5,27,15,22)(6,28,16,23)(7,34,46,39)(8,35,47,40)(9,36,48,41)(10,31,43,42)(11,32,44,37)(12,33,45,38), (1,12,4,9)(2,7,5,10)(3,8,6,11)(13,47,16,44)(14,48,17,45)(15,43,18,46)(19,34,22,31)(20,35,23,32)(21,36,24,33)(25,40,28,37)(26,41,29,38)(27,42,30,39) );`

`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),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,8),(2,7),(3,12),(4,11),(5,10),(6,9),(13,45),(14,44),(15,43),(16,48),(17,47),(18,46),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31)], [(1,29,17,24),(2,30,18,19),(3,25,13,20),(4,26,14,21),(5,27,15,22),(6,28,16,23),(7,34,46,39),(8,35,47,40),(9,36,48,41),(10,31,43,42),(11,32,44,37),(12,33,45,38)], [(1,12,4,9),(2,7,5,10),(3,8,6,11),(13,47,16,44),(14,48,17,45),(15,43,18,46),(19,34,22,31),(20,35,23,32),(21,36,24,33),(25,40,28,37),(26,41,29,38),(27,42,30,39)])`

Matrix representation of D6.D4 in GL4(𝔽13) generated by

 0 1 0 0 12 1 0 0 0 0 1 0 0 0 0 1
,
 3 3 0 0 6 10 0 0 0 0 12 0 0 0 0 12
,
 11 4 0 0 9 2 0 0 0 0 0 1 0 0 12 0
,
 3 7 0 0 6 10 0 0 0 0 12 0 0 0 0 1
`G:=sub<GL(4,GF(13))| [0,12,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[3,6,0,0,3,10,0,0,0,0,12,0,0,0,0,12],[11,9,0,0,4,2,0,0,0,0,0,12,0,0,1,0],[3,6,0,0,7,10,0,0,0,0,12,0,0,0,0,1] >;`

D6.D4 in GAP, Magma, Sage, TeX

`D_6.D_4`
`% in TeX`

`G:=Group("D6.D4");`
`// GroupNames label`

`G:=SmallGroup(96,101);`
`// by ID`

`G=gap.SmallGroup(96,101);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,55,218,188,86,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^2=c^4=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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