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## G = D6⋊3D4order 96 = 25·3

### 3rd semidirect product of D6 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D6⋊3D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — D6⋊3D4
 Lower central C3 — C2×C6 — D6⋊3D4
 Upper central C1 — C22 — C2×D4

Generators and relations for D63D4
G = < a,b,c,d | a6=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a3b, dcd=c-1 >

Subgroups: 226 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×10], S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, Dic3 [×3], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×6], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], C4⋊D4, C4⋊Dic3, C6.D4 [×2], S3×C2×C4, C2×C3⋊D4 [×2], C6×D4, D63D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, S3×D4, D42S3, C2×C3⋊D4, D63D4

Character table of D63D4

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 12A 12B size 1 1 1 1 4 4 6 6 2 2 2 6 6 12 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 2 -2 2 0 0 0 0 0 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 0 0 0 0 2 -2 2 0 0 0 0 -2 -2 2 0 0 0 0 -2 2 orthogonal lifted from D4 ρ11 2 2 2 2 2 -2 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ12 2 -2 -2 2 0 0 0 0 2 2 -2 0 0 0 0 -2 -2 2 0 0 0 0 2 -2 orthogonal lifted from D4 ρ13 2 2 2 2 2 2 0 0 -1 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 2 2 -2 -2 0 0 -1 2 2 0 0 0 0 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ15 2 2 2 2 -2 2 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 1 -1 -1 1 1 1 orthogonal lifted from D6 ρ16 2 -2 2 -2 0 0 -2 2 2 0 0 0 0 0 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 -2 -2 2 0 0 0 0 -1 -2 2 0 0 0 0 1 1 -1 -√-3 -√-3 √-3 √-3 1 -1 complex lifted from C3⋊D4 ρ18 2 -2 -2 2 0 0 0 0 -1 -2 2 0 0 0 0 1 1 -1 √-3 √-3 -√-3 -√-3 1 -1 complex lifted from C3⋊D4 ρ19 2 -2 -2 2 0 0 0 0 -1 2 -2 0 0 0 0 1 1 -1 √-3 -√-3 √-3 -√-3 -1 1 complex lifted from C3⋊D4 ρ20 2 -2 -2 2 0 0 0 0 -1 2 -2 0 0 0 0 1 1 -1 -√-3 √-3 -√-3 √-3 -1 1 complex lifted from C3⋊D4 ρ21 2 2 -2 -2 0 0 0 0 2 0 0 2i -2i 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 -2 -2 0 0 0 0 2 0 0 -2i 2i 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 4 -4 0 0 0 0 -2 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 -4 -4 0 0 0 0 -2 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2

Smallest permutation representation of D63D4
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 6)(2 5)(3 4)(8 12)(9 11)(13 14)(15 18)(16 17)(20 24)(21 23)(25 29)(26 28)(31 34)(32 33)(35 36)(37 38)(39 42)(40 41)(44 48)(45 47)
(1 41 17 36)(2 42 18 31)(3 37 13 32)(4 38 14 33)(5 39 15 34)(6 40 16 35)(7 27 46 22)(8 28 47 23)(9 29 48 24)(10 30 43 19)(11 25 44 20)(12 26 45 21)
(1 48)(2 43)(3 44)(4 45)(5 46)(6 47)(7 15)(8 16)(9 17)(10 18)(11 13)(12 14)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)```

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

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

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

Matrix representation of D63D4 in GL4(𝔽13) generated by

 0 1 0 0 12 1 0 0 0 0 12 0 0 0 0 12
,
 1 12 0 0 0 12 0 0 0 0 12 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 5 0 0 0 0 8
,
 2 9 0 0 4 11 0 0 0 0 0 8 0 0 5 0
`G:=sub<GL(4,GF(13))| [0,12,0,0,1,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,12,12,0,0,0,0,12,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,8],[2,4,0,0,9,11,0,0,0,0,0,5,0,0,8,0] >;`

D63D4 in GAP, Magma, Sage, TeX

`D_6\rtimes_3D_4`
`% in TeX`

`G:=Group("D6:3D4");`
`// GroupNames label`

`G:=SmallGroup(96,145);`
`// by ID`

`G=gap.SmallGroup(96,145);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,2309]);`
`// Polycyclic`

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

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