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## G = D6⋊3Q8order 96 = 25·3

### 3rd semidirect product of D6 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D6⋊3Q8
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — D6⋊3Q8
 Lower central C3 — C2×C6 — D6⋊3Q8
 Upper central C1 — C22 — C2×Q8

Generators and relations for D63Q8
G = < a,b,c,d | a6=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 162 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, Dic3 [×3], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, C22⋊Q8, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], S3×C2×C4, C6×Q8, D63Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, C3⋊D4 [×2], C22×S3, C22⋊Q8, S3×Q8, Q83S3, C2×C3⋊D4, D63Q8

Character table of D63Q8

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 12A 12B 12C 12D 12E 12F size 1 1 1 1 6 6 2 2 2 4 4 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 -1 -2 -2 -2 2 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ10 2 2 -2 -2 0 0 2 2 -2 0 0 0 0 0 0 -2 2 -2 -2 0 0 2 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 -1 2 2 -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 2 -2 2 0 0 0 0 0 0 -2 2 -2 2 0 0 -2 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 2 2 0 0 -1 -2 -2 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 2 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 -2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 2 -2 -2 0 0 -1 -2 2 0 0 0 0 0 0 1 -1 1 -1 -√-3 -√-3 1 √-3 √-3 complex lifted from C3⋊D4 ρ18 2 2 -2 -2 0 0 -1 2 -2 0 0 0 0 0 0 1 -1 1 1 -√-3 √-3 -1 -√-3 √-3 complex lifted from C3⋊D4 ρ19 2 2 -2 -2 0 0 -1 -2 2 0 0 0 0 0 0 1 -1 1 -1 √-3 √-3 1 -√-3 -√-3 complex lifted from C3⋊D4 ρ20 2 2 -2 -2 0 0 -1 2 -2 0 0 0 0 0 0 1 -1 1 1 √-3 -√-3 -1 √-3 -√-3 complex lifted from C3⋊D4 ρ21 2 -2 -2 2 0 0 2 0 0 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 2 0 0 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 -2 0 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 -2 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2

Smallest permutation representation of D63Q8
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 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 44)(8 43)(9 48)(10 47)(11 46)(12 45)(19 28)(20 27)(21 26)(22 25)(23 30)(24 29)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)
(1 29 17 22)(2 30 18 23)(3 25 13 24)(4 26 14 19)(5 27 15 20)(6 28 16 21)(7 32 44 39)(8 33 45 40)(9 34 46 41)(10 35 47 42)(11 36 48 37)(12 31 43 38)
(1 41 17 34)(2 42 18 35)(3 37 13 36)(4 38 14 31)(5 39 15 32)(6 40 16 33)(7 27 44 20)(8 28 45 21)(9 29 46 22)(10 30 47 23)(11 25 48 24)(12 26 43 19)```

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

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

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

Matrix representation of D63Q8 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 9 1 0 0 0 0 0 0 1 12 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 7 12
,
 7 3 0 0 0 0 10 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 3 0 0 0 0 3 4
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 9 8

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,9,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,7,0,0,0,0,0,12],[7,10,0,0,0,0,3,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,3,0,0,0,0,3,4],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,9,0,0,0,0,0,8] >;`

D63Q8 in GAP, Magma, Sage, TeX

`D_6\rtimes_3Q_8`
`% in TeX`

`G:=Group("D6:3Q8");`
`// GroupNames label`

`G:=SmallGroup(96,153);`
`// by ID`

`G=gap.SmallGroup(96,153);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,86,2309]);`
`// Polycyclic`

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

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