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## G = D8⋊3S3order 96 = 25·3

### The semidirect product of D8 and S3 acting through Inn(D8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D8⋊3S3
 Chief series C1 — C3 — C6 — C12 — C4×S3 — D4⋊2S3 — D8⋊3S3
 Lower central C3 — C6 — C12 — D8⋊3S3
 Upper central C1 — C2 — C4 — D8

Generators and relations for D83S3
G = < a,b,c,d | a8=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 146 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Dic3, Dic3, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C4○D8, S3×C8, Dic12, D4.S3, C3×D8, D42S3, D83S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C4○D8, S3×D4, D83S3

Character table of D83S3

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 8A 8B 8C 8D 12 24A 24B size 1 1 4 4 6 2 2 3 3 12 12 2 8 8 2 2 6 6 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 trivial ρ2 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 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 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 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 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 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 linear of order 2 ρ9 2 2 0 0 -2 2 -2 2 2 0 0 2 0 0 0 0 0 0 -2 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 0 -1 2 0 0 0 0 -1 -1 1 -2 -2 0 0 -1 1 1 orthogonal lifted from D6 ρ11 2 2 -2 -2 0 -1 2 0 0 0 0 -1 1 1 2 2 0 0 -1 -1 -1 orthogonal lifted from D6 ρ12 2 2 0 0 2 2 -2 -2 -2 0 0 2 0 0 0 0 0 0 -2 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 -1 2 0 0 0 0 -1 -1 -1 2 2 0 0 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 -2 2 0 -1 2 0 0 0 0 -1 1 -1 -2 -2 0 0 -1 1 1 orthogonal lifted from D6 ρ15 2 -2 0 0 0 2 0 -2i 2i 0 0 -2 0 0 √2 -√2 √-2 -√-2 0 √2 -√2 complex lifted from C4○D8 ρ16 2 -2 0 0 0 2 0 2i -2i 0 0 -2 0 0 -√2 √2 √-2 -√-2 0 -√2 √2 complex lifted from C4○D8 ρ17 2 -2 0 0 0 2 0 -2i 2i 0 0 -2 0 0 -√2 √2 -√-2 √-2 0 -√2 √2 complex lifted from C4○D8 ρ18 2 -2 0 0 0 2 0 2i -2i 0 0 -2 0 0 √2 -√2 -√-2 √-2 0 √2 -√2 complex lifted from C4○D8 ρ19 4 4 0 0 0 -2 -4 0 0 0 0 -2 0 0 0 0 0 0 2 0 0 orthogonal lifted from S3×D4 ρ20 4 -4 0 0 0 -2 0 0 0 0 0 2 0 0 2√2 -2√2 0 0 0 -√2 √2 symplectic faithful, Schur index 2 ρ21 4 -4 0 0 0 -2 0 0 0 0 0 2 0 0 -2√2 2√2 0 0 0 √2 -√2 symplectic faithful, Schur index 2

Smallest permutation representation of D83S3
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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(25 42)(26 41)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)
(1 35 46)(2 36 47)(3 37 48)(4 38 41)(5 39 42)(6 40 43)(7 33 44)(8 34 45)(9 26 20)(10 27 21)(11 28 22)(12 29 23)(13 30 24)(14 31 17)(15 32 18)(16 25 19)
(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 21)(18 22)(19 23)(20 24)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)```

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

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

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

Matrix representation of D83S3 in GL4(𝔽7) generated by

 0 2 5 5 3 5 6 5 1 1 5 5 6 1 4 3
,
 4 4 2 0 2 0 4 6 6 6 2 2 4 3 5 1
,
 0 0 5 2 5 3 0 3 2 5 0 3 5 5 1 2
,
 6 0 5 3 5 5 2 2 1 4 6 1 3 5 1 4
`G:=sub<GL(4,GF(7))| [0,3,1,6,2,5,1,1,5,6,5,4,5,5,5,3],[4,2,6,4,4,0,6,3,2,4,2,5,0,6,2,1],[0,5,2,5,0,3,5,5,5,0,0,1,2,3,3,2],[6,5,1,3,0,5,4,5,5,2,6,1,3,2,1,4] >;`

D83S3 in GAP, Magma, Sage, TeX

`D_8\rtimes_3S_3`
`% in TeX`

`G:=Group("D8:3S3");`
`// GroupNames label`

`G:=SmallGroup(96,119);`
`// by ID`

`G=gap.SmallGroup(96,119);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,362,116,297,159,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

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