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## G = D8.S3order 96 = 25·3

### The non-split extension by D8 of S3 acting via S3/C3=C2

Aliases: D8.S3, C8.5D6, C6.9D8, C32SD32, C12.4D4, Dic123C2, C24.3C22, C3⋊C162C2, (C3×D8).1C2, C2.5(D4⋊S3), C4.2(C3⋊D4), SmallGroup(96,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — D8.S3
 Chief series C1 — C3 — C6 — C12 — C24 — Dic12 — D8.S3
 Lower central C3 — C6 — C12 — C24 — D8.S3
 Upper central C1 — C2 — C4 — C8 — D8

Generators and relations for D8.S3
G = < a,b,c,d | a8=b2=c3=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Character table of D8.S3

 class 1 2A 2B 3 4A 4B 6A 6B 6C 8A 8B 12 16A 16B 16C 16D 24A 24B size 1 1 8 2 2 24 2 8 8 2 2 4 6 6 6 6 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 2 -1 2 0 -1 -1 -1 2 2 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ6 2 2 0 2 2 0 2 0 0 -2 -2 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ7 2 2 -2 -1 2 0 -1 1 1 2 2 -1 0 0 0 0 -1 -1 orthogonal lifted from D6 ρ8 2 2 0 2 -2 0 2 0 0 0 0 -2 √2 -√2 √2 -√2 0 0 orthogonal lifted from D8 ρ9 2 2 0 2 -2 0 2 0 0 0 0 -2 -√2 √2 -√2 √2 0 0 orthogonal lifted from D8 ρ10 2 2 0 -1 2 0 -1 -√-3 √-3 -2 -2 -1 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ11 2 2 0 -1 2 0 -1 √-3 -√-3 -2 -2 -1 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ12 2 -2 0 2 0 0 -2 0 0 √2 -√2 0 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 √2 -√2 complex lifted from SD32 ρ13 2 -2 0 2 0 0 -2 0 0 -√2 √2 0 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 -√2 √2 complex lifted from SD32 ρ14 2 -2 0 2 0 0 -2 0 0 -√2 √2 0 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 -√2 √2 complex lifted from SD32 ρ15 2 -2 0 2 0 0 -2 0 0 √2 -√2 0 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 √2 -√2 complex lifted from SD32 ρ16 4 4 0 -2 -4 0 -2 0 0 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ17 4 -4 0 -2 0 0 2 0 0 -2√2 2√2 0 0 0 0 0 √2 -√2 symplectic faithful, Schur index 2 ρ18 4 -4 0 -2 0 0 2 0 0 2√2 -2√2 0 0 0 0 0 -√2 √2 symplectic faithful, Schur index 2

Smallest permutation representation of D8.S3
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 6)(4 5)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)(25 32)(26 31)(27 30)(28 29)(33 34)(35 40)(36 39)(37 38)(42 48)(43 47)(44 46)
(1 34 25)(2 35 26)(3 36 27)(4 37 28)(5 38 29)(6 39 30)(7 40 31)(8 33 32)(9 22 48)(10 23 41)(11 24 42)(12 17 43)(13 18 44)(14 19 45)(15 20 46)(16 21 47)
(1 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 47 37 43)(34 46 38 42)(35 45 39 41)(36 44 40 48)```

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

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

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

D8.S3 is a maximal subgroup of
D8⋊D6  D163S3  S3×SD32  SD32⋊S3  D8.D6  Q16.D6  D8.9D6  D8.D9  D24.S3  C323SD32  C328SD32  C40.D6  D40.S3  D8.D15
D8.S3 is a maximal quotient of
C6.SD32  C6.Q32  D81Dic3  D8.D9  D24.S3  C323SD32  C328SD32  C40.D6  D40.S3  D8.D15

Matrix representation of D8.S3 in GL4(𝔽7) generated by

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

D8.S3 in GAP, Magma, Sage, TeX

`D_8.S_3`
`% in TeX`

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

`G:=SmallGroup(96,34);`
`// by ID`

`G=gap.SmallGroup(96,34);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,73,218,116,122,579,297,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^3=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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