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## G = D8⋊S3order 96 = 25·3

### 2nd semidirect product of D8 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for D8⋊S3
G = < a,b,c,d | a8=b2=c3=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 194 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, Dic3, Dic3, C12, D6, D6, C2×C6, M4(2), D8, D8, SD16, C2×D4, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C8⋊C22, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, D8⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8⋊C22, S3×D4, D8⋊S3

Character table of D8⋊S3

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 6A 6B 6C 8A 8B 12 24A 24B size 1 1 4 4 6 12 2 2 6 12 2 8 8 4 12 4 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 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 0 0 2 0 2 -2 -2 0 2 0 0 0 0 -2 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -1 2 0 0 -1 -1 -1 2 0 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 -2 -2 0 0 -1 2 0 0 -1 1 1 2 0 -1 -1 -1 orthogonal lifted from D6 ρ12 2 2 2 -2 0 0 -1 2 0 0 -1 -1 1 -2 0 -1 1 1 orthogonal lifted from D6 ρ13 2 2 -2 2 0 0 -1 2 0 0 -1 1 -1 -2 0 -1 1 1 orthogonal lifted from D6 ρ14 2 2 0 0 -2 0 2 -2 2 0 2 0 0 0 0 -2 0 0 orthogonal lifted from D4 ρ15 4 4 0 0 0 0 -2 -4 0 0 -2 0 0 0 0 2 0 0 orthogonal lifted from S3×D4 ρ16 4 -4 0 0 0 0 4 0 0 0 -4 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ17 4 -4 0 0 0 0 -2 0 0 0 2 0 0 0 0 0 √-6 -√-6 complex faithful ρ18 4 -4 0 0 0 0 -2 0 0 0 2 0 0 0 0 0 -√-6 √-6 complex faithful

Permutation representations of D8⋊S3
On 24 points - transitive group 24T141
Generators in S24
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 17 16)(8 18 9)
(1 5)(3 7)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)```

`G:=sub<Sym(24)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,17,16)(8,18,9), (1,5)(3,7)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)>;`

`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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,17,16)(8,18,9), (1,5)(3,7)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21) );`

`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)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(17,21),(18,20),(22,24)], [(1,19,10),(2,20,11),(3,21,12),(4,22,13),(5,23,14),(6,24,15),(7,17,16),(8,18,9)], [(1,5),(3,7),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21)]])`

`G:=TransitiveGroup(24,141);`

Matrix representation of D8⋊S3 in GL4(𝔽5) generated by

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

D8⋊S3 in GAP, Magma, Sage, TeX

`D_8\rtimes S_3`
`% in TeX`

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

`G:=SmallGroup(96,118);`
`// by ID`

`G=gap.SmallGroup(96,118);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,362,116,297,159,69,2309]);`
`// 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,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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