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## G = D12.S3order 144 = 24·32

### 1st non-split extension by D12 of S3 acting via S3/C3=C2

Aliases: D12.1S3, C12.12D6, C6.13D12, C324SD16, C3⋊C82S3, C4.2S32, (C3×C6).9D4, C33(C24⋊C2), C31(D4.S3), (C3×D12).2C2, C6.2(C3⋊D4), C324Q82C2, (C3×C12).4C22, C2.5(C3⋊D12), (C3×C3⋊C8)⋊2C2, SmallGroup(144,59)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12.S3
 Lower central C32 — C3×C6 — C3×C12 — D12.S3
 Upper central C1 — C2 — C4

Generators and relations for D12.S3
G = < a,b,c,d | a12=b2=c3=1, d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Character table of D12.S3

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 24A 24B 24C 24D size 1 1 12 2 2 4 2 36 2 2 4 12 12 6 6 2 2 4 4 4 6 6 6 6 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 2 2 -2 2 -1 -1 2 0 2 -1 -1 1 1 0 0 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from D6 ρ6 2 2 2 2 -1 -1 2 0 2 -1 -1 -1 -1 0 0 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ7 2 2 0 2 2 2 -2 0 2 2 2 0 0 0 0 -2 -2 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ8 2 2 0 -1 2 -1 2 0 -1 2 -1 0 0 -2 -2 -1 -1 -1 -1 2 1 1 1 1 orthogonal lifted from D6 ρ9 2 2 0 -1 2 -1 2 0 -1 2 -1 0 0 2 2 -1 -1 -1 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 0 -1 2 -1 -2 0 -1 2 -1 0 0 0 0 1 1 1 1 -2 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ11 2 2 0 -1 2 -1 -2 0 -1 2 -1 0 0 0 0 1 1 1 1 -2 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ12 2 -2 0 2 2 2 0 0 -2 -2 -2 0 0 √-2 -√-2 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ13 2 -2 0 2 2 2 0 0 -2 -2 -2 0 0 -√-2 √-2 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ14 2 2 0 2 -1 -1 -2 0 2 -1 -1 √-3 -√-3 0 0 -2 -2 1 1 1 0 0 0 0 complex lifted from C3⋊D4 ρ15 2 2 0 2 -1 -1 -2 0 2 -1 -1 -√-3 √-3 0 0 -2 -2 1 1 1 0 0 0 0 complex lifted from C3⋊D4 ρ16 2 -2 0 -1 2 -1 0 0 1 -2 1 0 0 -√-2 √-2 √3 -√3 √3 -√3 0 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ3+ζ87-ζ85ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ32+ζ83-ζ8ζ32 complex lifted from C24⋊C2 ρ17 2 -2 0 -1 2 -1 0 0 1 -2 1 0 0 √-2 -√-2 √3 -√3 √3 -√3 0 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ32+ζ87-ζ85ζ32 complex lifted from C24⋊C2 ρ18 2 -2 0 -1 2 -1 0 0 1 -2 1 0 0 -√-2 √-2 -√3 √3 -√3 √3 0 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ3+ζ83-ζ8ζ3 complex lifted from C24⋊C2 ρ19 2 -2 0 -1 2 -1 0 0 1 -2 1 0 0 √-2 -√-2 -√3 √3 -√3 √3 0 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ32+ζ83-ζ8ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ3+ζ87-ζ85ζ3 complex lifted from C24⋊C2 ρ20 4 4 0 -2 -2 1 -4 0 -2 -2 1 0 0 0 0 2 2 -1 -1 2 0 0 0 0 orthogonal lifted from C3⋊D12 ρ21 4 4 0 -2 -2 1 4 0 -2 -2 1 0 0 0 0 -2 -2 1 1 -2 0 0 0 0 orthogonal lifted from S32 ρ22 4 -4 0 4 -2 -2 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ23 4 -4 0 -2 -2 1 0 0 2 2 -1 0 0 0 0 2√3 -2√3 -√3 √3 0 0 0 0 0 symplectic faithful, Schur index 2 ρ24 4 -4 0 -2 -2 1 0 0 2 2 -1 0 0 0 0 -2√3 2√3 √3 -√3 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D12.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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 40)(14 39)(15 38)(16 37)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 19 10 16 7 13 4 22)(2 20 11 17 8 14 5 23)(3 21 12 18 9 15 6 24)(25 41 34 38 31 47 28 44)(26 42 35 39 32 48 29 45)(27 43 36 40 33 37 30 46)```

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

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

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

Matrix representation of D12.S3 in GL6(𝔽73)

 46 0 0 0 0 0 27 27 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 46 19 0 0 0 0 27 27 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 22 0 0 0 0 0 57 63 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(73))| [46,27,0,0,0,0,0,27,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,27,0,0,0,0,19,27,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[22,57,0,0,0,0,0,63,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D12.S3 in GAP, Magma, Sage, TeX

`D_{12}.S_3`
`% in TeX`

`G:=Group("D12.S3");`
`// GroupNames label`

`G:=SmallGroup(144,59);`
`// by ID`

`G=gap.SmallGroup(144,59);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,31,218,50,490,3461]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^3=1,d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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