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## G = S3×Q16order 96 = 25·3

### Direct product of S3 and Q16

Aliases: S3×Q16, C8.9D6, Q8.8D6, D6.14D4, Dic125C2, C12.8C23, C24.7C22, Dic3.5D4, Dic6.4C22, C32(C2×Q16), (S3×Q8).C2, (S3×C8).1C2, (C3×Q16)⋊2C2, C3⋊Q163C2, C6.34(C2×D4), C2.22(S3×D4), C3⋊C8.7C22, C4.8(C22×S3), (C3×Q8).3C22, (C4×S3).11C22, SmallGroup(96,124)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — S3×Q16
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×Q8 — S3×Q16
 Lower central C3 — C6 — C12 — S3×Q16
 Upper central C1 — C2 — C4 — Q16

Generators and relations for S3×Q16
G = < a,b,c,d | a3=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 130 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×5], C22, S3 [×2], C6, C8, C8, C2×C4 [×3], Q8 [×2], Q8 [×4], Dic3, Dic3 [×2], C12, C12 [×2], D6, C2×C8, Q16, Q16 [×3], C2×Q8 [×2], C3⋊C8, C24, Dic6 [×2], Dic6 [×2], C4×S3, C4×S3 [×2], C3×Q8 [×2], C2×Q16, S3×C8, Dic12, C3⋊Q16 [×2], C3×Q16, S3×Q8 [×2], S3×Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], Q16 [×2], C2×D4, C22×S3, C2×Q16, S3×D4, S3×Q16

Character table of S3×Q16

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6 8A 8B 8C 8D 12A 12B 12C 24A 24B size 1 1 3 3 2 2 4 4 6 12 12 2 2 2 6 6 4 8 8 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 2 2 2 -2 0 0 -2 0 0 2 0 0 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 0 -1 2 2 2 0 0 0 -1 2 2 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 -2 -2 2 -2 0 0 2 0 0 2 0 0 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 0 0 -1 2 -2 2 0 0 0 -1 -2 -2 0 0 -1 -1 1 1 1 orthogonal lifted from D6 ρ13 2 2 0 0 -1 2 -2 -2 0 0 0 -1 2 2 0 0 -1 1 1 -1 -1 orthogonal lifted from D6 ρ14 2 2 0 0 -1 2 2 -2 0 0 0 -1 -2 -2 0 0 -1 1 -1 1 1 orthogonal lifted from D6 ρ15 2 -2 2 -2 2 0 0 0 0 0 0 -2 -√2 √2 -√2 √2 0 0 0 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ16 2 -2 2 -2 2 0 0 0 0 0 0 -2 √2 -√2 √2 -√2 0 0 0 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ17 2 -2 -2 2 2 0 0 0 0 0 0 -2 √2 -√2 -√2 √2 0 0 0 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ18 2 -2 -2 2 2 0 0 0 0 0 0 -2 -√2 √2 √2 -√2 0 0 0 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ19 4 4 0 0 -2 -4 0 0 0 0 0 -2 0 0 0 0 2 0 0 0 0 orthogonal lifted from S3×D4 ρ20 4 -4 0 0 -2 0 0 0 0 0 0 2 2√2 -2√2 0 0 0 0 0 √2 -√2 symplectic faithful, Schur index 2 ρ21 4 -4 0 0 -2 0 0 0 0 0 0 2 -2√2 2√2 0 0 0 0 0 -√2 √2 symplectic faithful, Schur index 2

Smallest permutation representation of S3×Q16
On 48 points
Generators in S48
(1 31 46)(2 32 47)(3 25 48)(4 26 41)(5 27 42)(6 28 43)(7 29 44)(8 30 45)(9 33 24)(10 34 17)(11 35 18)(12 36 19)(13 37 20)(14 38 21)(15 39 22)(16 40 23)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)
(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 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 47 21 43)(18 46 22 42)(19 45 23 41)(20 44 24 48)(25 37 29 33)(26 36 30 40)(27 35 31 39)(28 34 32 38)

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

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

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

S3×Q16 is a maximal subgroup of
SD32⋊S3  Q32⋊S3  D12.30D4  D8.10D6  SD16.D6  C24.23D6  Dic6.9D6  Dic10.D6  D15⋊Q16
S3×Q16 is a maximal quotient of
Dic34Q16  Dic3.1Q16  Q83Dic6  Dic3⋊Q16  D6⋊Q16  D6.Q16  D61Q16  Dic35Q16  C242Q8  Dic3.Q16  D6.2Q16  D62Q16  Dic33Q16  C24.26D4  D65Q16  D63Q16  C24.23D6  Dic6.9D6  Dic10.D6  D15⋊Q16

Matrix representation of S3×Q16 in GL4(𝔽7) generated by

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

S3×Q16 in GAP, Magma, Sage, TeX

S_3\times Q_{16}
% in TeX

G:=Group("S3xQ16");
// GroupNames label

G:=SmallGroup(96,124);
// by ID

G=gap.SmallGroup(96,124);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,116,86,297,159,69,2309]);
// Polycyclic

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

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