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## G = S3×SD16order 96 = 25·3

### Direct product of S3 and SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — S3×SD16
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×D4 — S3×SD16
 Lower central C3 — C6 — C12 — S3×SD16
 Upper central C1 — C2 — C4 — SD16

Generators and relations for S3×SD16
G = < a,b,c,d | a3=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 186 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, SD16, SD16, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C2×SD16, S3×C8, C24⋊C2, D4.S3, Q82S3, C3×SD16, S3×D4, S3×Q8, S3×SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C22×S3, C2×SD16, S3×D4, S3×SD16

Character table of S3×SD16

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 12A 12B 24A 24B size 1 1 3 3 4 12 2 2 4 6 12 2 8 2 2 6 6 4 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 0 0 2 -2 0 -2 0 2 0 0 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 0 -2 0 -1 2 2 0 0 -1 1 -2 -2 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 0 0 2 0 -1 2 2 0 0 -1 -1 2 2 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 0 0 -2 0 -1 2 -2 0 0 -1 1 2 2 0 0 -1 1 -1 -1 orthogonal lifted from D6 ρ13 2 2 0 0 2 0 -1 2 -2 0 0 -1 -1 -2 -2 0 0 -1 1 1 1 orthogonal lifted from D6 ρ14 2 2 -2 -2 0 0 2 -2 0 2 0 2 0 0 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ15 2 -2 2 -2 0 0 2 0 0 0 0 -2 0 √-2 -√-2 √-2 -√-2 0 0 √-2 -√-2 complex lifted from SD16 ρ16 2 -2 -2 2 0 0 2 0 0 0 0 -2 0 √-2 -√-2 -√-2 √-2 0 0 √-2 -√-2 complex lifted from SD16 ρ17 2 -2 -2 2 0 0 2 0 0 0 0 -2 0 -√-2 √-2 √-2 -√-2 0 0 -√-2 √-2 complex lifted from SD16 ρ18 2 -2 2 -2 0 0 2 0 0 0 0 -2 0 -√-2 √-2 -√-2 √-2 0 0 -√-2 √-2 complex lifted from SD16 ρ19 4 4 0 0 0 0 -2 -4 0 0 0 -2 0 0 0 0 0 2 0 0 0 orthogonal lifted from S3×D4 ρ20 4 -4 0 0 0 0 -2 0 0 0 0 2 0 2√-2 -2√-2 0 0 0 0 -√-2 √-2 complex faithful ρ21 4 -4 0 0 0 0 -2 0 0 0 0 2 0 -2√-2 2√-2 0 0 0 0 √-2 -√-2 complex faithful

Permutation representations of S3×SD16
On 24 points - transitive group 24T142
Generators in S24
(1 10 21)(2 11 22)(3 12 23)(4 13 24)(5 14 17)(6 15 18)(7 16 19)(8 9 20)
(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(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 4)(3 7)(6 8)(9 15)(11 13)(12 16)(18 20)(19 23)(22 24)

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

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

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

G:=TransitiveGroup(24,142);

Matrix representation of S3×SD16 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 72 72 0 0 1 0
,
 72 0 0 0 0 72 0 0 0 0 1 0 0 0 72 72
,
 0 4 0 0 55 12 0 0 0 0 1 0 0 0 0 1
,
 1 48 0 0 0 72 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,1,0,0,72,0],[72,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72],[0,55,0,0,4,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,48,72,0,0,0,0,1,0,0,0,0,1] >;

S3×SD16 in GAP, Magma, Sage, TeX

S_3\times {\rm SD}_{16}
% in TeX

G:=Group("S3xSD16");
// GroupNames label

G:=SmallGroup(96,120);
// by ID

G=gap.SmallGroup(96,120);
# by ID

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

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

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