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## G = S3×D8order 96 = 25·3

### Direct product of S3 and D8

Aliases: S3×D8, C84D6, D41D6, D244C2, C242C22, D6.12D4, D121C22, C12.1C23, Dic3.3D4, C32(C2×D8), (S3×C8)⋊1C2, D4⋊S31C2, (C3×D8)⋊2C2, (S3×D4)⋊1C2, C3⋊C85C22, C2.15(S3×D4), C6.27(C2×D4), (C3×D4)⋊1C22, C4.1(C22×S3), (C4×S3).7C22, SmallGroup(96,117)

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×D8
G = < a,b,c,d | a3=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 242 in 76 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2×C4, D4, D4, C23, Dic3, C12, D6, D6, C2×C6, C2×C8, D8, D8, C2×D4, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C2×D8, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, S3×D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C22×S3, C2×D8, S3×D4, S3×D8

Character table of S3×D8

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 8A 8B 8C 8D 12 24A 24B size 1 1 3 3 4 4 12 12 2 2 6 2 8 8 2 2 6 6 4 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 0 0 2 -2 -2 2 0 0 0 0 0 0 -2 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 0 0 0 0 2 -2 2 2 0 0 0 0 0 0 -2 0 0 orthogonal lifted from D4 ρ11 2 2 0 0 -2 2 0 0 -1 2 0 -1 1 -1 -2 -2 0 0 -1 1 1 orthogonal lifted from D6 ρ12 2 2 0 0 -2 -2 0 0 -1 2 0 -1 1 1 2 2 0 0 -1 -1 -1 orthogonal lifted from D6 ρ13 2 2 0 0 2 -2 0 0 -1 2 0 -1 -1 1 -2 -2 0 0 -1 1 1 orthogonal lifted from D6 ρ14 2 2 0 0 2 2 0 0 -1 2 0 -1 -1 -1 2 2 0 0 -1 -1 -1 orthogonal lifted from S3 ρ15 2 -2 -2 2 0 0 0 0 2 0 0 -2 0 0 √2 -√2 √2 -√2 0 √2 -√2 orthogonal lifted from D8 ρ16 2 -2 2 -2 0 0 0 0 2 0 0 -2 0 0 -√2 √2 √2 -√2 0 -√2 √2 orthogonal lifted from D8 ρ17 2 -2 2 -2 0 0 0 0 2 0 0 -2 0 0 √2 -√2 -√2 √2 0 √2 -√2 orthogonal lifted from D8 ρ18 2 -2 -2 2 0 0 0 0 2 0 0 -2 0 0 -√2 √2 -√2 √2 0 -√2 √2 orthogonal lifted from D8 ρ19 4 4 0 0 0 0 0 0 -2 -4 0 -2 0 0 0 0 0 0 2 0 0 orthogonal lifted from S3×D4 ρ20 4 -4 0 0 0 0 0 0 -2 0 0 2 0 0 -2√2 2√2 0 0 0 √2 -√2 orthogonal faithful ρ21 4 -4 0 0 0 0 0 0 -2 0 0 2 0 0 2√2 -2√2 0 0 0 -√2 √2 orthogonal faithful

Permutation representations of S3×D8
On 24 points - transitive group 24T139
Generators in S24
(1 23 10)(2 24 11)(3 17 12)(4 18 13)(5 19 14)(6 20 15)(7 21 16)(8 22 9)
(1 5)(2 6)(3 7)(4 8)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(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)

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

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

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

G:=TransitiveGroup(24,139);

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

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

S3×D8 in GAP, Magma, Sage, TeX

S_3\times D_8
% in TeX

G:=Group("S3xD8");
// GroupNames label

G:=SmallGroup(96,117);
// by ID

G=gap.SmallGroup(96,117);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,116,297,159,69,2309]);
// 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^-1>;
// generators/relations

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