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## G = S3×F8order 336 = 24·3·7

### Direct product of S3 and F8

Aliases: S3×F8, C3⋊(C2×F8), (C3×F8)⋊C2, (S3×C23)⋊C7, C23⋊(S3×C7), (C22×C6)⋊C14, SmallGroup(336,211)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — S3×F8
 Chief series C1 — C3 — C22×C6 — C3×F8 — S3×F8
 Lower central C22×C6 — S3×F8
 Upper central C1

Generators and relations for S3×F8
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f7=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=ed=de, fdf-1=c, fef-1=d >

3C2
7C2
21C2
8C7
7C22
21C22
21C22
21C22
21C22
7C6
7S3
24C14
8C21
21C23
21C23
7D6
7D6
7D6
7D6
3C24

Character table of S3×F8

 class 1 2A 2B 2C 3 6 7A 7B 7C 7D 7E 7F 14A 14B 14C 14D 14E 14F 21A 21B 21C 21D 21E 21F size 1 3 7 21 2 14 8 8 8 8 8 8 24 24 24 24 24 24 16 16 16 16 16 16 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 ζ76 ζ72 ζ74 ζ73 ζ7 ζ75 ζ73 ζ74 ζ7 ζ75 ζ72 ζ76 ζ76 ζ75 ζ73 ζ7 ζ72 ζ74 linear of order 7 ρ4 1 1 1 1 1 1 ζ74 ζ76 ζ75 ζ72 ζ73 ζ7 ζ72 ζ75 ζ73 ζ7 ζ76 ζ74 ζ74 ζ7 ζ72 ζ73 ζ76 ζ75 linear of order 7 ρ5 1 1 1 1 1 1 ζ75 ζ74 ζ7 ζ76 ζ72 ζ73 ζ76 ζ7 ζ72 ζ73 ζ74 ζ75 ζ75 ζ73 ζ76 ζ72 ζ74 ζ7 linear of order 7 ρ6 1 1 1 1 1 1 ζ7 ζ75 ζ73 ζ74 ζ76 ζ72 ζ74 ζ73 ζ76 ζ72 ζ75 ζ7 ζ7 ζ72 ζ74 ζ76 ζ75 ζ73 linear of order 7 ρ7 1 -1 1 -1 1 1 ζ75 ζ74 ζ7 ζ76 ζ72 ζ73 -ζ76 -ζ7 -ζ72 -ζ73 -ζ74 -ζ75 ζ75 ζ73 ζ76 ζ72 ζ74 ζ7 linear of order 14 ρ8 1 -1 1 -1 1 1 ζ73 ζ7 ζ72 ζ75 ζ74 ζ76 -ζ75 -ζ72 -ζ74 -ζ76 -ζ7 -ζ73 ζ73 ζ76 ζ75 ζ74 ζ7 ζ72 linear of order 14 ρ9 1 -1 1 -1 1 1 ζ74 ζ76 ζ75 ζ72 ζ73 ζ7 -ζ72 -ζ75 -ζ73 -ζ7 -ζ76 -ζ74 ζ74 ζ7 ζ72 ζ73 ζ76 ζ75 linear of order 14 ρ10 1 1 1 1 1 1 ζ72 ζ73 ζ76 ζ7 ζ75 ζ74 ζ7 ζ76 ζ75 ζ74 ζ73 ζ72 ζ72 ζ74 ζ7 ζ75 ζ73 ζ76 linear of order 7 ρ11 1 -1 1 -1 1 1 ζ76 ζ72 ζ74 ζ73 ζ7 ζ75 -ζ73 -ζ74 -ζ7 -ζ75 -ζ72 -ζ76 ζ76 ζ75 ζ73 ζ7 ζ72 ζ74 linear of order 14 ρ12 1 1 1 1 1 1 ζ73 ζ7 ζ72 ζ75 ζ74 ζ76 ζ75 ζ72 ζ74 ζ76 ζ7 ζ73 ζ73 ζ76 ζ75 ζ74 ζ7 ζ72 linear of order 7 ρ13 1 -1 1 -1 1 1 ζ72 ζ73 ζ76 ζ7 ζ75 ζ74 -ζ7 -ζ76 -ζ75 -ζ74 -ζ73 -ζ72 ζ72 ζ74 ζ7 ζ75 ζ73 ζ76 linear of order 14 ρ14 1 -1 1 -1 1 1 ζ7 ζ75 ζ73 ζ74 ζ76 ζ72 -ζ74 -ζ73 -ζ76 -ζ72 -ζ75 -ζ7 ζ7 ζ72 ζ74 ζ76 ζ75 ζ73 linear of order 14 ρ15 2 0 2 0 -1 -1 2 2 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ16 2 0 2 0 -1 -1 2ζ76 2ζ72 2ζ74 2ζ73 2ζ7 2ζ75 0 0 0 0 0 0 -ζ76 -ζ75 -ζ73 -ζ7 -ζ72 -ζ74 complex lifted from S3×C7 ρ17 2 0 2 0 -1 -1 2ζ72 2ζ73 2ζ76 2ζ7 2ζ75 2ζ74 0 0 0 0 0 0 -ζ72 -ζ74 -ζ7 -ζ75 -ζ73 -ζ76 complex lifted from S3×C7 ρ18 2 0 2 0 -1 -1 2ζ75 2ζ74 2ζ7 2ζ76 2ζ72 2ζ73 0 0 0 0 0 0 -ζ75 -ζ73 -ζ76 -ζ72 -ζ74 -ζ7 complex lifted from S3×C7 ρ19 2 0 2 0 -1 -1 2ζ74 2ζ76 2ζ75 2ζ72 2ζ73 2ζ7 0 0 0 0 0 0 -ζ74 -ζ7 -ζ72 -ζ73 -ζ76 -ζ75 complex lifted from S3×C7 ρ20 2 0 2 0 -1 -1 2ζ73 2ζ7 2ζ72 2ζ75 2ζ74 2ζ76 0 0 0 0 0 0 -ζ73 -ζ76 -ζ75 -ζ74 -ζ7 -ζ72 complex lifted from S3×C7 ρ21 2 0 2 0 -1 -1 2ζ7 2ζ75 2ζ73 2ζ74 2ζ76 2ζ72 0 0 0 0 0 0 -ζ7 -ζ72 -ζ74 -ζ76 -ζ75 -ζ73 complex lifted from S3×C7 ρ22 7 -7 -1 1 7 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×F8 ρ23 7 7 -1 -1 7 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from F8 ρ24 14 0 -2 0 -7 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of S3×F8
On 24 points - transitive group 24T706
Generators in S24
(1 2 3)(4 21 11)(5 22 12)(6 23 13)(7 24 14)(8 18 15)(9 19 16)(10 20 17)
(2 3)(11 21)(12 22)(13 23)(14 24)(15 18)(16 19)(17 20)
(1 8)(2 18)(3 15)(4 10)(5 7)(6 9)(11 17)(12 14)(13 16)(19 23)(20 21)(22 24)
(1 9)(2 19)(3 16)(4 5)(6 8)(7 10)(11 12)(13 15)(14 17)(18 23)(20 24)(21 22)
(1 10)(2 20)(3 17)(4 8)(5 6)(7 9)(11 15)(12 13)(14 16)(18 21)(19 24)(22 23)
(4 5 6 7 8 9 10)(11 12 13 14 15 16 17)(18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,706);

Matrix representation of S3×F8 in GL9(𝔽43)

 42 42 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 42 42 0 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 0 42
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 42 0 1 0 0 0 0 0 0 42 1 0 0 0 0 0 0 0 42 0 0 0 1 0 0 0 0 42 0 0 1 0 0 0 0 0 42 0 0 0 0 0 1 0 0 42 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 42 0 0 0 0 0 0 1 0 42 0 0 0 0 0 1 0 0 42 0 0 0 0 1 0 0 0 42 0 0 0 1 0 0 0 0 42 0 0 1 0 0 0 0 0 42 0 0 0 0 0 0 0 0 42
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 42 0 0 0 0 0 0 1 0 42 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 42 0 0 0 1 0 0 0 0 42 0 0 1 0 0 0 0 0 42 0 1 0 0 0 0 0 0 42 1 0 0 0
,
 16 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0

G:=sub<GL(9,GF(43))| [42,1,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,0,42],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,42,42,42,42,42,42,42,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,42,42,42,42,42,42,42],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,42,42,42,42,42,42,42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0],[16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0] >;

S3×F8 in GAP, Magma, Sage, TeX

S_3\times F_8
% in TeX

G:=Group("S3xF8");
// GroupNames label

G:=SmallGroup(336,211);
// by ID

G=gap.SmallGroup(336,211);
# by ID

G:=PCGroup([6,-2,-7,-2,2,2,-3,764,177,430,8069]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^2=f^7=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f^-1=e*d=d*e,f*d*f^-1=c,f*e*f^-1=d>;
// generators/relations

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