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G = S3×F8  order 336 = 24·3·7

Direct product of S3 and F8

direct product, metabelian, soluble, monomial, A-group

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 >

C1
3C2
7C2
21C2
C3
8C7
7C22
21C22
21C22
21C22
21C22
S3
7C6
7S3
24C14
8C21
C23
21C23
21C23
7D6
7D6
7D6
7C2×C6
7D6
8S3×C7
3C24
C22×C6
7C22×S3
7C22×S3
F8
S3×C23
3C2×F8
C3×F8
S3×F8

Character table of S3×F8

 class 12A2B2C367A7B7C7D7E7F14A14B14C14D14E14F21A21B21C21D21E21F
 size 13721214888888242424242424161616161616
ρ1111111111111111111111111    trivial
ρ21-11-111111111-1-1-1-1-1-1111111    linear of order 2
ρ3111111ζ76ζ72ζ74ζ73ζ7ζ75ζ73ζ74ζ7ζ75ζ72ζ76ζ76ζ75ζ73ζ7ζ72ζ74    linear of order 7
ρ4111111ζ74ζ76ζ75ζ72ζ73ζ7ζ72ζ75ζ73ζ7ζ76ζ74ζ74ζ7ζ72ζ73ζ76ζ75    linear of order 7
ρ5111111ζ75ζ74ζ7ζ76ζ72ζ73ζ76ζ7ζ72ζ73ζ74ζ75ζ75ζ73ζ76ζ72ζ74ζ7    linear of order 7
ρ6111111ζ7ζ75ζ73ζ74ζ76ζ72ζ74ζ73ζ76ζ72ζ75ζ7ζ7ζ72ζ74ζ76ζ75ζ73    linear of order 7
ρ71-11-111ζ75ζ74ζ7ζ76ζ72ζ73-ζ76-ζ7-ζ72-ζ73-ζ74-ζ75ζ75ζ73ζ76ζ72ζ74ζ7    linear of order 14
ρ81-11-111ζ73ζ7ζ72ζ75ζ74ζ76-ζ75-ζ72-ζ74-ζ76-ζ7-ζ73ζ73ζ76ζ75ζ74ζ7ζ72    linear of order 14
ρ91-11-111ζ74ζ76ζ75ζ72ζ73ζ7-ζ72-ζ75-ζ73-ζ7-ζ76-ζ74ζ74ζ7ζ72ζ73ζ76ζ75    linear of order 14
ρ10111111ζ72ζ73ζ76ζ7ζ75ζ74ζ7ζ76ζ75ζ74ζ73ζ72ζ72ζ74ζ7ζ75ζ73ζ76    linear of order 7
ρ111-11-111ζ76ζ72ζ74ζ73ζ7ζ75-ζ73-ζ74-ζ7-ζ75-ζ72-ζ76ζ76ζ75ζ73ζ7ζ72ζ74    linear of order 14
ρ12111111ζ73ζ7ζ72ζ75ζ74ζ76ζ75ζ72ζ74ζ76ζ7ζ73ζ73ζ76ζ75ζ74ζ7ζ72    linear of order 7
ρ131-11-111ζ72ζ73ζ76ζ7ζ75ζ74-ζ7-ζ76-ζ75-ζ74-ζ73-ζ72ζ72ζ74ζ7ζ75ζ73ζ76    linear of order 14
ρ141-11-111ζ7ζ75ζ73ζ74ζ76ζ72-ζ74-ζ73-ζ76-ζ72-ζ75-ζ7ζ7ζ72ζ74ζ76ζ75ζ73    linear of order 14
ρ152020-1-1222222000000-1-1-1-1-1-1    orthogonal lifted from S3
ρ162020-1-12ζ762ζ722ζ742ζ732ζ72ζ75000000-ζ76-ζ75-ζ73-ζ7-ζ72-ζ74    complex lifted from S3×C7
ρ172020-1-12ζ722ζ732ζ762ζ72ζ752ζ74000000-ζ72-ζ74-ζ7-ζ75-ζ73-ζ76    complex lifted from S3×C7
ρ182020-1-12ζ752ζ742ζ72ζ762ζ722ζ73000000-ζ75-ζ73-ζ76-ζ72-ζ74-ζ7    complex lifted from S3×C7
ρ192020-1-12ζ742ζ762ζ752ζ722ζ732ζ7000000-ζ74-ζ7-ζ72-ζ73-ζ76-ζ75    complex lifted from S3×C7
ρ202020-1-12ζ732ζ72ζ722ζ752ζ742ζ76000000-ζ73-ζ76-ζ75-ζ74-ζ7-ζ72    complex lifted from S3×C7
ρ212020-1-12ζ72ζ752ζ732ζ742ζ762ζ72000000-ζ7-ζ72-ζ74-ζ76-ζ75-ζ73    complex lifted from S3×C7
ρ227-7-117-1000000000000000000    orthogonal lifted from C2×F8
ρ2377-1-17-1000000000000000000    orthogonal lifted from F8
ρ24140-20-71000000000000000000    orthogonal faithful

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

(2 3)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)

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

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

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

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

42420000000
100000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
,
100000000
42420000000
0042000000
0004200000
0000420000
0000042000
0000004200
0000000420
0000000042
,
100000000
010000000
0042000000
0042010000
0042100000
0042000100
0042001000
0042000001
0042000010
,
100000000
010000000
0000000142
0000001042
0000010042
0000100042
0001000042
0010000042
0000000042
,
100000000
010000000
0001420000
0010420000
0000420000
0000420001
0000420010
0000420100
0000421000
,
1600000000
0160000000
000001000
000010000
000000001
000000010
000100000
000000100
001000000

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
 

Export

Subgroup lattice of S3×F8 in TeX
Character table of S3×F8 in TeX

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