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G = S3×F8order 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

C1C22×C6 — S3×F8
C1C3C22×C6C3×F8 — S3×F8
C22×C6 — S3×F8
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
7C2×C6
7D6
8S3×C7
3C24
7C22×S3
7C22×S3
3C2×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
ρ71-11-111ζ75ζ74ζ7ζ76ζ72ζ7376772737475ζ75ζ73ζ76ζ72ζ74ζ7    linear of order 14
ρ81-11-111ζ73ζ7ζ72ζ75ζ74ζ7675727476773ζ73ζ76ζ75ζ74ζ7ζ72    linear of order 14
ρ91-11-111ζ74ζ76ζ75ζ72ζ73ζ772757377674ζ74ζ7ζ72ζ73ζ76ζ75    linear of order 14
ρ10111111ζ72ζ73ζ76ζ7ζ75ζ74ζ7ζ76ζ75ζ74ζ73ζ72ζ72ζ74ζ7ζ75ζ73ζ76    linear of order 7
ρ111-11-111ζ76ζ72ζ74ζ73ζ7ζ7573747757276ζ76ζ75ζ73ζ7ζ72ζ74    linear of order 14
ρ12111111ζ73ζ7ζ72ζ75ζ74ζ76ζ75ζ72ζ74ζ76ζ7ζ73ζ73ζ76ζ75ζ74ζ7ζ72    linear of order 7
ρ131-11-111ζ72ζ73ζ76ζ7ζ75ζ7477675747372ζ72ζ74ζ7ζ75ζ73ζ76    linear of order 14
ρ141-11-111ζ7ζ75ζ73ζ74ζ76ζ7274737672757ζ7ζ72ζ74ζ76ζ75ζ73    linear of order 14
ρ152020-1-1222222000000-1-1-1-1-1-1    orthogonal lifted from S3
ρ162020-1-17672747377500000076757377274    complex lifted from S3×C7
ρ172020-1-17273767757400000072747757376    complex lifted from S3×C7
ρ182020-1-17574776727300000075737672747    complex lifted from S3×C7
ρ192020-1-17476757273700000074772737675    complex lifted from S3×C7
ρ202020-1-17377275747600000073767574772    complex lifted from S3×C7
ρ212020-1-17757374767200000077274767573    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 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)

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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