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## G = S3×D7order 84 = 22·3·7

### Direct product of S3 and D7

Aliases: S3×D7, C71D6, D21⋊C2, C31D14, C21⋊C22, (S3×C7)⋊C2, (C3×D7)⋊C2, SmallGroup(84,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — S3×D7
 Chief series C1 — C7 — C21 — C3×D7 — S3×D7
 Lower central C21 — S3×D7
 Upper central C1

Generators and relations for S3×D7
G = < a,b,c,d | a3=b2=c7=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of S3×D7

 class 1 2A 2B 2C 3 6 7A 7B 7C 14A 14B 14C 21A 21B 21C size 1 3 7 21 2 14 2 2 2 6 6 6 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 0 -2 0 -1 1 2 2 2 0 0 0 -1 -1 -1 orthogonal lifted from D6 ρ6 2 0 2 0 -1 -1 2 2 2 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ7 2 -2 0 0 2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 orthogonal lifted from D14 ρ8 2 -2 0 0 2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 orthogonal lifted from D14 ρ9 2 2 0 0 2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 orthogonal lifted from D7 ρ10 2 2 0 0 2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 orthogonal lifted from D7 ρ11 2 -2 0 0 2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 orthogonal lifted from D14 ρ12 2 2 0 0 2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 orthogonal lifted from D7 ρ13 4 0 0 0 -2 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 0 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 orthogonal faithful ρ14 4 0 0 0 -2 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 0 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 orthogonal faithful ρ15 4 0 0 0 -2 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 0 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 orthogonal faithful

Permutation representations of S3×D7
On 21 points - transitive group 21T8
Generators in S21
(1 13 20)(2 14 21)(3 8 15)(4 9 16)(5 10 17)(6 11 18)(7 12 19)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)

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

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

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

G:=TransitiveGroup(21,8);

S3×D7 is a maximal subgroup of   D21⋊S3  D15⋊D7
S3×D7 is a maximal quotient of   D21⋊C4  C21⋊D4  C3⋊D28  C7⋊D12  C21⋊Q8  D21⋊S3  D15⋊D7

Matrix representation of S3×D7 in GL4(𝔽43) generated by

 0 42 0 0 1 42 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 42 1 0 0 22 20
,
 1 0 0 0 0 1 0 0 0 0 42 0 0 0 22 1
G:=sub<GL(4,GF(43))| [0,1,0,0,42,42,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,42,22,0,0,1,20],[1,0,0,0,0,1,0,0,0,0,42,22,0,0,0,1] >;

S3×D7 in GAP, Magma, Sage, TeX

S_3\times D_7
% in TeX

G:=Group("S3xD7");
// GroupNames label

G:=SmallGroup(84,8);
// by ID

G=gap.SmallGroup(84,8);
# by ID

G:=PCGroup([4,-2,-2,-3,-7,54,1155]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^7=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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