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G = S3×D7  order 84 = 22·3·7

Direct product of S3 and D7

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3×D7, C7⋊1D6, D21⋊C2, C3⋊1D14, 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 >

C1
3C2
7C2
21C2
C3
C7
21C22
S3
7C6
7S3
D7
3C14
3D7
C21
7D6
3D14
C3×D7
D21
S3×C7
S3×D7

Character table of S3×D7

 class 12A2B2C367A7B7C14A14B14C21A21B21C
 size 13721214222666444
ρ1111111111111111    trivial
ρ21-11-111111-1-1-1111    linear of order 2
ρ311-1-11-1111111111    linear of order 2
ρ41-1-111-1111-1-1-1111    linear of order 2
ρ520-20-11222000-1-1-1    orthogonal lifted from D6
ρ62020-1-1222000-1-1-1    orthogonal lifted from S3
ρ72-20020ζ74+ζ73ζ76+ζ7ζ75+ζ72-ζ76-ζ7-ζ75-ζ72-ζ74-ζ73ζ76+ζ7ζ74+ζ73ζ75+ζ72    orthogonal lifted from D14
ρ82-20020ζ76+ζ7ζ75+ζ72ζ74+ζ73-ζ75-ζ72-ζ74-ζ73-ζ76-ζ7ζ75+ζ72ζ76+ζ7ζ74+ζ73    orthogonal lifted from D14
ρ9220020ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ75+ζ72ζ76+ζ7    orthogonal lifted from D7
ρ10220020ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ74+ζ73ζ75+ζ72    orthogonal lifted from D7
ρ112-20020ζ75+ζ72ζ74+ζ73ζ76+ζ7-ζ74-ζ73-ζ76-ζ7-ζ75-ζ72ζ74+ζ73ζ75+ζ72ζ76+ζ7    orthogonal lifted from D14
ρ12220020ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ76+ζ7ζ74+ζ73    orthogonal lifted from D7
ρ134000-202ζ75+2ζ722ζ74+2ζ732ζ76+2ζ7000-ζ74-ζ73-ζ75-ζ72-ζ76-ζ7    orthogonal faithful
ρ144000-202ζ74+2ζ732ζ76+2ζ72ζ75+2ζ72000-ζ76-ζ7-ζ74-ζ73-ζ75-ζ72    orthogonal faithful
ρ154000-202ζ76+2ζ72ζ75+2ζ722ζ74+2ζ73000-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

04200
14200
0010
0001
,
0100
1000
0010
0001
,
1000
0100
00421
002220
,
1000
0100
00420
00221
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
 

Export

Subgroup lattice of S3×D7 in TeX
Character table of S3×D7 in TeX

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