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G = S3×C7  order 42 = 2·3·7

Direct product of C7 and S3

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: S3×C7, C3⋊C14, C21⋊3C2, SmallGroup(42,3)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for S3×C7
 G = < a,b,c | a7=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

C1
3C2
C3
C7
S3
3C14
C21
S3×C7

Character table of S3×C7

 class 1237A7B7C7D7E7F14A14B14C14D14E14F21A21B21C21D21E21F
 size 132111111333333222222
ρ1111111111111111111111    trivial
ρ21-11111111-1-1-1-1-1-1111111    linear of order 2
ρ31-11ζ75ζ73ζ74ζ7ζ76ζ72-ζ76-ζ7-ζ72-ζ73-ζ74-ζ75ζ7ζ75ζ73ζ76ζ72ζ74    linear of order 14
ρ4111ζ74ζ7ζ76ζ75ζ72ζ73ζ72ζ75ζ73ζ7ζ76ζ74ζ75ζ74ζ7ζ72ζ73ζ76    linear of order 7
ρ51-11ζ72ζ74ζ73ζ76ζ7ζ75-ζ7-ζ76-ζ75-ζ74-ζ73-ζ72ζ76ζ72ζ74ζ7ζ75ζ73    linear of order 14
ρ6111ζ76ζ75ζ72ζ74ζ73ζ7ζ73ζ74ζ7ζ75ζ72ζ76ζ74ζ76ζ75ζ73ζ7ζ72    linear of order 7
ρ71-11ζ74ζ7ζ76ζ75ζ72ζ73-ζ72-ζ75-ζ73-ζ7-ζ76-ζ74ζ75ζ74ζ7ζ72ζ73ζ76    linear of order 14
ρ81-11ζ76ζ75ζ72ζ74ζ73ζ7-ζ73-ζ74-ζ7-ζ75-ζ72-ζ76ζ74ζ76ζ75ζ73ζ7ζ72    linear of order 14
ρ91-11ζ7ζ72ζ75ζ73ζ74ζ76-ζ74-ζ73-ζ76-ζ72-ζ75-ζ7ζ73ζ7ζ72ζ74ζ76ζ75    linear of order 14
ρ101-11ζ73ζ76ζ7ζ72ζ75ζ74-ζ75-ζ72-ζ74-ζ76-ζ7-ζ73ζ72ζ73ζ76ζ75ζ74ζ7    linear of order 14
ρ11111ζ73ζ76ζ7ζ72ζ75ζ74ζ75ζ72ζ74ζ76ζ7ζ73ζ72ζ73ζ76ζ75ζ74ζ7    linear of order 7
ρ12111ζ72ζ74ζ73ζ76ζ7ζ75ζ7ζ76ζ75ζ74ζ73ζ72ζ76ζ72ζ74ζ7ζ75ζ73    linear of order 7
ρ13111ζ75ζ73ζ74ζ7ζ76ζ72ζ76ζ7ζ72ζ73ζ74ζ75ζ7ζ75ζ73ζ76ζ72ζ74    linear of order 7
ρ14111ζ7ζ72ζ75ζ73ζ74ζ76ζ74ζ73ζ76ζ72ζ75ζ7ζ73ζ7ζ72ζ74ζ76ζ75    linear of order 7
ρ1520-1222222000000-1-1-1-1-1-1    orthogonal lifted from S3
ρ1620-12ζ742ζ72ζ762ζ752ζ722ζ73000000-ζ75-ζ74-ζ7-ζ72-ζ73-ζ76    complex faithful
ρ1720-12ζ762ζ752ζ722ζ742ζ732ζ7000000-ζ74-ζ76-ζ75-ζ73-ζ7-ζ72    complex faithful
ρ1820-12ζ732ζ762ζ72ζ722ζ752ζ74000000-ζ72-ζ73-ζ76-ζ75-ζ74-ζ7    complex faithful
ρ1920-12ζ72ζ722ζ752ζ732ζ742ζ76000000-ζ73-ζ7-ζ72-ζ74-ζ76-ζ75    complex faithful
ρ2020-12ζ722ζ742ζ732ζ762ζ72ζ75000000-ζ76-ζ72-ζ74-ζ7-ζ75-ζ73    complex faithful
ρ2120-12ζ752ζ732ζ742ζ72ζ762ζ72000000-ζ7-ζ75-ζ73-ζ76-ζ72-ζ74    complex faithful

Permutation representations of S3×C7
►On 21 points - transitive group 21T6
Generators in S21
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)

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

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

Matrix representation of S3×C7 ►in GL2(𝔽29) generated by

160
016
,
022
2528
,
122
028
G:=sub<GL(2,GF(29))| [16,0,0,16],[0,25,22,28],[1,0,22,28] >;
 

S3×C7 in GAP, Magma, Sage, TeX 

S_3\times C_7
 
% in TeX
 
G:=Group("S3xC7");
 
// GroupNames label
 
G:=SmallGroup(42,3);
 
// by ID
 
G=gap.SmallGroup(42,3);
 
# by ID
 
G:=PCGroup([3,-2,-7,-3,254]);
 
// Polycyclic
 
G:=Group<a,b,c|a^7=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of S3×C7 in TeX
Character table of S3×C7 in TeX

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