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G = C3×D7  order 42 = 2·3·7

Direct product of C3 and D7

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

Aliases: C3×D7, C7⋊3C6, C21⋊2C2, SmallGroup(42,4)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C3×D7
 G = < a,b,c | a3=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

C1
7C2
C3
C7
7C6
D7
C21
C3×D7

Character table of C3×D7

 class 123A3B6A6B7A7B7C21A21B21C21D21E21F
 size 171177222222222
ρ1111111111111111    trivial
ρ21-111-1-1111111111    linear of order 2
ρ311ζ3ζ32ζ3ζ32111ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ41-1ζ3ζ32ζ65ζ6111ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 6
ρ511ζ32ζ3ζ32ζ3111ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ61-1ζ32ζ3ζ6ζ65111ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 6
ρ7202200ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72    orthogonal lifted from D7
ρ8202200ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73    orthogonal lifted from D7
ρ9202200ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7    orthogonal lifted from D7
ρ1020-1-√-3-1+√-300ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ3ζ75+ζ3ζ72ζ32ζ75+ζ32ζ72ζ32ζ74+ζ32ζ73ζ3ζ74+ζ3ζ73ζ32ζ76+ζ32ζ7ζ3ζ76+ζ3ζ7    complex faithful
ρ1120-1+√-3-1-√-300ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ32ζ74+ζ32ζ73ζ3ζ74+ζ3ζ73ζ3ζ76+ζ3ζ7ζ32ζ76+ζ32ζ7ζ3ζ75+ζ3ζ72ζ32ζ75+ζ32ζ72    complex faithful
ρ1220-1-√-3-1+√-300ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ3ζ76+ζ3ζ7ζ32ζ76+ζ32ζ7ζ32ζ75+ζ32ζ72ζ3ζ75+ζ3ζ72ζ32ζ74+ζ32ζ73ζ3ζ74+ζ3ζ73    complex faithful
ρ1320-1+√-3-1-√-300ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ32ζ75+ζ32ζ72ζ3ζ75+ζ3ζ72ζ3ζ74+ζ3ζ73ζ32ζ74+ζ32ζ73ζ3ζ76+ζ3ζ7ζ32ζ76+ζ32ζ7    complex faithful
ρ1420-1+√-3-1-√-300ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ32ζ76+ζ32ζ7ζ3ζ76+ζ3ζ7ζ3ζ75+ζ3ζ72ζ32ζ75+ζ32ζ72ζ3ζ74+ζ3ζ73ζ32ζ74+ζ32ζ73    complex faithful
ρ1520-1-√-3-1+√-300ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ3ζ74+ζ3ζ73ζ32ζ74+ζ32ζ73ζ32ζ76+ζ32ζ7ζ3ζ76+ζ3ζ7ζ32ζ75+ζ32ζ72ζ3ζ75+ζ3ζ72    complex faithful

Permutation representations of C3×D7
►On 21 points - transitive group 21T3
Generators in S21
(1 20 13)(2 21 14)(3 15 8)(4 16 9)(5 17 10)(6 18 11)(7 19 12)

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

C3×D7 is a maximal subgroup of   C7⋊C18  C7⋊5F7
C3×D7 is a maximal quotient of   C7⋊5F7

Matrix representation of C3×D7 ►in GL2(𝔽13) generated by

30
03
,
98
212
,
120
111
G:=sub<GL(2,GF(13))| [3,0,0,3],[9,2,8,12],[12,11,0,1] >;
 

C3×D7 in GAP, Magma, Sage, TeX 

C_3\times D_7
 
% in TeX
 
G:=Group("C3xD7");
 
// GroupNames label
 
G:=SmallGroup(42,4);
 
// by ID
 
G=gap.SmallGroup(42,4);
 
# by ID
 
G:=PCGroup([3,-2,-3,-7,326]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C3×D7 in TeX
Character table of C3×D7 in TeX

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