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## G = C3×D7order 42 = 2·3·7

### Direct product of C3 and D7

Aliases: C3×D7, C73C6, C212C2, 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 >

Character table of C3×D7

 class 1 2 3A 3B 6A 6B 7A 7B 7C 21A 21B 21C 21D 21E 21F size 1 7 1 1 7 7 2 2 2 2 2 2 2 2 2 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 ζ3 ζ32 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ4 1 -1 ζ3 ζ32 ζ65 ζ6 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 6 ρ5 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ6 1 -1 ζ32 ζ3 ζ6 ζ65 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 6 ρ7 2 0 2 2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 orthogonal lifted from D7 ρ8 2 0 2 2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 orthogonal lifted from D7 ρ9 2 0 2 2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 orthogonal lifted from D7 ρ10 2 0 -1-√-3 -1+√-3 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ3ζ75+ζ3ζ72 ζ32ζ75+ζ32ζ72 ζ32ζ74+ζ32ζ73 ζ3ζ74+ζ3ζ73 ζ32ζ76+ζ32ζ7 ζ3ζ76+ζ3ζ7 complex faithful ρ11 2 0 -1+√-3 -1-√-3 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ32ζ74+ζ32ζ73 ζ3ζ74+ζ3ζ73 ζ3ζ76+ζ3ζ7 ζ32ζ76+ζ32ζ7 ζ3ζ75+ζ3ζ72 ζ32ζ75+ζ32ζ72 complex faithful ρ12 2 0 -1-√-3 -1+√-3 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ3ζ76+ζ3ζ7 ζ32ζ76+ζ32ζ7 ζ32ζ75+ζ32ζ72 ζ3ζ75+ζ3ζ72 ζ32ζ74+ζ32ζ73 ζ3ζ74+ζ3ζ73 complex faithful ρ13 2 0 -1+√-3 -1-√-3 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ32ζ75+ζ32ζ72 ζ3ζ75+ζ3ζ72 ζ3ζ74+ζ3ζ73 ζ32ζ74+ζ32ζ73 ζ3ζ76+ζ3ζ7 ζ32ζ76+ζ32ζ7 complex faithful ρ14 2 0 -1+√-3 -1-√-3 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ32ζ76+ζ32ζ7 ζ3ζ76+ζ3ζ7 ζ3ζ75+ζ3ζ72 ζ32ζ75+ζ32ζ72 ζ3ζ74+ζ3ζ73 ζ32ζ74+ζ32ζ73 complex faithful ρ15 2 0 -1-√-3 -1+√-3 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ3ζ74+ζ3ζ73 ζ32ζ74+ζ32ζ73 ζ32ζ76+ζ32ζ7 ζ3ζ76+ζ3ζ7 ζ32ζ75+ζ32ζ72 ζ3ζ75+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  C75F7
C3×D7 is a maximal quotient of   C75F7

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

 3 0 0 3
,
 9 8 2 12
,
 12 0 11 1
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

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