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## G = S3×C7order 42 = 2·3·7

### Direct product of C7 and S3

Aliases: S3×C7, C3⋊C14, C213C2, 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 >

Character table of S3×C7

 class 1 2 3 7A 7B 7C 7D 7E 7F 14A 14B 14C 14D 14E 14F 21A 21B 21C 21D 21E 21F size 1 3 2 1 1 1 1 1 1 3 3 3 3 3 3 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 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 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 1 ζ75 ζ73 ζ74 ζ7 ζ76 ζ72 -ζ76 -ζ7 -ζ72 -ζ73 -ζ74 -ζ75 ζ7 ζ75 ζ73 ζ76 ζ72 ζ74 linear of order 14 ρ4 1 1 1 ζ74 ζ7 ζ76 ζ75 ζ72 ζ73 ζ72 ζ75 ζ73 ζ7 ζ76 ζ74 ζ75 ζ74 ζ7 ζ72 ζ73 ζ76 linear of order 7 ρ5 1 -1 1 ζ72 ζ74 ζ73 ζ76 ζ7 ζ75 -ζ7 -ζ76 -ζ75 -ζ74 -ζ73 -ζ72 ζ76 ζ72 ζ74 ζ7 ζ75 ζ73 linear of order 14 ρ6 1 1 1 ζ76 ζ75 ζ72 ζ74 ζ73 ζ7 ζ73 ζ74 ζ7 ζ75 ζ72 ζ76 ζ74 ζ76 ζ75 ζ73 ζ7 ζ72 linear of order 7 ρ7 1 -1 1 ζ74 ζ7 ζ76 ζ75 ζ72 ζ73 -ζ72 -ζ75 -ζ73 -ζ7 -ζ76 -ζ74 ζ75 ζ74 ζ7 ζ72 ζ73 ζ76 linear of order 14 ρ8 1 -1 1 ζ76 ζ75 ζ72 ζ74 ζ73 ζ7 -ζ73 -ζ74 -ζ7 -ζ75 -ζ72 -ζ76 ζ74 ζ76 ζ75 ζ73 ζ7 ζ72 linear of order 14 ρ9 1 -1 1 ζ7 ζ72 ζ75 ζ73 ζ74 ζ76 -ζ74 -ζ73 -ζ76 -ζ72 -ζ75 -ζ7 ζ73 ζ7 ζ72 ζ74 ζ76 ζ75 linear of order 14 ρ10 1 -1 1 ζ73 ζ76 ζ7 ζ72 ζ75 ζ74 -ζ75 -ζ72 -ζ74 -ζ76 -ζ7 -ζ73 ζ72 ζ73 ζ76 ζ75 ζ74 ζ7 linear of order 14 ρ11 1 1 1 ζ73 ζ76 ζ7 ζ72 ζ75 ζ74 ζ75 ζ72 ζ74 ζ76 ζ7 ζ73 ζ72 ζ73 ζ76 ζ75 ζ74 ζ7 linear of order 7 ρ12 1 1 1 ζ72 ζ74 ζ73 ζ76 ζ7 ζ75 ζ7 ζ76 ζ75 ζ74 ζ73 ζ72 ζ76 ζ72 ζ74 ζ7 ζ75 ζ73 linear of order 7 ρ13 1 1 1 ζ75 ζ73 ζ74 ζ7 ζ76 ζ72 ζ76 ζ7 ζ72 ζ73 ζ74 ζ75 ζ7 ζ75 ζ73 ζ76 ζ72 ζ74 linear of order 7 ρ14 1 1 1 ζ7 ζ72 ζ75 ζ73 ζ74 ζ76 ζ74 ζ73 ζ76 ζ72 ζ75 ζ7 ζ73 ζ7 ζ72 ζ74 ζ76 ζ75 linear of order 7 ρ15 2 0 -1 2 2 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ16 2 0 -1 2ζ74 2ζ7 2ζ76 2ζ75 2ζ72 2ζ73 0 0 0 0 0 0 -ζ75 -ζ74 -ζ7 -ζ72 -ζ73 -ζ76 complex faithful ρ17 2 0 -1 2ζ76 2ζ75 2ζ72 2ζ74 2ζ73 2ζ7 0 0 0 0 0 0 -ζ74 -ζ76 -ζ75 -ζ73 -ζ7 -ζ72 complex faithful ρ18 2 0 -1 2ζ73 2ζ76 2ζ7 2ζ72 2ζ75 2ζ74 0 0 0 0 0 0 -ζ72 -ζ73 -ζ76 -ζ75 -ζ74 -ζ7 complex faithful ρ19 2 0 -1 2ζ7 2ζ72 2ζ75 2ζ73 2ζ74 2ζ76 0 0 0 0 0 0 -ζ73 -ζ7 -ζ72 -ζ74 -ζ76 -ζ75 complex faithful ρ20 2 0 -1 2ζ72 2ζ74 2ζ73 2ζ76 2ζ7 2ζ75 0 0 0 0 0 0 -ζ76 -ζ72 -ζ74 -ζ7 -ζ75 -ζ73 complex faithful ρ21 2 0 -1 2ζ75 2ζ73 2ζ74 2ζ7 2ζ76 2ζ72 0 0 0 0 0 0 -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 14 18)(2 8 19)(3 9 20)(4 10 21)(5 11 15)(6 12 16)(7 13 17)
(8 19)(9 20)(10 21)(11 15)(12 16)(13 17)(14 18)

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

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

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

G:=TransitiveGroup(21,6);

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

 16 0 0 16
,
 0 22 25 28
,
 1 22 0 28
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

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