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## G = S3×C7⋊C3order 126 = 2·32·7

### Direct product of S3 and C7⋊C3

Aliases: S3×C7⋊C3, C213C6, (S3×C7)⋊C3, C72(C3×S3), C3⋊(C2×C7⋊C3), (C3×C7⋊C3)⋊3C2, SmallGroup(126,8)

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C7⋊C3
G = < a,b,c,d | a3=b2=c7=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Character table of S3×C7⋊C3

 class 1 2 3A 3B 3C 3D 3E 6A 6B 7A 7B 14A 14B 21A 21B size 1 3 2 7 7 14 14 21 21 3 3 9 9 6 6 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 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ4 1 -1 1 ζ3 ζ32 ζ3 ζ32 ζ6 ζ65 1 1 -1 -1 1 1 linear of order 6 ρ5 1 -1 1 ζ32 ζ3 ζ32 ζ3 ζ65 ζ6 1 1 -1 -1 1 1 linear of order 6 ρ6 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ7 2 0 -1 2 2 -1 -1 0 0 2 2 0 0 -1 -1 orthogonal lifted from S3 ρ8 2 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 2 2 0 0 -1 -1 complex lifted from C3×S3 ρ9 2 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 2 2 0 0 -1 -1 complex lifted from C3×S3 ρ10 3 3 3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ11 3 -3 3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 1-√-7/2 1+√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C2×C7⋊C3 ρ12 3 3 3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ13 3 -3 3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 1+√-7/2 1-√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C2×C7⋊C3 ρ14 6 0 -3 0 0 0 0 0 0 -1-√-7 -1+√-7 0 0 1-√-7/2 1+√-7/2 complex faithful ρ15 6 0 -3 0 0 0 0 0 0 -1+√-7 -1-√-7 0 0 1+√-7/2 1-√-7/2 complex faithful

Permutation representations of S3×C7⋊C3
On 21 points - transitive group 21T11
Generators in S21
(1 8 15)(2 9 16)(3 10 17)(4 11 18)(5 12 19)(6 13 20)(7 14 21)
(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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)

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

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

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

G:=TransitiveGroup(21,11);

S3×C7⋊C3 is a maximal subgroup of   C63⋊C6  C636C6  C7⋊He3⋊C2
S3×C7⋊C3 is a maximal quotient of   C63⋊C6  C636C6  C7⋊He3⋊C2

Matrix representation of S3×C7⋊C3 in GL5(𝔽43)

 42 42 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 42 42 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 42 42 18 0 0 1 0 1 0 0 0 1 25
,
 36 0 0 0 0 0 36 0 0 0 0 0 0 19 1 0 0 1 1 25 0 0 0 25 42

G:=sub<GL(5,GF(43))| [42,1,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,42,0,0,0,0,42,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,42,1,0,0,0,42,0,1,0,0,18,1,25],[36,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,19,1,25,0,0,1,25,42] >;

S3×C7⋊C3 in GAP, Magma, Sage, TeX

S_3\times C_7\rtimes C_3
% in TeX

G:=Group("S3xC7:C3");
// GroupNames label

G:=SmallGroup(126,8);
// by ID

G=gap.SmallGroup(126,8);
# by ID

G:=PCGroup([4,-2,-3,-3,-7,146,295]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^7=d^3=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^-1=c^4>;
// generators/relations

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