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G = S3×F7order 252 = 22·32·7

Direct product of S3 and F7

Aliases: S3×F7, D21⋊C6, C3⋊F7⋊C2, (S3×C7)⋊C6, C21⋊(C2×C6), C71(S3×C6), (C3×D7)⋊C6, (S3×D7)⋊C3, D7⋊(C3×S3), C7⋊C31D6, (C3×F7)⋊C2, C31(C2×F7), (S3×C7⋊C3)⋊C2, (C3×C7⋊C3)⋊C22, Aut(D21), Hol(C21), SmallGroup(252,26)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — S3×F7
 Chief series C1 — C7 — C21 — C3×C7⋊C3 — C3×F7 — S3×F7
 Lower central C21 — S3×F7
 Upper central C1

Generators and relations for S3×F7
G = < a,b,c,d | a3=b2=c7=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Character table of S3×F7

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I 7 14 21 size 1 3 7 21 2 7 7 14 14 7 7 14 14 14 21 21 21 21 6 18 12 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 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ4 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 ρ5 1 -1 1 -1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ65 ζ6 ζ6 ζ65 1 -1 1 linear of order 6 ρ6 1 -1 -1 1 1 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 ζ65 ζ6 -1 ζ32 ζ65 ζ3 ζ6 1 -1 1 linear of order 6 ρ7 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ3 1 1 1 linear of order 3 ρ8 1 -1 1 -1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ6 ζ65 ζ65 ζ6 1 -1 1 linear of order 6 ρ9 1 1 -1 -1 1 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 ζ6 ζ65 -1 ζ65 ζ32 ζ6 ζ3 1 1 1 linear of order 6 ρ10 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ32 1 1 1 linear of order 3 ρ11 1 -1 -1 1 1 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 ζ6 ζ65 -1 ζ3 ζ6 ζ32 ζ65 1 -1 1 linear of order 6 ρ12 1 1 -1 -1 1 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 ζ65 ζ6 -1 ζ6 ζ3 ζ65 ζ32 1 1 1 linear of order 6 ρ13 2 0 -2 0 -1 2 2 -1 -1 -2 -2 1 1 1 0 0 0 0 2 0 -1 orthogonal lifted from D6 ρ14 2 0 2 0 -1 2 2 -1 -1 2 2 -1 -1 -1 0 0 0 0 2 0 -1 orthogonal lifted from S3 ρ15 2 0 -2 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 1+√-3 1-√-3 ζ32 ζ3 1 0 0 0 0 2 0 -1 complex lifted from S3×C6 ρ16 2 0 2 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 0 0 2 0 -1 complex lifted from C3×S3 ρ17 2 0 2 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 0 0 2 0 -1 complex lifted from C3×S3 ρ18 2 0 -2 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 1-√-3 1+√-3 ζ3 ζ32 1 0 0 0 0 2 0 -1 complex lifted from S3×C6 ρ19 6 -6 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 -1 orthogonal lifted from C2×F7 ρ20 6 6 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from F7 ρ21 12 0 0 0 -6 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 1 orthogonal faithful

Permutation representations of S3×F7
On 21 points - transitive group 21T15
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 4 3 7 5 6)(9 11 10 14 12 13)(16 18 17 21 19 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), (2,4,3,7,5,6)(9,11,10,14,12,13)(16,18,17,21,19,20)>;

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

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

G:=TransitiveGroup(21,15);

Matrix representation of S3×F7 in GL8(𝔽43)

 0 42 0 0 0 0 0 0 1 42 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 42 0 0 1 0 0 0 0 42 0 0 0 1 0 0 0 42 0 0 0 0 1 0 0 42 0 0 0 0 0 1 0 42 0 0 0 0 0 0 1 42
,
 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0

G:=sub<GL(8,GF(43))| [0,1,0,0,0,0,0,0,42,42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,42,42,42,42,42,42],[6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;

S3×F7 in GAP, Magma, Sage, TeX

S_3\times F_7
% in TeX

G:=Group("S3xF7");
// GroupNames label

G:=SmallGroup(252,26);
// by ID

G=gap.SmallGroup(252,26);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7,248,5404,914]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^7=d^6=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^5>;
// generators/relations

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