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## G = S3×C14order 84 = 22·3·7

### Direct product of C14 and S3

Aliases: S3×C14, C6⋊C14, C423C2, C214C22, C3⋊(C2×C14), SmallGroup(84,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C14
 Chief series C1 — C3 — C21 — S3×C7 — S3×C14
 Lower central C3 — S3×C14
 Upper central C1 — C14

Generators and relations for S3×C14
G = < a,b,c | a14=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of S3×C14
On 42 points
Generators in S42
(1 2 3 4 5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42)
(1 19 31)(2 20 32)(3 21 33)(4 22 34)(5 23 35)(6 24 36)(7 25 37)(8 26 38)(9 27 39)(10 28 40)(11 15 41)(12 16 42)(13 17 29)(14 18 30)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 29)(25 30)(26 31)(27 32)(28 33)

G:=sub<Sym(42)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42), (1,19,31)(2,20,32)(3,21,33)(4,22,34)(5,23,35)(6,24,36)(7,25,37)(8,26,38)(9,27,39)(10,28,40)(11,15,41)(12,16,42)(13,17,29)(14,18,30), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,29)(25,30)(26,31)(27,32)(28,33)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42), (1,19,31)(2,20,32)(3,21,33)(4,22,34)(5,23,35)(6,24,36)(7,25,37)(8,26,38)(9,27,39)(10,28,40)(11,15,41)(12,16,42)(13,17,29)(14,18,30), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,29)(25,30)(26,31)(27,32)(28,33) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42)], [(1,19,31),(2,20,32),(3,21,33),(4,22,34),(5,23,35),(6,24,36),(7,25,37),(8,26,38),(9,27,39),(10,28,40),(11,15,41),(12,16,42),(13,17,29),(14,18,30)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,40),(22,41),(23,42),(24,29),(25,30),(26,31),(27,32),(28,33)])

S3×C14 is a maximal subgroup of   C21⋊D4  C7⋊D12

42 conjugacy classes

 class 1 2A 2B 2C 3 6 7A ··· 7F 14A ··· 14F 14G ··· 14R 21A ··· 21F 42A ··· 42F order 1 2 2 2 3 6 7 ··· 7 14 ··· 14 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 3 3 2 2 1 ··· 1 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C7 C14 C14 S3 D6 S3×C7 S3×C14 kernel S3×C14 S3×C7 C42 D6 S3 C6 C14 C7 C2 C1 # reps 1 2 1 6 12 6 1 1 6 6

Matrix representation of S3×C14 in GL2(𝔽29) generated by

 4 0 0 4
,
 28 22 25 0
,
 28 22 0 1
G:=sub<GL(2,GF(29))| [4,0,0,4],[28,25,22,0],[28,0,22,1] >;

S3×C14 in GAP, Magma, Sage, TeX

S_3\times C_{14}
% in TeX

G:=Group("S3xC14");
// GroupNames label

G:=SmallGroup(84,13);
// by ID

G=gap.SmallGroup(84,13);
# by ID

G:=PCGroup([4,-2,-2,-7,-3,899]);
// Polycyclic

G:=Group<a,b,c|a^14=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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