direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C14, C6⋊C14, C42⋊3C2, C21⋊4C22, C3⋊(C2×C14), SmallGroup(84,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×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 23 37)(2 24 38)(3 25 39)(4 26 40)(5 27 41)(6 28 42)(7 15 29)(8 16 30)(9 17 31)(10 18 32)(11 19 33)(12 20 34)(13 21 35)(14 22 36)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(28 35)
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), (1,23,37)(2,24,38)(3,25,39)(4,26,40)(5,27,41)(6,28,42)(7,15,29)(8,16,30)(9,17,31)(10,18,32)(11,19,33)(12,20,34)(13,21,35)(14,22,36), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(28,35)>;
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), (1,23,37)(2,24,38)(3,25,39)(4,26,40)(5,27,41)(6,28,42)(7,15,29)(8,16,30)(9,17,31)(10,18,32)(11,19,33)(12,20,34)(13,21,35)(14,22,36), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(28,35) );
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)], [(1,23,37),(2,24,38),(3,25,39),(4,26,40),(5,27,41),(6,28,42),(7,15,29),(8,16,30),(9,17,31),(10,18,32),(11,19,33),(12,20,34),(13,21,35),(14,22,36)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,29),(23,30),(24,31),(25,32),(26,33),(27,34),(28,35)]])
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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