direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C28, C84⋊6C2, D6.C14, C12⋊2C14, C14.14D6, Dic3⋊2C14, C42.19C22, C3⋊1(C2×C28), C21⋊6(C2×C4), C2.1(S3×C14), C6.2(C2×C14), (S3×C14).2C2, (C7×Dic3)⋊5C2, SmallGroup(168,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C28 |
Generators and relations for S3×C28
G = < a,b,c | a28=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C28
►On 84 points
Generators in S84
(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 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)
(1 39 72)(2 40 73)(3 41 74)(4 42 75)(5 43 76)(6 44 77)(7 45 78)(8 46 79)(9 47 80)(10 48 81)(11 49 82)(12 50 83)(13 51 84)(14 52 57)(15 53 58)(16 54 59)(17 55 60)(18 56 61)(19 29 62)(20 30 63)(21 31 64)(22 32 65)(23 33 66)(24 34 67)(25 35 68)(26 36 69)(27 37 70)(28 38 71)
(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 57)(53 58)(54 59)(55 60)(56 61)
G:=sub<Sym(84)| (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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84), (1,39,72)(2,40,73)(3,41,74)(4,42,75)(5,43,76)(6,44,77)(7,45,78)(8,46,79)(9,47,80)(10,48,81)(11,49,82)(12,50,83)(13,51,84)(14,52,57)(15,53,58)(16,54,59)(17,55,60)(18,56,61)(19,29,62)(20,30,63)(21,31,64)(22,32,65)(23,33,66)(24,34,67)(25,35,68)(26,36,69)(27,37,70)(28,38,71), (29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,57)(53,58)(54,59)(55,60)(56,61)>;
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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84), (1,39,72)(2,40,73)(3,41,74)(4,42,75)(5,43,76)(6,44,77)(7,45,78)(8,46,79)(9,47,80)(10,48,81)(11,49,82)(12,50,83)(13,51,84)(14,52,57)(15,53,58)(16,54,59)(17,55,60)(18,56,61)(19,29,62)(20,30,63)(21,31,64)(22,32,65)(23,33,66)(24,34,67)(25,35,68)(26,36,69)(27,37,70)(28,38,71), (29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,57)(53,58)(54,59)(55,60)(56,61) );
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),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)], [(1,39,72),(2,40,73),(3,41,74),(4,42,75),(5,43,76),(6,44,77),(7,45,78),(8,46,79),(9,47,80),(10,48,81),(11,49,82),(12,50,83),(13,51,84),(14,52,57),(15,53,58),(16,54,59),(17,55,60),(18,56,61),(19,29,62),(20,30,63),(21,31,64),(22,32,65),(23,33,66),(24,34,67),(25,35,68),(26,36,69),(27,37,70),(28,38,71)], [(29,62),(30,63),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,81),(49,82),(50,83),(51,84),(52,57),(53,58),(54,59),(55,60),(56,61)]])
S3×C28 is a maximal subgroup of
D6.Dic7 D28⋊5S3 D84⋊C2 D6.D14
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 7A | ··· | 7F | 12A | 12B | 14A | ··· | 14F | 14G | ··· | 14R | 21A | ··· | 21F | 28A | ··· | 28L | 28M | ··· | 28X | 42A | ··· | 42F | 84A | ··· | 84L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 7 | ··· | 7 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C28 | S3 | D6 | C4×S3 | S3×C7 | S3×C14 | S3×C28 |
| kernel | S3×C28 | C7×Dic3 | C84 | S3×C14 | S3×C7 | C4×S3 | Dic3 | C12 | D6 | S3 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 24 | 1 | 1 | 2 | 6 | 6 | 12 |
Matrix representation of S3×C28 ►in GL2(𝔽29) generated by
| 3 | 0 |
| 0 | 3 |
| 0 | 10 |
| 26 | 28 |
| 28 | 0 |
| 3 | 1 |
G:=sub<GL(2,GF(29))| [3,0,0,3],[0,26,10,28],[28,3,0,1] >;
S3×C28 in GAP, Magma, Sage, TeX
S_3\times C_{28} % in TeX
G:=Group("S3xC28"); // GroupNames label
G:=SmallGroup(168,30);
// by ID
G=gap.SmallGroup(168,30);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-3,146,2804]);
// Polycyclic
G:=Group<a,b,c|a^28=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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