direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C42, C6⋊C42, C42⋊7C6, C3⋊(C2×C42), C21⋊9(C2×C6), (C3×C42)⋊4C2, (C3×C6)⋊1C14, (C3×C21)⋊9C22, C32⋊2(C2×C14), SmallGroup(252,42)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C42 |
Generators and relations for S3×C42
G = < a,b,c | a42=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C42
►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 15 29)(2 16 30)(3 17 31)(4 18 32)(5 19 33)(6 20 34)(7 21 35)(8 22 36)(9 23 37)(10 24 38)(11 25 39)(12 26 40)(13 27 41)(14 28 42)(43 71 57)(44 72 58)(45 73 59)(46 74 60)(47 75 61)(48 76 62)(49 77 63)(50 78 64)(51 79 65)(52 80 66)(53 81 67)(54 82 68)(55 83 69)(56 84 70)
(1 81)(2 82)(3 83)(4 84)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)
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), (1,15,29)(2,16,30)(3,17,31)(4,18,32)(5,19,33)(6,20,34)(7,21,35)(8,22,36)(9,23,37)(10,24,38)(11,25,39)(12,26,40)(13,27,41)(14,28,42)(43,71,57)(44,72,58)(45,73,59)(46,74,60)(47,75,61)(48,76,62)(49,77,63)(50,78,64)(51,79,65)(52,80,66)(53,81,67)(54,82,68)(55,83,69)(56,84,70), (1,81)(2,82)(3,83)(4,84)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)>;
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), (1,15,29)(2,16,30)(3,17,31)(4,18,32)(5,19,33)(6,20,34)(7,21,35)(8,22,36)(9,23,37)(10,24,38)(11,25,39)(12,26,40)(13,27,41)(14,28,42)(43,71,57)(44,72,58)(45,73,59)(46,74,60)(47,75,61)(48,76,62)(49,77,63)(50,78,64)(51,79,65)(52,80,66)(53,81,67)(54,82,68)(55,83,69)(56,84,70), (1,81)(2,82)(3,83)(4,84)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80) );
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)], [(1,15,29),(2,16,30),(3,17,31),(4,18,32),(5,19,33),(6,20,34),(7,21,35),(8,22,36),(9,23,37),(10,24,38),(11,25,39),(12,26,40),(13,27,41),(14,28,42),(43,71,57),(44,72,58),(45,73,59),(46,74,60),(47,75,61),(48,76,62),(49,77,63),(50,78,64),(51,79,65),(52,80,66),(53,81,67),(54,82,68),(55,83,69),(56,84,70)], [(1,81),(2,82),(3,83),(4,84),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(41,79),(42,80)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 7A | ··· | 7F | 14A | ··· | 14F | 14G | ··· | 14R | 21A | ··· | 21L | 21M | ··· | 21AD | 42A | ··· | 42L | 42M | ··· | 42AD | 42AE | ··· | 42BB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 21 | ··· | 21 | 42 | ··· | 42 | 42 | ··· | 42 | 42 | ··· | 42 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C7 | C14 | C14 | C21 | C42 | C42 | S3 | D6 | C3×S3 | S3×C6 | S3×C7 | S3×C14 | S3×C21 | S3×C42 |
| kernel | S3×C42 | S3×C21 | C3×C42 | S3×C14 | S3×C7 | C42 | S3×C6 | C3×S3 | C3×C6 | D6 | S3 | C6 | C42 | C21 | C14 | C7 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 12 | 6 | 12 | 24 | 12 | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 |
Matrix representation of S3×C42 ►in GL2(𝔽43) generated by
| 26 | 0 |
| 0 | 26 |
| 6 | 0 |
| 24 | 36 |
| 28 | 35 |
| 28 | 15 |
G:=sub<GL(2,GF(43))| [26,0,0,26],[6,24,0,36],[28,28,35,15] >;
S3×C42 in GAP, Magma, Sage, TeX
S_3\times C_{42} % in TeX
G:=Group("S3xC42"); // GroupNames label
G:=SmallGroup(252,42);
// by ID
G=gap.SmallGroup(252,42);
# by ID
G:=PCGroup([5,-2,-2,-3,-7,-3,4204]);
// Polycyclic
G:=Group<a,b,c|a^42=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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