direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C31, C3⋊C62, C93⋊3C2, SmallGroup(186,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C31 |
Generators and relations for S3×C31
G = < a,b,c | a31=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C31
►On 93 points
Generators in S93
(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 85 86 87 88 89 90 91 92 93)
(1 55 90)(2 56 91)(3 57 92)(4 58 93)(5 59 63)(6 60 64)(7 61 65)(8 62 66)(9 32 67)(10 33 68)(11 34 69)(12 35 70)(13 36 71)(14 37 72)(15 38 73)(16 39 74)(17 40 75)(18 41 76)(19 42 77)(20 43 78)(21 44 79)(22 45 80)(23 46 81)(24 47 82)(25 48 83)(26 49 84)(27 50 85)(28 51 86)(29 52 87)(30 53 88)(31 54 89)
(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 63)(60 64)(61 65)(62 66)
G:=sub<Sym(93)| (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)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93), (1,55,90)(2,56,91)(3,57,92)(4,58,93)(5,59,63)(6,60,64)(7,61,65)(8,62,66)(9,32,67)(10,33,68)(11,34,69)(12,35,70)(13,36,71)(14,37,72)(15,38,73)(16,39,74)(17,40,75)(18,41,76)(19,42,77)(20,43,78)(21,44,79)(22,45,80)(23,46,81)(24,47,82)(25,48,83)(26,49,84)(27,50,85)(28,51,86)(29,52,87)(30,53,88)(31,54,89), (32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,63)(60,64)(61,65)(62,66)>;
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)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93), (1,55,90)(2,56,91)(3,57,92)(4,58,93)(5,59,63)(6,60,64)(7,61,65)(8,62,66)(9,32,67)(10,33,68)(11,34,69)(12,35,70)(13,36,71)(14,37,72)(15,38,73)(16,39,74)(17,40,75)(18,41,76)(19,42,77)(20,43,78)(21,44,79)(22,45,80)(23,46,81)(24,47,82)(25,48,83)(26,49,84)(27,50,85)(28,51,86)(29,52,87)(30,53,88)(31,54,89), (32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,63)(60,64)(61,65)(62,66) );
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),(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93)], [(1,55,90),(2,56,91),(3,57,92),(4,58,93),(5,59,63),(6,60,64),(7,61,65),(8,62,66),(9,32,67),(10,33,68),(11,34,69),(12,35,70),(13,36,71),(14,37,72),(15,38,73),(16,39,74),(17,40,75),(18,41,76),(19,42,77),(20,43,78),(21,44,79),(22,45,80),(23,46,81),(24,47,82),(25,48,83),(26,49,84),(27,50,85),(28,51,86),(29,52,87),(30,53,88),(31,54,89)], [(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(42,77),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,84),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91),(57,92),(58,93),(59,63),(60,64),(61,65),(62,66)]])
93 conjugacy classes
| class | 1 | 2 | 3 | 31A | ··· | 31AD | 62A | ··· | 62AD | 93A | ··· | 93AD |
| order | 1 | 2 | 3 | 31 | ··· | 31 | 62 | ··· | 62 | 93 | ··· | 93 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
93 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C31 | C62 | S3 | S3×C31 |
| kernel | S3×C31 | C93 | S3 | C3 | C31 | C1 |
| # reps | 1 | 1 | 30 | 30 | 1 | 30 |
Matrix representation of S3×C31 ►in GL2(𝔽373) generated by
| 213 | 0 |
| 0 | 213 |
| 0 | 372 |
| 1 | 372 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(373))| [213,0,0,213],[0,1,372,372],[0,1,1,0] >;
S3×C31 in GAP, Magma, Sage, TeX
S_3\times C_{31} % in TeX
G:=Group("S3xC31"); // GroupNames label
G:=SmallGroup(186,3);
// by ID
G=gap.SmallGroup(186,3);
# by ID
G:=PCGroup([3,-2,-31,-3,1118]);
// Polycyclic
G:=Group<a,b,c|a^31=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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