direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C35, C3⋊C70, C105⋊7C2, C15⋊3C14, C21⋊3C10, SmallGroup(210,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C35 |
Generators and relations for S3×C35
G = < a,b,c | a35=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C35
►On 105 points
Generators in S105
(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 94 95 96 97 98 99 100 101 102 103 104 105)
(1 47 87)(2 48 88)(3 49 89)(4 50 90)(5 51 91)(6 52 92)(7 53 93)(8 54 94)(9 55 95)(10 56 96)(11 57 97)(12 58 98)(13 59 99)(14 60 100)(15 61 101)(16 62 102)(17 63 103)(18 64 104)(19 65 105)(20 66 71)(21 67 72)(22 68 73)(23 69 74)(24 70 75)(25 36 76)(26 37 77)(27 38 78)(28 39 79)(29 40 80)(30 41 81)(31 42 82)(32 43 83)(33 44 84)(34 45 85)(35 46 86)
(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 97)(58 98)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(65 105)(66 71)(67 72)(68 73)(69 74)(70 75)
G:=sub<Sym(105)| (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)(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105), (1,47,87)(2,48,88)(3,49,89)(4,50,90)(5,51,91)(6,52,92)(7,53,93)(8,54,94)(9,55,95)(10,56,96)(11,57,97)(12,58,98)(13,59,99)(14,60,100)(15,61,101)(16,62,102)(17,63,103)(18,64,104)(19,65,105)(20,66,71)(21,67,72)(22,68,73)(23,69,74)(24,70,75)(25,36,76)(26,37,77)(27,38,78)(28,39,79)(29,40,80)(30,41,81)(31,42,82)(32,43,83)(33,44,84)(34,45,85)(35,46,86), (36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,71)(67,72)(68,73)(69,74)(70,75)>;
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)(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105), (1,47,87)(2,48,88)(3,49,89)(4,50,90)(5,51,91)(6,52,92)(7,53,93)(8,54,94)(9,55,95)(10,56,96)(11,57,97)(12,58,98)(13,59,99)(14,60,100)(15,61,101)(16,62,102)(17,63,103)(18,64,104)(19,65,105)(20,66,71)(21,67,72)(22,68,73)(23,69,74)(24,70,75)(25,36,76)(26,37,77)(27,38,78)(28,39,79)(29,40,80)(30,41,81)(31,42,82)(32,43,83)(33,44,84)(34,45,85)(35,46,86), (36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,71)(67,72)(68,73)(69,74)(70,75) );
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),(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)], [(1,47,87),(2,48,88),(3,49,89),(4,50,90),(5,51,91),(6,52,92),(7,53,93),(8,54,94),(9,55,95),(10,56,96),(11,57,97),(12,58,98),(13,59,99),(14,60,100),(15,61,101),(16,62,102),(17,63,103),(18,64,104),(19,65,105),(20,66,71),(21,67,72),(22,68,73),(23,69,74),(24,70,75),(25,36,76),(26,37,77),(27,38,78),(28,39,79),(29,40,80),(30,41,81),(31,42,82),(32,43,83),(33,44,84),(34,45,85),(35,46,86)], [(36,76),(37,77),(38,78),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(56,96),(57,97),(58,98),(59,99),(60,100),(61,101),(62,102),(63,103),(64,104),(65,105),(66,71),(67,72),(68,73),(69,74),(70,75)]])
105 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 7A | ··· | 7F | 10A | 10B | 10C | 10D | 14A | ··· | 14F | 15A | 15B | 15C | 15D | 21A | ··· | 21F | 35A | ··· | 35X | 70A | ··· | 70X | 105A | ··· | 105X |
| order | 1 | 2 | 3 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 10 | 10 | 10 | 10 | 14 | ··· | 14 | 15 | 15 | 15 | 15 | 21 | ··· | 21 | 35 | ··· | 35 | 70 | ··· | 70 | 105 | ··· | 105 |
| size | 1 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C5 | C7 | C10 | C14 | C35 | C70 | S3 | C5×S3 | S3×C7 | S3×C35 |
| kernel | S3×C35 | C105 | S3×C7 | C5×S3 | C21 | C15 | S3 | C3 | C35 | C7 | C5 | C1 |
| # reps | 1 | 1 | 4 | 6 | 4 | 6 | 24 | 24 | 1 | 4 | 6 | 24 |
Matrix representation of S3×C35 ►in GL3(𝔽211) generated by
| 107 | 0 | 0 |
| 0 | 58 | 0 |
| 0 | 0 | 58 |
| 1 | 0 | 0 |
| 0 | 210 | 210 |
| 0 | 1 | 0 |
| 210 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(211))| [107,0,0,0,58,0,0,0,58],[1,0,0,0,210,1,0,210,0],[210,0,0,0,0,1,0,1,0] >;
S3×C35 in GAP, Magma, Sage, TeX
S_3\times C_{35} % in TeX
G:=Group("S3xC35"); // GroupNames label
G:=SmallGroup(210,8);
// by ID
G=gap.SmallGroup(210,8);
# by ID
G:=PCGroup([4,-2,-5,-7,-3,2243]);
// Polycyclic
G:=Group<a,b,c|a^35=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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