direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×D35, C35⋊4D6, C3⋊1D70, C15⋊2D14, C21⋊2D10, D105⋊1C2, C105⋊1C22, (C5×S3)⋊D7, (S3×C7)⋊D5, C7⋊1(S3×D5), C5⋊1(S3×D7), (S3×C35)⋊1C2, (C3×D35)⋊1C2, SmallGroup(420,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C105 — S3×D35 |
Generators and relations for S3×D35
G = < a,b,c,d | a3=b2=c35=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
3C2
35C2
105C2
105C22
35S3
35C6
3C10
7D5
21D5
3C14
5D7
15D7
35D6
21D10
15D14
7D15
5D21
3C70
3D35
3D70
Smallest permutation representation of S3×D35
►On 105 points
Generators in S105
(1 63 103)(2 64 104)(3 65 105)(4 66 71)(5 67 72)(6 68 73)(7 69 74)(8 70 75)(9 36 76)(10 37 77)(11 38 78)(12 39 79)(13 40 80)(14 41 81)(15 42 82)(16 43 83)(17 44 84)(18 45 85)(19 46 86)(20 47 87)(21 48 88)(22 49 89)(23 50 90)(24 51 91)(25 52 92)(26 53 93)(27 54 94)(28 55 95)(29 56 96)(30 57 97)(31 58 98)(32 59 99)(33 60 100)(34 61 101)(35 62 102)
(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)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(36 54)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)(100 105)(101 104)(102 103)
G:=sub<Sym(105)| (1,63,103)(2,64,104)(3,65,105)(4,66,71)(5,67,72)(6,68,73)(7,69,74)(8,70,75)(9,36,76)(10,37,77)(11,38,78)(12,39,79)(13,40,80)(14,41,81)(15,42,82)(16,43,83)(17,44,84)(18,45,85)(19,46,86)(20,47,87)(21,48,88)(22,49,89)(23,50,90)(24,51,91)(25,52,92)(26,53,93)(27,54,94)(28,55,95)(29,56,96)(30,57,97)(31,58,98)(32,59,99)(33,60,100)(34,61,101)(35,62,102), (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), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)(100,105)(101,104)(102,103)>;
G:=Group( (1,63,103)(2,64,104)(3,65,105)(4,66,71)(5,67,72)(6,68,73)(7,69,74)(8,70,75)(9,36,76)(10,37,77)(11,38,78)(12,39,79)(13,40,80)(14,41,81)(15,42,82)(16,43,83)(17,44,84)(18,45,85)(19,46,86)(20,47,87)(21,48,88)(22,49,89)(23,50,90)(24,51,91)(25,52,92)(26,53,93)(27,54,94)(28,55,95)(29,56,96)(30,57,97)(31,58,98)(32,59,99)(33,60,100)(34,61,101)(35,62,102), (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), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)(100,105)(101,104)(102,103) );
G=PermutationGroup([[(1,63,103),(2,64,104),(3,65,105),(4,66,71),(5,67,72),(6,68,73),(7,69,74),(8,70,75),(9,36,76),(10,37,77),(11,38,78),(12,39,79),(13,40,80),(14,41,81),(15,42,82),(16,43,83),(17,44,84),(18,45,85),(19,46,86),(20,47,87),(21,48,88),(22,49,89),(23,50,90),(24,51,91),(25,52,92),(26,53,93),(27,54,94),(28,55,95),(29,56,96),(30,57,97),(31,58,98),(32,59,99),(33,60,100),(34,61,101),(35,62,102)], [(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)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(36,54),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(55,70),(56,69),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(81,89),(82,88),(83,87),(84,86),(100,105),(101,104),(102,103)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 6 | 7A | 7B | 7C | 10A | 10B | 14A | 14B | 14C | 15A | 15B | 21A | 21B | 21C | 35A | ··· | 35L | 70A | ··· | 70L | 105A | ··· | 105L |
| order | 1 | 2 | 2 | 2 | 3 | 5 | 5 | 6 | 7 | 7 | 7 | 10 | 10 | 14 | 14 | 14 | 15 | 15 | 21 | 21 | 21 | 35 | ··· | 35 | 70 | ··· | 70 | 105 | ··· | 105 |
| size | 1 | 3 | 35 | 105 | 2 | 2 | 2 | 70 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D5 | D6 | D7 | D10 | D14 | D35 | D70 | S3×D5 | S3×D7 | S3×D35 |
| kernel | S3×D35 | C3×D35 | S3×C35 | D105 | D35 | S3×C7 | C35 | C5×S3 | C21 | C15 | S3 | C3 | C7 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 3 | 12 | 12 | 2 | 3 | 12 |
Matrix representation of S3×D35 ►in GL4(𝔽211) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 209 | 29 |
| 0 | 0 | 29 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 182 | 210 |
| 79 | 9 | 0 | 0 |
| 193 | 22 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 18 | 56 | 0 | 0 |
| 209 | 193 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(211))| [1,0,0,0,0,1,0,0,0,0,209,29,0,0,29,1],[1,0,0,0,0,1,0,0,0,0,1,182,0,0,0,210],[79,193,0,0,9,22,0,0,0,0,1,0,0,0,0,1],[18,209,0,0,56,193,0,0,0,0,1,0,0,0,0,1] >;
S3×D35 in GAP, Magma, Sage, TeX
S_3\times D_{35} % in TeX
G:=Group("S3xD35"); // GroupNames label
G:=SmallGroup(420,29);
// by ID
G=gap.SmallGroup(420,29);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-7,67,963,9004]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^35=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export