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## G = S3×D35order 420 = 22·3·5·7

### Direct product of S3 and D35

Aliases: S3×D35, C354D6, C31D70, C152D14, C212D10, D1051C2, C1051C22, (C5×S3)⋊D7, (S3×C7)⋊D5, C71(S3×D5), C51(S3×D7), (S3×C35)⋊1C2, (C3×D35)⋊1C2, SmallGroup(420,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C105 — S3×D35
 Chief series C1 — C7 — C35 — C105 — C3×D35 — S3×D35
 Lower central C105 — S3×D35
 Upper central C1

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

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