Copied to
clipboard

## G = S3×C35order 210 = 2·3·5·7

### Direct product of C35 and S3

Aliases: S3×C35, C3⋊C70, C1057C2, C153C14, C213C10, SmallGroup(210,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C35
 Chief series C1 — C3 — C21 — C105 — S3×C35
 Lower central C3 — S3×C35
 Upper central C1 — 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 84 65)(2 85 66)(3 86 67)(4 87 68)(5 88 69)(6 89 70)(7 90 36)(8 91 37)(9 92 38)(10 93 39)(11 94 40)(12 95 41)(13 96 42)(14 97 43)(15 98 44)(16 99 45)(17 100 46)(18 101 47)(19 102 48)(20 103 49)(21 104 50)(22 105 51)(23 71 52)(24 72 53)(25 73 54)(26 74 55)(27 75 56)(28 76 57)(29 77 58)(30 78 59)(31 79 60)(32 80 61)(33 81 62)(34 82 63)(35 83 64)
(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)

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), (1,84,65)(2,85,66)(3,86,67)(4,87,68)(5,88,69)(6,89,70)(7,90,36)(8,91,37)(9,92,38)(10,93,39)(11,94,40)(12,95,41)(13,96,42)(14,97,43)(15,98,44)(16,99,45)(17,100,46)(18,101,47)(19,102,48)(20,103,49)(21,104,50)(22,105,51)(23,71,52)(24,72,53)(25,73,54)(26,74,55)(27,75,56)(28,76,57)(29,77,58)(30,78,59)(31,79,60)(32,80,61)(33,81,62)(34,82,63)(35,83,64), (36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)>;

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), (1,84,65)(2,85,66)(3,86,67)(4,87,68)(5,88,69)(6,89,70)(7,90,36)(8,91,37)(9,92,38)(10,93,39)(11,94,40)(12,95,41)(13,96,42)(14,97,43)(15,98,44)(16,99,45)(17,100,46)(18,101,47)(19,102,48)(20,103,49)(21,104,50)(22,105,51)(23,71,52)(24,72,53)(25,73,54)(26,74,55)(27,75,56)(28,76,57)(29,77,58)(30,78,59)(31,79,60)(32,80,61)(33,81,62)(34,82,63)(35,83,64), (36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89) );

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)], [(1,84,65),(2,85,66),(3,86,67),(4,87,68),(5,88,69),(6,89,70),(7,90,36),(8,91,37),(9,92,38),(10,93,39),(11,94,40),(12,95,41),(13,96,42),(14,97,43),(15,98,44),(16,99,45),(17,100,46),(18,101,47),(19,102,48),(20,103,49),(21,104,50),(22,105,51),(23,71,52),(24,72,53),(25,73,54),(26,74,55),(27,75,56),(28,76,57),(29,77,58),(30,78,59),(31,79,60),(32,80,61),(33,81,62),(34,82,63),(35,83,64)], [(36,90),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89)])

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

Export

׿
×
𝔽