Copied to
clipboard

G = S3×C35order 210 = 2·3·5·7

Direct product of C35 and S3

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C3 — S3×C35
C1C3C21C105 — S3×C35
C3 — S3×C35
C1C35

Generators and relations for S3×C35
 G = < a,b,c | a35=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C10
3C14
3C70

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 5A5B5C5D7A···7F10A10B10C10D14A···14F15A15B15C15D21A···21F35A···35X70A···70X105A···105X
order12355557···71010101014···141515151521···2135···3570···70105···105
size13211111···133333···322222···21···13···32···2

105 irreducible representations

dim111111112222
type+++
imageC1C2C5C7C10C14C35C70S3C5×S3S3×C7S3×C35
kernelS3×C35C105S3×C7C5×S3C21C15S3C3C35C7C5C1
# reps114646242414624

Matrix representation of S3×C35 in GL3(𝔽211) generated by

10700
0580
0058
,
100
0210210
010
,
21000
001
010
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

Subgroup lattice of S3×C35 in TeX

׿
×
𝔽