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G = S3xD35order 420 = 22·3·5·7

Direct product of S3 and D35

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3xD35, C35:4D6, C3:1D70, C15:2D14, C21:2D10, D105:1C2, C105:1C22, (C5xS3):D7, (S3xC7):D5, C7:1(S3xD5), C5:1(S3xD7), (S3xC35):1C2, (C3xD35):1C2, SmallGroup(420,29)

Series: Derived Chief Lower central Upper central

C1C105 — S3xD35
C1C7C35C105C3xD35 — S3xD35
C105 — S3xD35
C1

Generators and relations for S3xD35
 G = < a,b,c,d | a3=b2=c35=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 504 in 40 conjugacy classes, 16 normal (all characteristic)
Quotients: C1, C2, C22, S3, D5, D6, D7, D10, D14, S3xD5, D35, S3xD7, D70, S3xD35
3C2
35C2
105C2
105C22
35S3
35C6
3C10
7D5
21D5
3C14
5D7
15D7
35D6
21D10
15D14
7D15
7C3xD5
5D21
5C3xD7
3C70
3D35
7S3xD5
5S3xD7
3D70

Smallest permutation representation of S3xD35
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 2A2B2C 3 5A5B 6 7A7B7C10A10B14A14B14C15A15B21A21B21C35A···35L70A···70L105A···105L
order122235567771010141414151521212135···3570···70105···105
size13351052227022266666444442···26···64···4

57 irreducible representations

dim111122222222444
type+++++++++++++++
imageC1C2C2C2S3D5D6D7D10D14D35D70S3xD5S3xD7S3xD35
kernelS3xD35C3xD35S3xC35D105D35S3xC7C35C5xS3C21C15S3C3C7C5C1
# reps111112132312122312

Matrix representation of S3xD35 in GL4(F211) generated by

1000
0100
0020929
00291
,
1000
0100
0010
00182210
,
79900
1932200
0010
0001
,
185600
20919300
0010
0001
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] >;

S3xD35 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

Subgroup lattice of S3xD35 in TeX

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