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G = S3×D29order 348 = 22·3·29

Direct product of S3 and D29

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

Aliases: S3×D29, D87⋊C2, C291D6, C31D58, C87⋊C22, (S3×C29)⋊C2, (C3×D29)⋊C2, SmallGroup(348,7)

Series: Derived Chief Lower central Upper central

C1C87 — S3×D29
C1C29C87C3×D29 — S3×D29
C87 — S3×D29
C1

Generators and relations for S3×D29
 G = < a,b,c,d | a3=b2=c29=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

3C2
29C2
87C2
87C22
29C6
29S3
3C58
3D29
29D6
3D58

Smallest permutation representation of S3×D29
On 87 points
Generators in S87
(1 50 80)(2 51 81)(3 52 82)(4 53 83)(5 54 84)(6 55 85)(7 56 86)(8 57 87)(9 58 59)(10 30 60)(11 31 61)(12 32 62)(13 33 63)(14 34 64)(15 35 65)(16 36 66)(17 37 67)(18 38 68)(19 39 69)(20 40 70)(21 41 71)(22 42 72)(23 43 73)(24 44 74)(25 45 75)(26 46 76)(27 47 77)(28 48 78)(29 49 79)
(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 59)
(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)
(1 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(30 40)(31 39)(32 38)(33 37)(34 36)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(72 87)(73 86)(74 85)(75 84)(76 83)(77 82)(78 81)(79 80)

G:=sub<Sym(87)| (1,50,80)(2,51,81)(3,52,82)(4,53,83)(5,54,84)(6,55,85)(7,56,86)(8,57,87)(9,58,59)(10,30,60)(11,31,61)(12,32,62)(13,33,63)(14,34,64)(15,35,65)(16,36,66)(17,37,67)(18,38,68)(19,39,69)(20,40,70)(21,41,71)(22,42,72)(23,43,73)(24,44,74)(25,45,75)(26,46,76)(27,47,77)(28,48,78)(29,49,79), (30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,59), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(30,40)(31,39)(32,38)(33,37)(34,36)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)>;

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

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

48 conjugacy classes

class 1 2A2B2C 3  6 29A···29N58A···58N87A···87N
order12223629···2958···5887···87
size1329872582···26···64···4

48 irreducible representations

dim111122224
type+++++++++
imageC1C2C2C2S3D6D29D58S3×D29
kernelS3×D29S3×C29C3×D29D87D29C29S3C3C1
# reps111111141414

Matrix representation of S3×D29 in GL4(𝔽349) generated by

1000
0100
00183
00206347
,
1000
0100
00348266
0001
,
56100
1583400
0010
0001
,
12614100
7822300
0010
0001
G:=sub<GL(4,GF(349))| [1,0,0,0,0,1,0,0,0,0,1,206,0,0,83,347],[1,0,0,0,0,1,0,0,0,0,348,0,0,0,266,1],[56,158,0,0,1,34,0,0,0,0,1,0,0,0,0,1],[126,78,0,0,141,223,0,0,0,0,1,0,0,0,0,1] >;

S3×D29 in GAP, Magma, Sage, TeX

S_3\times D_{29}
% in TeX

G:=Group("S3xD29");
// GroupNames label

G:=SmallGroup(348,7);
// by ID

G=gap.SmallGroup(348,7);
# by ID

G:=PCGroup([4,-2,-2,-3,-29,54,5379]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^29=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 S3×D29 in TeX

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