Copied to
clipboard

G = S3xD29order 348 = 22·3·29

Direct product of S3 and D29

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

Aliases: S3xD29, D87:C2, C29:1D6, C3:1D58, C87:C22, (S3xC29):C2, (C3xD29):C2, SmallGroup(348,7)

Series: Derived Chief Lower central Upper central

C1C87 — S3xD29
C1C29C87C3xD29 — S3xD29
C87 — S3xD29
C1

Generators and relations for S3xD29
 G = < a,b,c,d | a3=b2=c29=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 312 in 20 conjugacy classes, 10 normal (all characteristic)
Quotients: C1, C2, C22, S3, D6, D29, D58, S3xD29
3C2
29C2
87C2
87C22
29C6
29S3
3C58
3D29
29D6
3D58

Smallest permutation representation of S3xD29
On 87 points
Generators in S87
(1 34 86)(2 35 87)(3 36 59)(4 37 60)(5 38 61)(6 39 62)(7 40 63)(8 41 64)(9 42 65)(10 43 66)(11 44 67)(12 45 68)(13 46 69)(14 47 70)(15 48 71)(16 49 72)(17 50 73)(18 51 74)(19 52 75)(20 53 76)(21 54 77)(22 55 78)(23 56 79)(24 57 80)(25 58 81)(26 30 82)(27 31 83)(28 32 84)(29 33 85)
(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 81)
(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 37)(31 36)(32 35)(33 34)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(59 83)(60 82)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(84 87)(85 86)

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

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

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

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+++++++++
imageC1C2C2C2S3D6D29D58S3xD29
kernelS3xD29S3xC29C3xD29D87D29C29S3C3C1
# reps111111141414

Matrix representation of S3xD29 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] >;

S3xD29 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 S3xD29 in TeX

׿
x
:
Z
F
o
wr
Q
<