direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×D29, D87⋊C2, C29⋊1D6, C3⋊1D58, C87⋊C22, (S3×C29)⋊C2, (C3×D29)⋊C2, SmallGroup(348,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C87 — S3×D29 |
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 >
Smallest permutation representation of S3×D29
►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 | 2A | 2B | 2C | 3 | 6 | 29A | ··· | 29N | 58A | ··· | 58N | 87A | ··· | 87N |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 29 | ··· | 29 | 58 | ··· | 58 | 87 | ··· | 87 |
| size | 1 | 3 | 29 | 87 | 2 | 58 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D6 | D29 | D58 | S3×D29 |
| kernel | S3×D29 | S3×C29 | C3×D29 | D87 | D29 | C29 | S3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 14 | 14 | 14 |
Matrix representation of S3×D29 ►in GL4(𝔽349) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 83 |
| 0 | 0 | 206 | 347 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 348 | 266 |
| 0 | 0 | 0 | 1 |
| 56 | 1 | 0 | 0 |
| 158 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 126 | 141 | 0 | 0 |
| 78 | 223 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
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
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