direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D29, C29⋊C6, C87⋊2C2, SmallGroup(174,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — C3×D29 |
Generators and relations for C3×D29
G = < a,b,c | a3=b29=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D29
►On 87 points
Generators in S87
(1 82 57)(2 83 58)(3 84 30)(4 85 31)(5 86 32)(6 87 33)(7 59 34)(8 60 35)(9 61 36)(10 62 37)(11 63 38)(12 64 39)(13 65 40)(14 66 41)(15 67 42)(16 68 43)(17 69 44)(18 70 45)(19 71 46)(20 72 47)(21 73 48)(22 74 49)(23 75 50)(24 76 51)(25 77 52)(26 78 53)(27 79 54)(28 80 55)(29 81 56)
(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 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(55 58)(56 57)(59 75)(60 74)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(76 87)(77 86)(78 85)(79 84)(80 83)(81 82)
G:=sub<Sym(87)| (1,82,57)(2,83,58)(3,84,30)(4,85,31)(5,86,32)(6,87,33)(7,59,34)(8,60,35)(9,61,36)(10,62,37)(11,63,38)(12,64,39)(13,65,40)(14,66,41)(15,67,42)(16,68,43)(17,69,44)(18,70,45)(19,71,46)(20,72,47)(21,73,48)(22,74,49)(23,75,50)(24,76,51)(25,77,52)(26,78,53)(27,79,54)(28,80,55)(29,81,56), (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)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(55,58)(56,57)(59,75)(60,74)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)>;
G:=Group( (1,82,57)(2,83,58)(3,84,30)(4,85,31)(5,86,32)(6,87,33)(7,59,34)(8,60,35)(9,61,36)(10,62,37)(11,63,38)(12,64,39)(13,65,40)(14,66,41)(15,67,42)(16,68,43)(17,69,44)(18,70,45)(19,71,46)(20,72,47)(21,73,48)(22,74,49)(23,75,50)(24,76,51)(25,77,52)(26,78,53)(27,79,54)(28,80,55)(29,81,56), (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)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(55,58)(56,57)(59,75)(60,74)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82) );
G=PermutationGroup([[(1,82,57),(2,83,58),(3,84,30),(4,85,31),(5,86,32),(6,87,33),(7,59,34),(8,60,35),(9,61,36),(10,62,37),(11,63,38),(12,64,39),(13,65,40),(14,66,41),(15,67,42),(16,68,43),(17,69,44),(18,70,45),(19,71,46),(20,72,47),(21,73,48),(22,74,49),(23,75,50),(24,76,51),(25,77,52),(26,78,53),(27,79,54),(28,80,55),(29,81,56)], [(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),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(55,58),(56,57),(59,75),(60,74),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(76,87),(77,86),(78,85),(79,84),(80,83),(81,82)]])
C3×D29 is a maximal subgroup of
C87⋊C4
48 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 29A | ··· | 29N | 87A | ··· | 87AB |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 29 | ··· | 29 | 87 | ··· | 87 |
| size | 1 | 29 | 1 | 1 | 29 | 29 | 2 | ··· | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D29 | C3×D29 |
| kernel | C3×D29 | C87 | D29 | C29 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 14 | 28 |
Matrix representation of C3×D29 ►in GL2(𝔽349) generated by
| 226 | 0 |
| 0 | 226 |
| 18 | 1 |
| 348 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(349))| [226,0,0,226],[18,348,1,0],[0,1,1,0] >;
C3×D29 in GAP, Magma, Sage, TeX
C_3\times D_{29} % in TeX
G:=Group("C3xD29"); // GroupNames label
G:=SmallGroup(174,2);
// by ID
G=gap.SmallGroup(174,2);
# by ID
G:=PCGroup([3,-2,-3,-29,1514]);
// Polycyclic
G:=Group<a,b,c|a^3=b^29=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export