direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C29⋊C4, C29⋊C12, C87⋊2C4, D29.C6, (C3×D29).2C2, SmallGroup(348,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — C3×C29⋊C4 |
Generators and relations for C3×C29⋊C4
G = < a,b,c | a3=b29=c4=1, ab=ba, ac=ca, cbc-1=b17 >
Smallest permutation representation of C3×C29⋊C4
►On 87 points
Generators in S87
(1 59 30)(2 60 31)(3 61 32)(4 62 33)(5 63 34)(6 64 35)(7 65 36)(8 66 37)(9 67 38)(10 68 39)(11 69 40)(12 70 41)(13 71 42)(14 72 43)(15 73 44)(16 74 45)(17 75 46)(18 76 47)(19 77 48)(20 78 49)(21 79 50)(22 80 51)(23 81 52)(24 82 53)(25 83 54)(26 84 55)(27 85 56)(28 86 57)(29 87 58)
(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)
(2 13 29 18)(3 25 28 6)(4 8 27 23)(5 20 26 11)(7 15 24 16)(9 10 22 21)(12 17 19 14)(31 42 58 47)(32 54 57 35)(33 37 56 52)(34 49 55 40)(36 44 53 45)(38 39 51 50)(41 46 48 43)(60 71 87 76)(61 83 86 64)(62 66 85 81)(63 78 84 69)(65 73 82 74)(67 68 80 79)(70 75 77 72)
G:=sub<Sym(87)| (1,59,30)(2,60,31)(3,61,32)(4,62,33)(5,63,34)(6,64,35)(7,65,36)(8,66,37)(9,67,38)(10,68,39)(11,69,40)(12,70,41)(13,71,42)(14,72,43)(15,73,44)(16,74,45)(17,75,46)(18,76,47)(19,77,48)(20,78,49)(21,79,50)(22,80,51)(23,81,52)(24,82,53)(25,83,54)(26,84,55)(27,85,56)(28,86,57)(29,87,58), (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), (2,13,29,18)(3,25,28,6)(4,8,27,23)(5,20,26,11)(7,15,24,16)(9,10,22,21)(12,17,19,14)(31,42,58,47)(32,54,57,35)(33,37,56,52)(34,49,55,40)(36,44,53,45)(38,39,51,50)(41,46,48,43)(60,71,87,76)(61,83,86,64)(62,66,85,81)(63,78,84,69)(65,73,82,74)(67,68,80,79)(70,75,77,72)>;
G:=Group( (1,59,30)(2,60,31)(3,61,32)(4,62,33)(5,63,34)(6,64,35)(7,65,36)(8,66,37)(9,67,38)(10,68,39)(11,69,40)(12,70,41)(13,71,42)(14,72,43)(15,73,44)(16,74,45)(17,75,46)(18,76,47)(19,77,48)(20,78,49)(21,79,50)(22,80,51)(23,81,52)(24,82,53)(25,83,54)(26,84,55)(27,85,56)(28,86,57)(29,87,58), (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), (2,13,29,18)(3,25,28,6)(4,8,27,23)(5,20,26,11)(7,15,24,16)(9,10,22,21)(12,17,19,14)(31,42,58,47)(32,54,57,35)(33,37,56,52)(34,49,55,40)(36,44,53,45)(38,39,51,50)(41,46,48,43)(60,71,87,76)(61,83,86,64)(62,66,85,81)(63,78,84,69)(65,73,82,74)(67,68,80,79)(70,75,77,72) );
G=PermutationGroup([[(1,59,30),(2,60,31),(3,61,32),(4,62,33),(5,63,34),(6,64,35),(7,65,36),(8,66,37),(9,67,38),(10,68,39),(11,69,40),(12,70,41),(13,71,42),(14,72,43),(15,73,44),(16,74,45),(17,75,46),(18,76,47),(19,77,48),(20,78,49),(21,79,50),(22,80,51),(23,81,52),(24,82,53),(25,83,54),(26,84,55),(27,85,56),(28,86,57),(29,87,58)], [(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)], [(2,13,29,18),(3,25,28,6),(4,8,27,23),(5,20,26,11),(7,15,24,16),(9,10,22,21),(12,17,19,14),(31,42,58,47),(32,54,57,35),(33,37,56,52),(34,49,55,40),(36,44,53,45),(38,39,51,50),(41,46,48,43),(60,71,87,76),(61,83,86,64),(62,66,85,81),(63,78,84,69),(65,73,82,74),(67,68,80,79),(70,75,77,72)]])
33 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 29A | ··· | 29G | 87A | ··· | 87N |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 29 | ··· | 29 | 87 | ··· | 87 |
| size | 1 | 29 | 1 | 1 | 29 | 29 | 29 | 29 | 29 | 29 | 29 | 29 | 4 | ··· | 4 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | C29⋊C4 | C3×C29⋊C4 |
| kernel | C3×C29⋊C4 | C3×D29 | C29⋊C4 | C87 | D29 | C29 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 7 | 14 |
Matrix representation of C3×C29⋊C4 ►in GL4(𝔽349) generated by
| 122 | 0 | 0 | 0 |
| 0 | 122 | 0 | 0 |
| 0 | 0 | 122 | 0 |
| 0 | 0 | 0 | 122 |
| 301 | 129 | 301 | 348 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 312 | 6 | 322 | 55 |
| 48 | 209 | 42 | 339 |
| 167 | 301 | 10 | 300 |
G:=sub<GL(4,GF(349))| [122,0,0,0,0,122,0,0,0,0,122,0,0,0,0,122],[301,1,0,0,129,0,1,0,301,0,0,1,348,0,0,0],[1,312,48,167,0,6,209,301,0,322,42,10,0,55,339,300] >;
C3×C29⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_{29}\rtimes C_4 % in TeX
G:=Group("C3xC29:C4"); // GroupNames label
G:=SmallGroup(348,5);
// by ID
G=gap.SmallGroup(348,5);
# by ID
G:=PCGroup([4,-2,-3,-2,-29,24,2307,907]);
// Polycyclic
G:=Group<a,b,c|a^3=b^29=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^17>;
// generators/relations
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