direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D5xC30, C10:C30, C30:2C10, C5:(C2xC30), (C5xC10):3C6, (C5xC30):3C2, C52:4(C2xC6), C15:3(C2xC10), (C5xC15):8C22, SmallGroup(300,44)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5xC30 |
Generators and relations for D5xC30
G = < a,b,c | a30=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C3, C22, C5, C6, D5, C10, C2xC6, C15, D10, C2xC10, C3xD5, C30, C5xD5, C6xD5, C2xC30, D5xC10, D5xC15, D5xC30
Smallest permutation representation of D5xC30
►On 60 points
Generators in S60
(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)
(1 25 19 13 7)(2 26 20 14 8)(3 27 21 15 9)(4 28 22 16 10)(5 29 23 17 11)(6 30 24 18 12)(31 37 43 49 55)(32 38 44 50 56)(33 39 45 51 57)(34 40 46 52 58)(35 41 47 53 59)(36 42 48 54 60)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)
G:=sub<Sym(60)| (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), (1,25,19,13,7)(2,26,20,14,8)(3,27,21,15,9)(4,28,22,16,10)(5,29,23,17,11)(6,30,24,18,12)(31,37,43,49,55)(32,38,44,50,56)(33,39,45,51,57)(34,40,46,52,58)(35,41,47,53,59)(36,42,48,54,60), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)>;
G:=Group( (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), (1,25,19,13,7)(2,26,20,14,8)(3,27,21,15,9)(4,28,22,16,10)(5,29,23,17,11)(6,30,24,18,12)(31,37,43,49,55)(32,38,44,50,56)(33,39,45,51,57)(34,40,46,52,58)(35,41,47,53,59)(36,42,48,54,60), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44) );
G=PermutationGroup([[(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)], [(1,25,19,13,7),(2,26,20,14,8),(3,27,21,15,9),(4,28,22,16,10),(5,29,23,17,11),(6,30,24,18,12),(31,37,43,49,55),(32,38,44,50,56),(33,39,45,51,57),(34,40,46,52,58),(35,41,47,53,59),(36,42,48,54,60)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42),(29,43),(30,44)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 6A | 6B | 6C | 6D | 6E | 6F | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 10O | ··· | 10V | 15A | ··· | 15H | 15I | ··· | 15AB | 30A | ··· | 30H | 30I | ··· | 30AB | 30AC | ··· | 30AR |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | ··· | 15 | 15 | ··· | 15 | 30 | ··· | 30 | 30 | ··· | 30 | 30 | ··· | 30 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | D5 | D10 | C3xD5 | C5xD5 | C6xD5 | D5xC10 | D5xC15 | D5xC30 |
| kernel | D5xC30 | D5xC15 | C5xC30 | D5xC10 | C6xD5 | C5xD5 | C5xC10 | C3xD5 | C30 | D10 | D5 | C10 | C30 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 4 | 8 | 16 | 8 | 2 | 2 | 4 | 8 | 4 | 8 | 16 | 16 |
Matrix representation of D5xC30 ►in GL2(F31) generated by
| 12 | 0 |
| 0 | 12 |
| 19 | 30 |
| 20 | 30 |
| 30 | 0 |
| 11 | 1 |
G:=sub<GL(2,GF(31))| [12,0,0,12],[19,20,30,30],[30,11,0,1] >;
D5xC30 in GAP, Magma, Sage, TeX
D_5\times C_{30} % in TeX
G:=Group("D5xC30"); // GroupNames label
G:=SmallGroup(300,44);
// by ID
G=gap.SmallGroup(300,44);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5,6004]);
// Polycyclic
G:=Group<a,b,c|a^30=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export