direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C10xD15, C30:3D5, C30:1C10, C52:7D6, C15:8D10, C6:(C5xD5), C10:(C5xS3), C3:2(D5xC10), C5:2(S3xC10), (C5xC10):2S3, (C5xC30):2C2, C15:2(C2xC10), (C5xC15):7C22, SmallGroup(300,47)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C10xD15 |
Generators and relations for C10xD15
G = < a,b,c | a10=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C22, C5, S3, D5, C10, D6, D10, C2xC10, C5xS3, D15, C5xD5, S3xC10, D30, D5xC10, C5xD15, C10xD15
Smallest permutation representation of C10xD15
►On 60 points
Generators in S60
(1 26 13 23 10 20 7 17 4 29)(2 27 14 24 11 21 8 18 5 30)(3 28 15 25 12 22 9 19 6 16)(31 46 34 49 37 52 40 55 43 58)(32 47 35 50 38 53 41 56 44 59)(33 48 36 51 39 54 42 57 45 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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 60)(9 59)(10 58)(11 57)(12 56)(13 55)(14 54)(15 53)(16 35)(17 34)(18 33)(19 32)(20 31)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(29 37)(30 36)
G:=sub<Sym(60)| (1,26,13,23,10,20,7,17,4,29)(2,27,14,24,11,21,8,18,5,30)(3,28,15,25,12,22,9,19,6,16)(31,46,34,49,37,52,40,55,43,58)(32,47,35,50,38,53,41,56,44,59)(33,48,36,51,39,54,42,57,45,60), (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,35)(17,34)(18,33)(19,32)(20,31)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)>;
G:=Group( (1,26,13,23,10,20,7,17,4,29)(2,27,14,24,11,21,8,18,5,30)(3,28,15,25,12,22,9,19,6,16)(31,46,34,49,37,52,40,55,43,58)(32,47,35,50,38,53,41,56,44,59)(33,48,36,51,39,54,42,57,45,60), (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,35)(17,34)(18,33)(19,32)(20,31)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36) );
G=PermutationGroup([[(1,26,13,23,10,20,7,17,4,29),(2,27,14,24,11,21,8,18,5,30),(3,28,15,25,12,22,9,19,6,16),(31,46,34,49,37,52,40,55,43,58),(32,47,35,50,38,53,41,56,44,59),(33,48,36,51,39,54,42,57,45,60)], [(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)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,60),(9,59),(10,58),(11,57),(12,56),(13,55),(14,54),(15,53),(16,35),(17,34),(18,33),(19,32),(20,31),(21,45),(22,44),(23,43),(24,42),(25,41),(26,40),(27,39),(28,38),(29,37),(30,36)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 6 | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 10O | ··· | 10V | 15A | ··· | 15X | 30A | ··· | 30X |
| order | 1 | 2 | 2 | 2 | 3 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 15 | 15 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 15 | ··· | 15 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D5 | D6 | D10 | C5xS3 | D15 | C5xD5 | S3xC10 | D30 | D5xC10 | C5xD15 | C10xD15 |
| kernel | C10xD15 | C5xD15 | C5xC30 | D30 | D15 | C30 | C5xC10 | C30 | C52 | C15 | C10 | C10 | C6 | C5 | C5 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 2 | 1 | 2 | 4 | 4 | 8 | 4 | 4 | 8 | 16 | 16 |
Matrix representation of C10xD15 ►in GL2(F31) generated by
| 29 | 0 |
| 0 | 29 |
| 8 | 14 |
| 6 | 30 |
| 30 | 0 |
| 25 | 1 |
G:=sub<GL(2,GF(31))| [29,0,0,29],[8,6,14,30],[30,25,0,1] >;
C10xD15 in GAP, Magma, Sage, TeX
C_{10}\times D_{15} % in TeX
G:=Group("C10xD15"); // GroupNames label
G:=SmallGroup(300,47);
// by ID
G=gap.SmallGroup(300,47);
# by ID
G:=PCGroup([5,-2,-2,-5,-3,-5,803,6004]);
// Polycyclic
G:=Group<a,b,c|a^10=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export