direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C6xD15, C30:1C6, C30:2S3, C15:7D6, C32:5D10, C6:(C3xD5), C10:(C3xS3), C5:2(S3xC6), (C3xC6):1D5, C3:2(C6xD5), C15:2(C2xC6), (C3xC30):2C2, (C3xC15):7C22, SmallGroup(180,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C6xD15 |
Generators and relations for C6xD15
G = < a,b,c | a6=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C3, C22, S3, C6, D5, D6, C2xC6, C3xS3, D10, C3xD5, D15, S3xC6, C6xD5, D30, C3xD15, C6xD15
Smallest permutation representation of C6xD15
►On 60 points
Generators in S60
(1 27 6 17 11 22)(2 28 7 18 12 23)(3 29 8 19 13 24)(4 30 9 20 14 25)(5 16 10 21 15 26)(31 49 41 59 36 54)(32 50 42 60 37 55)(33 51 43 46 38 56)(34 52 44 47 39 57)(35 53 45 48 40 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)
(1 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 45)(29 44)(30 43)
G:=sub<Sym(60)| (1,27,6,17,11,22)(2,28,7,18,12,23)(3,29,8,19,13,24)(4,30,9,20,14,25)(5,16,10,21,15,26)(31,49,41,59,36,54)(32,50,42,60,37,55)(33,51,43,46,38,56)(34,52,44,47,39,57)(35,53,45,48,40,58), (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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,45)(29,44)(30,43)>;
G:=Group( (1,27,6,17,11,22)(2,28,7,18,12,23)(3,29,8,19,13,24)(4,30,9,20,14,25)(5,16,10,21,15,26)(31,49,41,59,36,54)(32,50,42,60,37,55)(33,51,43,46,38,56)(34,52,44,47,39,57)(35,53,45,48,40,58), (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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,45)(29,44)(30,43) );
G=PermutationGroup([[(1,27,6,17,11,22),(2,28,7,18,12,23),(3,29,8,19,13,24),(4,30,9,20,14,25),(5,16,10,21,15,26),(31,49,41,59,36,54),(32,50,42,60,37,55),(33,51,43,46,38,56),(34,52,44,47,39,57),(35,53,45,48,40,58)], [(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)], [(1,54),(2,53),(3,52),(4,51),(5,50),(6,49),(7,48),(8,47),(9,46),(10,60),(11,59),(12,58),(13,57),(14,56),(15,55),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,45),(29,44),(30,43)]])
C6xD15 is a maximal subgroup of
D6:D15 C3:D60 D30.S3 D30:S3 C32:3D20 S3xC6xD5
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 10A | 10B | 15A | ··· | 15P | 30A | ··· | 30P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 15 | 15 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 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 | C3 | C6 | C6 | S3 | D5 | D6 | C3xS3 | D10 | C3xD5 | D15 | S3xC6 | C6xD5 | D30 | C3xD15 | C6xD15 |
| kernel | C6xD15 | C3xD15 | C3xC30 | D30 | D15 | C30 | C30 | C3xC6 | C15 | C10 | C32 | C6 | C6 | C5 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C6xD15 ►in GL2(F31) generated by
| 6 | 0 |
| 0 | 6 |
| 28 | 0 |
| 22 | 10 |
| 4 | 8 |
| 2 | 27 |
G:=sub<GL(2,GF(31))| [6,0,0,6],[28,22,0,10],[4,2,8,27] >;
C6xD15 in GAP, Magma, Sage, TeX
C_6\times D_{15} % in TeX
G:=Group("C6xD15"); // GroupNames label
G:=SmallGroup(180,34);
// by ID
G=gap.SmallGroup(180,34);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-5,483,3604]);
// Polycyclic
G:=Group<a,b,c|a^6=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export