direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3xS3xD5, D15:C6, C15:4D6, C32:3D10, (C5xS3):C6, C15:(C2xC6), C5:1(S3xC6), (C3xD5):C6, C3:1(C6xD5), (S3xC15):2C2, (C3xD15):1C2, (C3xC15):1C22, (C32xD5):1C2, SmallGroup(180,26)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C3xS3xD5 |
Generators and relations for C3xS3xD5
G = < a,b,c,d,e | a3=b3=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Quotients: C1, C2, C3, C22, S3, C6, D5, D6, C2xC6, C3xS3, D10, C3xD5, S3xC6, S3xD5, C6xD5, C3xS3xD5
Permutation representations of C3xS3xD5
►On 30 points - transitive group 30T44
Generators in S30
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)
(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)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)
G:=sub<Sym(30)| (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)>;
G:=Group( (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29) );
G=PermutationGroup([[(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30)], [(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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29)]])
G:=TransitiveGroup(30,44);
36 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 | 15B | 15C | 15D | 15E | ··· | 15J | 30A | 30B | 30C | 30D |
| 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 | 15 | 15 | 15 | ··· | 15 | 30 | 30 | 30 | 30 |
| size | 1 | 3 | 5 | 15 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 10 | 10 | 10 | 15 | 15 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D5 | D6 | C3xS3 | D10 | C3xD5 | S3xC6 | C6xD5 | S3xD5 | C3xS3xD5 |
| kernel | C3xS3xD5 | C32xD5 | S3xC15 | C3xD15 | S3xD5 | C5xS3 | C3xD5 | D15 | C3xD5 | C3xS3 | C15 | D5 | C32 | S3 | C5 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 4 |
Matrix representation of C3xS3xD5 ►in GL4(F31) generated by
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 0 | 30 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 30 |
G:=sub<GL(4,GF(31))| [5,0,0,0,0,5,0,0,0,0,1,0,0,0,0,1],[5,0,0,0,0,25,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,30,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,12,30] >;
C3xS3xD5 in GAP, Magma, Sage, TeX
C_3\times S_3\times D_5
% in TeX
G:=Group("C3xS3xD5"); // GroupNames label
G:=SmallGroup(180,26);
// by ID
G=gap.SmallGroup(180,26);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-5,248,3604]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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