direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C15, C3⋊C30, C15⋊3C6, C32⋊1C10, (C3×C15)⋊4C2, SmallGroup(90,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C15 |
Generators and relations for S3×C15
G = < a,b,c | a15=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Permutation representations of S3×C15
►On 30 points - transitive group 30T15
Generators in S30
(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 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)
G:=sub<Sym(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), (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)>;
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), (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30) );
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)], [(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30)]])
G:=TransitiveGroup(30,15);
45 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 15A | ··· | 15H | 15I | ··· | 15T | 30A | ··· | 30H |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | S3 | C3×S3 | C5×S3 | S3×C15 |
| kernel | S3×C15 | C3×C15 | C5×S3 | C3×S3 | C15 | C32 | S3 | C3 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of S3×C15 ►in GL2(𝔽31) generated by
| 10 | 0 |
| 0 | 10 |
| 5 | 0 |
| 0 | 25 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(31))| [10,0,0,10],[5,0,0,25],[0,1,1,0] >;
S3×C15 in GAP, Magma, Sage, TeX
S_3\times C_{15} % in TeX
G:=Group("S3xC15"); // GroupNames label
G:=SmallGroup(90,6);
// by ID
G=gap.SmallGroup(90,6);
# by ID
G:=PCGroup([4,-2,-3,-5,-3,963]);
// Polycyclic
G:=Group<a,b,c|a^15=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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