direct product, abelian, monomial, 3-elementary
Aliases: C3xC15, SmallGroup(45,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3xC15 |
| C1 — C3xC15 |
| C1 — C3xC15 |
Generators and relations for C3xC15
G = < a,b | a3=b15=1, ab=ba >
Quotients: C1, C3, C5, C32, C15, C3xC15
Smallest permutation representation of C3xC15
►Regular action on 45 points
Generators in S45
(1 38 16)(2 39 17)(3 40 18)(4 41 19)(5 42 20)(6 43 21)(7 44 22)(8 45 23)(9 31 24)(10 32 25)(11 33 26)(12 34 27)(13 35 28)(14 36 29)(15 37 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)(31 32 33 34 35 36 37 38 39 40 41 42 43 44 45)
G:=sub<Sym(45)| (1,38,16)(2,39,17)(3,40,18)(4,41,19)(5,42,20)(6,43,21)(7,44,22)(8,45,23)(9,31,24)(10,32,25)(11,33,26)(12,34,27)(13,35,28)(14,36,29)(15,37,30), (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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)>;
G:=Group( (1,38,16)(2,39,17)(3,40,18)(4,41,19)(5,42,20)(6,43,21)(7,44,22)(8,45,23)(9,31,24)(10,32,25)(11,33,26)(12,34,27)(13,35,28)(14,36,29)(15,37,30), (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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45) );
G=PermutationGroup([[(1,38,16),(2,39,17),(3,40,18),(4,41,19),(5,42,20),(6,43,21),(7,44,22),(8,45,23),(9,31,24),(10,32,25),(11,33,26),(12,34,27),(13,35,28),(14,36,29),(15,37,30)], [(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),(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)]])
C3xC15 is a maximal subgroup of
C3:D15
45 conjugacy classes
| class | 1 | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 15A | ··· | 15AF |
| order | 1 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 15 | ··· | 15 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C5 | C15 |
| kernel | C3xC15 | C15 | C32 | C3 |
| # reps | 1 | 8 | 4 | 32 |
Matrix representation of C3xC15 ►in GL2(F31) generated by
| 5 | 0 |
| 0 | 5 |
| 10 | 0 |
| 0 | 7 |
G:=sub<GL(2,GF(31))| [5,0,0,5],[10,0,0,7] >;
C3xC15 in GAP, Magma, Sage, TeX
C_3\times C_{15} % in TeX
G:=Group("C3xC15"); // GroupNames label
G:=SmallGroup(45,2);
// by ID
G=gap.SmallGroup(45,2);
# by ID
G:=PCGroup([3,-3,-3,-5]);
// Polycyclic
G:=Group<a,b|a^3=b^15=1,a*b=b*a>;
// generators/relations
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