direct product, abelian, monomial, 3-elementary
Aliases: C3×C15, SmallGroup(45,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C15 |
| C1 — C3×C15 |
| C1 — C3×C15 |
Generators and relations for C3×C15
G = < a,b | a3=b15=1, ab=ba >
Smallest permutation representation of C3×C15
►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)]])
C3×C15 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 | C3×C15 | C15 | C32 | C3 |
| # reps | 1 | 8 | 4 | 32 |
Matrix representation of C3×C15 ►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] >;
C3×C15 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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