direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C15×Dic5, C5⋊2C60, C10.C30, C15⋊4C20, C52⋊8C12, C30.8D5, C30.2C10, C2.(D5×C15), (C5×C15)⋊11C4, C6.2(C5×D5), (C5×C10).3C6, (C5×C30).3C2, C10.4(C3×D5), SmallGroup(300,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C15×Dic5 |
Generators and relations for C15×Dic5
G = < a,b,c | a15=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C15×Dic5
►On 60 points
Generators in S60
(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 48 10 57 4 51 13 60 7 54)(2 49 11 58 5 52 14 46 8 55)(3 50 12 59 6 53 15 47 9 56)(16 35 22 41 28 32 19 38 25 44)(17 36 23 42 29 33 20 39 26 45)(18 37 24 43 30 34 21 40 27 31)
(1 37 51 21)(2 38 52 22)(3 39 53 23)(4 40 54 24)(5 41 55 25)(6 42 56 26)(7 43 57 27)(8 44 58 28)(9 45 59 29)(10 31 60 30)(11 32 46 16)(12 33 47 17)(13 34 48 18)(14 35 49 19)(15 36 50 20)
G:=sub<Sym(60)| (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), (1,48,10,57,4,51,13,60,7,54)(2,49,11,58,5,52,14,46,8,55)(3,50,12,59,6,53,15,47,9,56)(16,35,22,41,28,32,19,38,25,44)(17,36,23,42,29,33,20,39,26,45)(18,37,24,43,30,34,21,40,27,31), (1,37,51,21)(2,38,52,22)(3,39,53,23)(4,40,54,24)(5,41,55,25)(6,42,56,26)(7,43,57,27)(8,44,58,28)(9,45,59,29)(10,31,60,30)(11,32,46,16)(12,33,47,17)(13,34,48,18)(14,35,49,19)(15,36,50,20)>;
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)(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), (1,48,10,57,4,51,13,60,7,54)(2,49,11,58,5,52,14,46,8,55)(3,50,12,59,6,53,15,47,9,56)(16,35,22,41,28,32,19,38,25,44)(17,36,23,42,29,33,20,39,26,45)(18,37,24,43,30,34,21,40,27,31), (1,37,51,21)(2,38,52,22)(3,39,53,23)(4,40,54,24)(5,41,55,25)(6,42,56,26)(7,43,57,27)(8,44,58,28)(9,45,59,29)(10,31,60,30)(11,32,46,16)(12,33,47,17)(13,34,48,18)(14,35,49,19)(15,36,50,20) );
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),(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)], [(1,48,10,57,4,51,13,60,7,54),(2,49,11,58,5,52,14,46,8,55),(3,50,12,59,6,53,15,47,9,56),(16,35,22,41,28,32,19,38,25,44),(17,36,23,42,29,33,20,39,26,45),(18,37,24,43,30,34,21,40,27,31)], [(1,37,51,21),(2,38,52,22),(3,39,53,23),(4,40,54,24),(5,41,55,25),(6,42,56,26),(7,43,57,27),(8,44,58,28),(9,45,59,29),(10,31,60,30),(11,32,46,16),(12,33,47,17),(13,34,48,18),(14,35,49,19),(15,36,50,20)]])
120 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 6A | 6B | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 15I | ··· | 15AB | 20A | ··· | 20H | 30A | ··· | 30H | 30I | ··· | 30AB | 60A | ··· | 60P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C10 | C12 | C15 | C20 | C30 | C60 | D5 | Dic5 | C3×D5 | C5×D5 | C3×Dic5 | C5×Dic5 | D5×C15 | C15×Dic5 |
| kernel | C15×Dic5 | C5×C30 | C5×Dic5 | C5×C15 | C3×Dic5 | C5×C10 | C30 | C52 | Dic5 | C15 | C10 | C5 | C30 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 2 | 2 | 4 | 8 | 4 | 8 | 16 | 16 |
Matrix representation of C15×Dic5 ►in GL2(𝔽31) generated by
| 19 | 0 |
| 0 | 19 |
| 23 | 0 |
| 0 | 27 |
| 0 | 26 |
| 25 | 0 |
G:=sub<GL(2,GF(31))| [19,0,0,19],[23,0,0,27],[0,25,26,0] >;
C15×Dic5 in GAP, Magma, Sage, TeX
C_{15}\times {\rm Dic}_5 % in TeX
G:=Group("C15xDic5"); // GroupNames label
G:=SmallGroup(300,16);
// by ID
G=gap.SmallGroup(300,16);
# by ID
G:=PCGroup([5,-2,-3,-5,-2,-5,150,6004]);
// Polycyclic
G:=Group<a,b,c|a^15=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export