direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9×D5, C5⋊C18, C45⋊3C2, C15.C6, C3.(C3×D5), (C3×D5).C3, SmallGroup(90,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C9×D5 |
Generators and relations for C9×D5
G = < a,b,c | a9=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C9×D5
►On 45 points
Generators in S45
(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)
(1 40 13 28 24)(2 41 14 29 25)(3 42 15 30 26)(4 43 16 31 27)(5 44 17 32 19)(6 45 18 33 20)(7 37 10 34 21)(8 38 11 35 22)(9 39 12 36 23)
(1 24)(2 25)(3 26)(4 27)(5 19)(6 20)(7 21)(8 22)(9 23)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 37)(35 38)(36 39)
G:=sub<Sym(45)| (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), (1,40,13,28,24)(2,41,14,29,25)(3,42,15,30,26)(4,43,16,31,27)(5,44,17,32,19)(6,45,18,33,20)(7,37,10,34,21)(8,38,11,35,22)(9,39,12,36,23), (1,24)(2,25)(3,26)(4,27)(5,19)(6,20)(7,21)(8,22)(9,23)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39)>;
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), (1,40,13,28,24)(2,41,14,29,25)(3,42,15,30,26)(4,43,16,31,27)(5,44,17,32,19)(6,45,18,33,20)(7,37,10,34,21)(8,38,11,35,22)(9,39,12,36,23), (1,24)(2,25)(3,26)(4,27)(5,19)(6,20)(7,21)(8,22)(9,23)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39) );
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)], [(1,40,13,28,24),(2,41,14,29,25),(3,42,15,30,26),(4,43,16,31,27),(5,44,17,32,19),(6,45,18,33,20),(7,37,10,34,21),(8,38,11,35,22),(9,39,12,36,23)], [(1,24),(2,25),(3,26),(4,27),(5,19),(6,20),(7,21),(8,22),(9,23),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,37),(35,38),(36,39)]])
C9×D5 is a maximal subgroup of
C9⋊F5
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 9A | ··· | 9F | 15A | 15B | 15C | 15D | 18A | ··· | 18F | 45A | ··· | 45L |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 9 | ··· | 9 | 15 | 15 | 15 | 15 | 18 | ··· | 18 | 45 | ··· | 45 |
| size | 1 | 5 | 1 | 1 | 2 | 2 | 5 | 5 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | D5 | C3×D5 | C9×D5 |
| kernel | C9×D5 | C45 | C3×D5 | C15 | D5 | C5 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 4 | 12 |
Matrix representation of C9×D5 ►in GL2(𝔽19) generated by
| 4 | 0 |
| 0 | 4 |
| 18 | 1 |
| 13 | 5 |
| 5 | 15 |
| 6 | 14 |
G:=sub<GL(2,GF(19))| [4,0,0,4],[18,13,1,5],[5,6,15,14] >;
C9×D5 in GAP, Magma, Sage, TeX
C_9\times D_5
% in TeX
G:=Group("C9xD5"); // GroupNames label
G:=SmallGroup(90,2);
// by ID
G=gap.SmallGroup(90,2);
# by ID
G:=PCGroup([4,-2,-3,-3,-5,29,1155]);
// Polycyclic
G:=Group<a,b,c|a^9=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export