direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9×F5, C5⋊C36, C45⋊2C4, C15.C12, D5.C18, C3.(C3×F5), (C3×F5).C3, (C3×D5).2C6, (C9×D5).2C2, SmallGroup(180,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C9×F5 |
Generators and relations for C9×F5
G = < a,b,c | a9=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C9×F5
►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 29 22 16 40)(2 30 23 17 41)(3 31 24 18 42)(4 32 25 10 43)(5 33 26 11 44)(6 34 27 12 45)(7 35 19 13 37)(8 36 20 14 38)(9 28 21 15 39)
(10 32 25 43)(11 33 26 44)(12 34 27 45)(13 35 19 37)(14 36 20 38)(15 28 21 39)(16 29 22 40)(17 30 23 41)(18 31 24 42)
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), (1,29,22,16,40)(2,30,23,17,41)(3,31,24,18,42)(4,32,25,10,43)(5,33,26,11,44)(6,34,27,12,45)(7,35,19,13,37)(8,36,20,14,38)(9,28,21,15,39), (10,32,25,43)(11,33,26,44)(12,34,27,45)(13,35,19,37)(14,36,20,38)(15,28,21,39)(16,29,22,40)(17,30,23,41)(18,31,24,42)>;
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), (1,29,22,16,40)(2,30,23,17,41)(3,31,24,18,42)(4,32,25,10,43)(5,33,26,11,44)(6,34,27,12,45)(7,35,19,13,37)(8,36,20,14,38)(9,28,21,15,39), (10,32,25,43)(11,33,26,44)(12,34,27,45)(13,35,19,37)(14,36,20,38)(15,28,21,39)(16,29,22,40)(17,30,23,41)(18,31,24,42) );
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)], [(1,29,22,16,40),(2,30,23,17,41),(3,31,24,18,42),(4,32,25,10,43),(5,33,26,11,44),(6,34,27,12,45),(7,35,19,13,37),(8,36,20,14,38),(9,28,21,15,39)], [(10,32,25,43),(11,33,26,44),(12,34,27,45),(13,35,19,37),(14,36,20,38),(15,28,21,39),(16,29,22,40),(17,30,23,41),(18,31,24,42)]])
45 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5 | 6A | 6B | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 15A | 15B | 18A | ··· | 18F | 36A | ··· | 36L | 45A | ··· | 45F |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 15 | 15 | 18 | ··· | 18 | 36 | ··· | 36 | 45 | ··· | 45 |
| size | 1 | 5 | 1 | 1 | 5 | 5 | 4 | 5 | 5 | 1 | ··· | 1 | 5 | 5 | 5 | 5 | 4 | 4 | 5 | ··· | 5 | 5 | ··· | 5 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | F5 | C3×F5 | C9×F5 |
| kernel | C9×F5 | C9×D5 | C3×F5 | C45 | C3×D5 | F5 | C15 | D5 | C5 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 1 | 2 | 6 |
Matrix representation of C9×F5 ►in GL5(𝔽181)
| 39 | 0 | 0 | 0 | 0 |
| 0 | 132 | 0 | 0 | 0 |
| 0 | 0 | 132 | 0 | 0 |
| 0 | 0 | 0 | 132 | 0 |
| 0 | 0 | 0 | 0 | 132 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 180 | 180 | 180 | 180 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 19 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 180 | 180 | 180 | 180 |
G:=sub<GL(5,GF(181))| [39,0,0,0,0,0,132,0,0,0,0,0,132,0,0,0,0,0,132,0,0,0,0,0,132],[1,0,0,0,0,0,180,1,0,0,0,180,0,1,0,0,180,0,0,1,0,180,0,0,0],[19,0,0,0,0,0,1,0,0,180,0,0,0,1,180,0,0,0,0,180,0,0,1,0,180] >;
C9×F5 in GAP, Magma, Sage, TeX
C_9\times F_5
% in TeX
G:=Group("C9xF5"); // GroupNames label
G:=SmallGroup(180,5);
// by ID
G=gap.SmallGroup(180,5);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-5,30,66,1804,614]);
// Polycyclic
G:=Group<a,b,c|a^9=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export