direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary
Aliases: C5×3- 1+2, C9⋊C15, C45⋊C3, C32.C15, C15.2C32, (C3×C15).C3, C3.2(C3×C15), SmallGroup(135,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×3- 1+2
G = < a,b,c | a5=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C5×3- 1+2
►On 45 points
Generators in S45
(1 36 43 25 14)(2 28 44 26 15)(3 29 45 27 16)(4 30 37 19 17)(5 31 38 20 18)(6 32 39 21 10)(7 33 40 22 11)(8 34 41 23 12)(9 35 42 24 13)
(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)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(20 26 23)(21 24 27)(28 34 31)(29 32 35)(38 44 41)(39 42 45)
G:=sub<Sym(45)| (1,36,43,25,14)(2,28,44,26,15)(3,29,45,27,16)(4,30,37,19,17)(5,31,38,20,18)(6,32,39,21,10)(7,33,40,22,11)(8,34,41,23,12)(9,35,42,24,13), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(20,26,23)(21,24,27)(28,34,31)(29,32,35)(38,44,41)(39,42,45)>;
G:=Group( (1,36,43,25,14)(2,28,44,26,15)(3,29,45,27,16)(4,30,37,19,17)(5,31,38,20,18)(6,32,39,21,10)(7,33,40,22,11)(8,34,41,23,12)(9,35,42,24,13), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(20,26,23)(21,24,27)(28,34,31)(29,32,35)(38,44,41)(39,42,45) );
G=PermutationGroup([[(1,36,43,25,14),(2,28,44,26,15),(3,29,45,27,16),(4,30,37,19,17),(5,31,38,20,18),(6,32,39,21,10),(7,33,40,22,11),(8,34,41,23,12),(9,35,42,24,13)], [(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)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(20,26,23),(21,24,27),(28,34,31),(29,32,35),(38,44,41),(39,42,45)]])
C5×3- 1+2 is a maximal subgroup of
D45⋊C3
55 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 5A | 5B | 5C | 5D | 9A | ··· | 9F | 15A | ··· | 15H | 15I | ··· | 15P | 45A | ··· | 45X |
| order | 1 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 15 | ··· | 15 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||||
| image | C1 | C3 | C3 | C5 | C15 | C15 | 3- 1+2 | C5×3- 1+2 |
| kernel | C5×3- 1+2 | C45 | C3×C15 | 3- 1+2 | C9 | C32 | C5 | C1 |
| # reps | 1 | 6 | 2 | 4 | 24 | 8 | 2 | 8 |
Matrix representation of C5×3- 1+2 ►in GL3(𝔽181) generated by
| 135 | 0 | 0 |
| 0 | 135 | 0 |
| 0 | 0 | 135 |
| 0 | 1 | 0 |
| 0 | 0 | 132 |
| 1 | 0 | 0 |
| 1 | 0 | 0 |
| 0 | 132 | 0 |
| 0 | 0 | 48 |
G:=sub<GL(3,GF(181))| [135,0,0,0,135,0,0,0,135],[0,0,1,1,0,0,0,132,0],[1,0,0,0,132,0,0,0,48] >;
C5×3- 1+2 in GAP, Magma, Sage, TeX
C_5\times 3_-^{1+2} % in TeX
G:=Group("C5xES-(3,1)"); // GroupNames label
G:=SmallGroup(135,4);
// by ID
G=gap.SmallGroup(135,4);
# by ID
G:=PCGroup([4,-3,-3,-5,-3,180,385]);
// Polycyclic
G:=Group<a,b,c|a^5=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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