direct product, cyclic, abelian, monomial
Aliases: C45, also denoted Z45, SmallGroup(45,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C45 |
| C1 — C45 |
| C1 — C45 |
Generators and relations for C45
G = < a | a45=1 >
Smallest permutation representation of C45
►Regular action 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)
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)>;
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) );
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)]])
C45 is a maximal subgroup of
D45
45 conjugacy classes
| class | 1 | 3A | 3B | 5A | 5B | 5C | 5D | 9A | ··· | 9F | 15A | ··· | 15H | 45A | ··· | 45X |
| order | 1 | 3 | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||
| image | C1 | C3 | C5 | C9 | C15 | C45 |
| kernel | C45 | C15 | C9 | C5 | C3 | C1 |
| # reps | 1 | 2 | 4 | 6 | 8 | 24 |
Matrix representation of C45 ►in GL1(𝔽181) generated by
| 87 |
G:=sub<GL(1,GF(181))| [87] >;
C45 in GAP, Magma, Sage, TeX
C_{45} % in TeX
G:=Group("C45"); // GroupNames label
G:=SmallGroup(45,1);
// by ID
G=gap.SmallGroup(45,1);
# by ID
G:=PCGroup([3,-3,-5,-3,45]);
// Polycyclic
G:=Group<a|a^45=1>;
// generators/relations
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