direct product, cyclic, abelian, monomial
Aliases: C90, also denoted Z90, SmallGroup(90,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C90 |
| C1 — C90 |
| C1 — C90 |
Generators and relations for C90
G = < a | a90=1 >
Smallest permutation representation of C90
►Regular action on 90 points
Generators in S90
(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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)
G:=sub<Sym(90)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)]])
C90 is a maximal subgroup of
Dic45
90 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 6A | 6B | 9A | ··· | 9F | 10A | 10B | 10C | 10D | 15A | ··· | 15H | 18A | ··· | 18F | 30A | ··· | 30H | 45A | ··· | 45X | 90A | ··· | 90X |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 9 | ··· | 9 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 18 | ··· | 18 | 30 | ··· | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C9 | C10 | C15 | C18 | C30 | C45 | C90 |
| kernel | C90 | C45 | C30 | C18 | C15 | C10 | C9 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 6 | 4 | 8 | 6 | 8 | 24 | 24 |
Matrix representation of C90 ►in GL2(𝔽19) generated by
| 0 | 10 |
| 9 | 7 |
G:=sub<GL(2,GF(19))| [0,9,10,7] >;
C90 in GAP, Magma, Sage, TeX
C_{90} % in TeX
G:=Group("C90"); // GroupNames label
G:=SmallGroup(90,4);
// by ID
G=gap.SmallGroup(90,4);
# by ID
G:=PCGroup([4,-2,-3,-5,-3,125]);
// Polycyclic
G:=Group<a|a^90=1>;
// generators/relations
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