p-group, cyclic, elementary abelian, simple, monomial
Aliases: C89, also denoted Z89, SmallGroup(89,1)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C89 |
| C1 — C89 |
| C1 — C89 |
| C1 — C89 |
Generators and relations for C89
G = < a | a89=1 >
Smallest permutation representation of C89
►Regular action on 89 points
Generators in S89
(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)
G:=sub<Sym(89)| (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)>;
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) );
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)]])
C89 is a maximal subgroup of
D89
89 conjugacy classes
| class | 1 | 89A | ··· | 89CJ |
| order | 1 | 89 | ··· | 89 |
| size | 1 | 1 | ··· | 1 |
89 irreducible representations
| dim | 1 | 1 |
| type | + | |
| image | C1 | C89 |
| kernel | C89 | C1 |
| # reps | 1 | 88 |
Matrix representation of C89 ►in GL1(𝔽179) generated by
| 161 |
G:=sub<GL(1,GF(179))| [161] >;
C89 in GAP, Magma, Sage, TeX
C_{89} % in TeX
G:=Group("C89"); // GroupNames label
G:=SmallGroup(89,1);
// by ID
G=gap.SmallGroup(89,1);
# by ID
G:=PCGroup([1,-89]:ExponentLimit:=1);
// Polycyclic
G:=Group<a|a^89=1>;
// generators/relations
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