direct product, cyclic, abelian, monomial
Aliases: C82, also denoted Z82, SmallGroup(82,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C82 |
| C1 — C82 |
| C1 — C82 |
Generators and relations for C82
G = < a | a82=1 >
Smallest permutation representation of C82
►Regular action on 82 points
Generators in S82
(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)
G:=sub<Sym(82)| (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)>;
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) );
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)]])
C82 is a maximal subgroup of
Dic41
82 conjugacy classes
| class | 1 | 2 | 41A | ··· | 41AN | 82A | ··· | 82AN |
| order | 1 | 2 | 41 | ··· | 41 | 82 | ··· | 82 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C41 | C82 |
| kernel | C82 | C41 | C2 | C1 |
| # reps | 1 | 1 | 40 | 40 |
Matrix representation of C82 ►in GL1(𝔽83) generated by
| 67 |
G:=sub<GL(1,GF(83))| [67] >;
C82 in GAP, Magma, Sage, TeX
C_{82} % in TeX
G:=Group("C82"); // GroupNames label
G:=SmallGroup(82,2);
// by ID
G=gap.SmallGroup(82,2);
# by ID
G:=PCGroup([2,-2,-41]);
// Polycyclic
G:=Group<a|a^82=1>;
// generators/relations
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