direct product, cyclic, abelian, monomial
Aliases: C69, also denoted Z69, SmallGroup(69,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C69 |
| C1 — C69 |
| C1 — C69 |
Generators and relations for C69
G = < a | a69=1 >
Smallest permutation representation of C69
►Regular action on 69 points
Generators in S69
(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)
G:=sub<Sym(69)| (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)>;
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) );
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)]])
C69 is a maximal subgroup of
D69
69 conjugacy classes
| class | 1 | 3A | 3B | 23A | ··· | 23V | 69A | ··· | 69AR |
| order | 1 | 3 | 3 | 23 | ··· | 23 | 69 | ··· | 69 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
69 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C23 | C69 |
| kernel | C69 | C23 | C3 | C1 |
| # reps | 1 | 2 | 22 | 44 |
Matrix representation of C69 ►in GL1(𝔽139) generated by
| 37 |
G:=sub<GL(1,GF(139))| [37] >;
C69 in GAP, Magma, Sage, TeX
C_{69} % in TeX
G:=Group("C69"); // GroupNames label
G:=SmallGroup(69,1);
// by ID
G=gap.SmallGroup(69,1);
# by ID
G:=PCGroup([2,-3,-23]);
// Polycyclic
G:=Group<a|a^69=1>;
// generators/relations
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