direct product, cyclic, abelian, monomial
Aliases: C68, also denoted Z68, SmallGroup(68,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C68 |
| C1 — C68 |
| C1 — C68 |
Generators and relations for C68
G = < a | a68=1 >
Smallest permutation representation of C68
►Regular action on 68 points
Generators in S68
(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)
G:=sub<Sym(68)| (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)>;
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) );
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)]])
C68 is a maximal subgroup of
C17⋊3C8 Dic34 D68
68 conjugacy classes
| class | 1 | 2 | 4A | 4B | 17A | ··· | 17P | 34A | ··· | 34P | 68A | ··· | 68AF |
| order | 1 | 2 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C17 | C34 | C68 |
| kernel | C68 | C34 | C17 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 16 | 16 | 32 |
Matrix representation of C68 ►in GL1(𝔽137) generated by
| 118 |
G:=sub<GL(1,GF(137))| [118] >;
C68 in GAP, Magma, Sage, TeX
C_{68} % in TeX
G:=Group("C68"); // GroupNames label
G:=SmallGroup(68,2);
// by ID
G=gap.SmallGroup(68,2);
# by ID
G:=PCGroup([3,-2,-17,-2,102]);
// Polycyclic
G:=Group<a|a^68=1>;
// generators/relations
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