p-group, cyclic, elementary abelian, simple, monomial
Aliases: C37, also denoted Z37, SmallGroup(37,1)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C37 |
| C1 — C37 |
| C1 — C37 |
| C1 — C37 |
Generators and relations for C37
G = < a | a37=1 >
Smallest permutation representation of C37
►Regular action on 37 points
Generators in S37
(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)
G:=sub<Sym(37)| (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)>;
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) );
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)]])
C37 is a maximal subgroup of
D37 C37⋊C3
37 conjugacy classes
| class | 1 | 37A | ··· | 37AJ |
| order | 1 | 37 | ··· | 37 |
| size | 1 | 1 | ··· | 1 |
37 irreducible representations
| dim | 1 | 1 |
| type | + | |
| image | C1 | C37 |
| kernel | C37 | C1 |
| # reps | 1 | 36 |
Matrix representation of C37 ►in GL1(𝔽149) generated by
| 85 |
G:=sub<GL(1,GF(149))| [85] >;
C37 in GAP, Magma, Sage, TeX
C_{37} % in TeX
G:=Group("C37"); // GroupNames label
G:=SmallGroup(37,1);
// by ID
G=gap.SmallGroup(37,1);
# by ID
G:=PCGroup([1,-37]:ExponentLimit:=1);
// Polycyclic
G:=Group<a|a^37=1>;
// generators/relations
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