direct product, cyclic, abelian, monomial
Aliases: C42, also denoted Z42, SmallGroup(42,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C42 |
| C1 — C42 |
| C1 — C42 |
Generators and relations for C42
G = < a | a42=1 >
Smallest permutation representation of C42
►Regular action on 42 points
Generators in S42
(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)
G:=sub<Sym(42)| (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)>;
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) );
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)]])
C42 is a maximal subgroup of
Dic21
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21L | 42A | ··· | 42L |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 |
| kernel | C42 | C21 | C14 | C7 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 |
Matrix representation of C42 ►in GL1(𝔽43) generated by
| 18 |
G:=sub<GL(1,GF(43))| [18] >;
C42 in GAP, Magma, Sage, TeX
C_{42} % in TeX
G:=Group("C42"); // GroupNames label
G:=SmallGroup(42,6);
// by ID
G=gap.SmallGroup(42,6);
# by ID
G:=PCGroup([3,-2,-3,-7]);
// Polycyclic
G:=Group<a|a^42=1>;
// generators/relations
Export