direct product, abelian, monomial, 3-elementary
Aliases: C3xC21, SmallGroup(63,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3xC21 |
| C1 — C3xC21 |
| C1 — C3xC21 |
Generators and relations for C3xC21
G = < a,b | a3=b21=1, ab=ba >
Quotients: C1, C3, C7, C32, C21, C3xC21
Smallest permutation representation of C3xC21
►Regular action on 63 points
Generators in S63
(1 53 37)(2 54 38)(3 55 39)(4 56 40)(5 57 41)(6 58 42)(7 59 22)(8 60 23)(9 61 24)(10 62 25)(11 63 26)(12 43 27)(13 44 28)(14 45 29)(15 46 30)(16 47 31)(17 48 32)(18 49 33)(19 50 34)(20 51 35)(21 52 36)
(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)
G:=sub<Sym(63)| (1,53,37)(2,54,38)(3,55,39)(4,56,40)(5,57,41)(6,58,42)(7,59,22)(8,60,23)(9,61,24)(10,62,25)(11,63,26)(12,43,27)(13,44,28)(14,45,29)(15,46,30)(16,47,31)(17,48,32)(18,49,33)(19,50,34)(20,51,35)(21,52,36), (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)>;
G:=Group( (1,53,37)(2,54,38)(3,55,39)(4,56,40)(5,57,41)(6,58,42)(7,59,22)(8,60,23)(9,61,24)(10,62,25)(11,63,26)(12,43,27)(13,44,28)(14,45,29)(15,46,30)(16,47,31)(17,48,32)(18,49,33)(19,50,34)(20,51,35)(21,52,36), (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) );
G=PermutationGroup([[(1,53,37),(2,54,38),(3,55,39),(4,56,40),(5,57,41),(6,58,42),(7,59,22),(8,60,23),(9,61,24),(10,62,25),(11,63,26),(12,43,27),(13,44,28),(14,45,29),(15,46,30),(16,47,31),(17,48,32),(18,49,33),(19,50,34),(20,51,35),(21,52,36)], [(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)]])
C3xC21 is a maximal subgroup of
C3:D21 C21.C32 C7:He3
63 conjugacy classes
| class | 1 | 3A | ··· | 3H | 7A | ··· | 7F | 21A | ··· | 21AV |
| order | 1 | 3 | ··· | 3 | 7 | ··· | 7 | 21 | ··· | 21 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C7 | C21 |
| kernel | C3xC21 | C21 | C32 | C3 |
| # reps | 1 | 8 | 6 | 48 |
Matrix representation of C3xC21 ►in GL2(F43) generated by
| 36 | 0 |
| 0 | 1 |
| 17 | 0 |
| 0 | 6 |
G:=sub<GL(2,GF(43))| [36,0,0,1],[17,0,0,6] >;
C3xC21 in GAP, Magma, Sage, TeX
C_3\times C_{21} % in TeX
G:=Group("C3xC21"); // GroupNames label
G:=SmallGroup(63,4);
// by ID
G=gap.SmallGroup(63,4);
# by ID
G:=PCGroup([3,-3,-3,-7]);
// Polycyclic
G:=Group<a,b|a^3=b^21=1,a*b=b*a>;
// generators/relations
Export