direct product, abelian, monomial, 3-elementary
Aliases: C3×C21, SmallGroup(63,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C21 |
| C1 — C3×C21 |
| C1 — C3×C21 |
Generators and relations for C3×C21
G = < a,b | a3=b21=1, ab=ba >
Smallest permutation representation of C3×C21
►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)]])
C3×C21 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 | C3×C21 | C21 | C32 | C3 |
| # reps | 1 | 8 | 6 | 48 |
Matrix representation of C3×C21 ►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] >;
C3×C21 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
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