Copied to
clipboard

G = C3×C21order 63 = 32·7

Abelian group of type [3,21]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C21, SmallGroup(63,4)

Series: Derived Chief Lower central Upper central

C1 — C3×C21
C1C7C21 — 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 58 31)(2 59 32)(3 60 33)(4 61 34)(5 62 35)(6 63 36)(7 43 37)(8 44 38)(9 45 39)(10 46 40)(11 47 41)(12 48 42)(13 49 22)(14 50 23)(15 51 24)(16 52 25)(17 53 26)(18 54 27)(19 55 28)(20 56 29)(21 57 30)
(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)| (1,58,31)(2,59,32)(3,60,33)(4,61,34)(5,62,35)(6,63,36)(7,43,37)(8,44,38)(9,45,39)(10,46,40)(11,47,41)(12,48,42)(13,49,22)(14,50,23)(15,51,24)(16,52,25)(17,53,26)(18,54,27)(19,55,28)(20,56,29)(21,57,30), (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( (1,58,31)(2,59,32)(3,60,33)(4,61,34)(5,62,35)(6,63,36)(7,43,37)(8,44,38)(9,45,39)(10,46,40)(11,47,41)(12,48,42)(13,49,22)(14,50,23)(15,51,24)(16,52,25)(17,53,26)(18,54,27)(19,55,28)(20,56,29)(21,57,30), (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([(1,58,31),(2,59,32),(3,60,33),(4,61,34),(5,62,35),(6,63,36),(7,43,37),(8,44,38),(9,45,39),(10,46,40),(11,47,41),(12,48,42),(13,49,22),(14,50,23),(15,51,24),(16,52,25),(17,53,26),(18,54,27),(19,55,28),(20,56,29),(21,57,30)], [(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···3H7A···7F21A···21AV
order13···37···721···21
size11···11···11···1

63 irreducible representations

dim1111
type+
imageC1C3C7C21
kernelC3×C21C21C32C3
# reps18648

Matrix representation of C3×C21 in GL2(𝔽43) generated by

360
01
,
170
06
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

Export

Subgroup lattice of C3×C21 in TeX

׿
×
𝔽