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G = C7×D21order 294 = 2·3·72

Direct product of C7 and D21

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C7×D21, C213D7, C722S3, C211C14, C3⋊(C7×D7), C7⋊(S3×C7), (C7×C21)⋊2C2, SmallGroup(294,21)

Series: Derived Chief Lower central Upper central

C1C21 — C7×D21
C1C7C21C7×C21 — C7×D21
C21 — C7×D21
C1C7

Generators and relations for C7×D21
 G = < a,b,c | a7=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >

21C2
2C7
2C7
2C7
7S3
3D7
21C14
2C21
2C21
2C21
7S3×C7
3C7×D7

Smallest permutation representation of C7×D21
On 42 points
Generators in S42
(1 13 4 16 7 19 10)(2 14 5 17 8 20 11)(3 15 6 18 9 21 12)(22 31 40 28 37 25 34)(23 32 41 29 38 26 35)(24 33 42 30 39 27 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)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)

G:=sub<Sym(42)| (1,13,4,16,7,19,10)(2,14,5,17,8,20,11)(3,15,6,18,9,21,12)(22,31,40,28,37,25,34)(23,32,41,29,38,26,35)(24,33,42,30,39,27,36), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,42)(11,41)(12,40)(13,39)(14,38)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)>;

G:=Group( (1,13,4,16,7,19,10)(2,14,5,17,8,20,11)(3,15,6,18,9,21,12)(22,31,40,28,37,25,34)(23,32,41,29,38,26,35)(24,33,42,30,39,27,36), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,42)(11,41)(12,40)(13,39)(14,38)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31) );

G=PermutationGroup([(1,13,4,16,7,19,10),(2,14,5,17,8,20,11),(3,15,6,18,9,21,12),(22,31,40,28,37,25,34),(23,32,41,29,38,26,35),(24,33,42,30,39,27,36)], [(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,42),(11,41),(12,40),(13,39),(14,38),(15,37),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31)])

84 conjugacy classes

class 1  2  3 7A···7F7G···7AA14A···14F21A···21AV
order1237···77···714···1421···21
size12121···12···221···212···2

84 irreducible representations

dim1111222222
type+++++
imageC1C2C7C14S3D7S3×C7D21C7×D7C7×D21
kernelC7×D21C7×C21D21C21C72C21C7C7C3C1
# reps116613661836

Matrix representation of C7×D21 in GL2(𝔽43) generated by

40
04
,
250
631
,
3535
248
G:=sub<GL(2,GF(43))| [4,0,0,4],[25,6,0,31],[35,24,35,8] >;

C7×D21 in GAP, Magma, Sage, TeX

C_7\times D_{21}
% in TeX

G:=Group("C7xD21");
// GroupNames label

G:=SmallGroup(294,21);
// by ID

G=gap.SmallGroup(294,21);
# by ID

G:=PCGroup([4,-2,-7,-3,-7,338,4035]);
// Polycyclic

G:=Group<a,b,c|a^7=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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Subgroup lattice of C7×D21 in TeX

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