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G = C3×D21order 126 = 2·32·7

Direct product of C3 and D21

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

Aliases: C3×D21, C215C6, C212S3, C321D7, C3⋊(C3×D7), C73(C3×S3), (C3×C21)⋊2C2, SmallGroup(126,13)

Series: Derived Chief Lower central Upper central

C1C21 — C3×D21
C1C7C21C3×C21 — C3×D21
C21 — C3×D21
C1C3

Generators and relations for C3×D21
 G = < a,b,c | a3=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >

21C2
2C3
7S3
21C6
3D7
2C21
7C3×S3
3C3×D7

Smallest permutation representation of C3×D21
On 42 points
Generators in S42
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 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)
(1 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 42)(21 41)

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

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

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

C3×D21 is a maximal subgroup of   C3×S3×D7  D21⋊S3  D21⋊C9  He3⋊D7  D63⋊C3  C32⋊D21
C3×D21 is a maximal quotient of   He3⋊D7  D63⋊C3

36 conjugacy classes

class 1  2 3A3B3C3D3E6A6B7A7B7C21A···21X
order12333336677721···21
size1211122221212222···2

36 irreducible representations

dim1111222222
type+++++
imageC1C2C3C6S3D7C3×S3C3×D7D21C3×D21
kernelC3×D21C3×C21D21C21C21C32C7C3C3C1
# reps11221326612

Matrix representation of C3×D21 in GL2(𝔽43) generated by

360
036
,
230
015
,
015
230
G:=sub<GL(2,GF(43))| [36,0,0,36],[23,0,0,15],[0,23,15,0] >;

C3×D21 in GAP, Magma, Sage, TeX

C_3\times D_{21}
% in TeX

G:=Group("C3xD21");
// GroupNames label

G:=SmallGroup(126,13);
// by ID

G=gap.SmallGroup(126,13);
# by ID

G:=PCGroup([4,-2,-3,-3,-7,146,1731]);
// Polycyclic

G:=Group<a,b,c|a^3=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 C3×D21 in TeX

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