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## G = C3×D21order 126 = 2·32·7

### Direct product of C3 and D21

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

 Derived series C1 — C21 — C3×D21
 Chief series C1 — C7 — C21 — C3×C21 — C3×D21
 Lower central C21 — C3×D21
 Upper central C1 — C3

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

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

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

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

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

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 3A 3B 3C 3D 3E 6A 6B 7A 7B 7C 21A ··· 21X order 1 2 3 3 3 3 3 6 6 7 7 7 21 ··· 21 size 1 21 1 1 2 2 2 21 21 2 2 2 2 ··· 2

36 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C6 S3 D7 C3×S3 C3×D7 D21 C3×D21 kernel C3×D21 C3×C21 D21 C21 C21 C32 C7 C3 C3 C1 # reps 1 1 2 2 1 3 2 6 6 12

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

 36 0 0 36
,
 23 0 0 15
,
 0 15 23 0
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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