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## G = S3×D21order 252 = 22·32·7

### Direct product of S3 and D21

Aliases: S3×D21, C31D42, C212D6, C321D14, C71S32, (S3×C7)⋊S3, (C3×S3)⋊D7, C31(S3×D7), C3⋊D212C2, (S3×C21)⋊1C2, (C3×D21)⋊2C2, (C3×C21)⋊3C22, SmallGroup(252,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C21 — S3×D21
 Chief series C1 — C7 — C21 — C3×C21 — C3×D21 — S3×D21
 Lower central C3×C21 — S3×D21
 Upper central C1

Generators and relations for S3×D21
G = < a,b,c,d | a3=b2=c21=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

3C2
21C2
63C2
2C3
63C22
3C6
7S3
21S3
21C6
21S3
42S3
3C14
3D7
9D7
2C21
21D6
21D6
9D14
3D21
3D21
3C42
6D21
7S32
3D42

Smallest permutation representation of S3×D21
On 42 points
Generators in S42
(1 8 15)(2 9 16)(3 10 17)(4 11 18)(5 12 19)(6 13 20)(7 14 21)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 6A 6B 7A 7B 7C 14A 14B 14C 21A ··· 21F 21G ··· 21O 42A ··· 42F order 1 2 2 2 3 3 3 6 6 7 7 7 14 14 14 21 ··· 21 21 ··· 21 42 ··· 42 size 1 3 21 63 2 2 4 6 42 2 2 2 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

36 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 D6 D7 D14 D21 D42 S32 S3×D7 S3×D21 kernel S3×D21 S3×C21 C3×D21 C3⋊D21 S3×C7 D21 C21 C3×S3 C32 S3 C3 C7 C3 C1 # reps 1 1 1 1 1 1 2 3 3 6 6 1 3 6

Matrix representation of S3×D21 in GL6(𝔽43)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 42 42
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 42 0 0 0 0 0 0 42 0 0 0 0 0 0 42 0 0 0 0 0 1 1
,
 0 35 0 0 0 0 27 19 0 0 0 0 0 0 0 42 0 0 0 0 1 42 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 42 20 0 0 0 0 0 1 0 0 0 0 0 0 42 1 0 0 0 0 0 1 0 0 0 0 0 0 42 0 0 0 0 0 0 42

G:=sub<GL(6,GF(43))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,42,0,0,0,0,1,42],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,1,0,0,0,0,0,1],[0,27,0,0,0,0,35,19,0,0,0,0,0,0,0,1,0,0,0,0,42,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,0,0,0,0,0,20,1,0,0,0,0,0,0,42,0,0,0,0,0,1,1,0,0,0,0,0,0,42,0,0,0,0,0,0,42] >;

S3×D21 in GAP, Magma, Sage, TeX

S_3\times D_{21}
% in TeX

G:=Group("S3xD21");
// GroupNames label

G:=SmallGroup(252,36);
// by ID

G=gap.SmallGroup(252,36);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7,67,483,5404]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^21=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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