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## G = Dic3×C21order 252 = 22·32·7

### Direct product of C21 and Dic3

Aliases: Dic3×C21, C3⋊C84, C6.C42, C217C12, C42.8S3, C322C28, C42.13C6, (C3×C21)⋊6C4, C2.(S3×C21), C6.4(S3×C7), C14.4(C3×S3), (C3×C6).1C14, (C3×C42).4C2, SmallGroup(252,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C21
 Chief series C1 — C3 — C6 — C42 — C3×C42 — Dic3×C21
 Lower central C3 — Dic3×C21
 Upper central C1 — C42

Generators and relations for Dic3×C21
G = < a,b,c | a21=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of Dic3×C21
On 84 points
Generators in S84
(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)(64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)
(1 68 8 75 15 82)(2 69 9 76 16 83)(3 70 10 77 17 84)(4 71 11 78 18 64)(5 72 12 79 19 65)(6 73 13 80 20 66)(7 74 14 81 21 67)(22 60 36 53 29 46)(23 61 37 54 30 47)(24 62 38 55 31 48)(25 63 39 56 32 49)(26 43 40 57 33 50)(27 44 41 58 34 51)(28 45 42 59 35 52)
(1 60 75 29)(2 61 76 30)(3 62 77 31)(4 63 78 32)(5 43 79 33)(6 44 80 34)(7 45 81 35)(8 46 82 36)(9 47 83 37)(10 48 84 38)(11 49 64 39)(12 50 65 40)(13 51 66 41)(14 52 67 42)(15 53 68 22)(16 54 69 23)(17 55 70 24)(18 56 71 25)(19 57 72 26)(20 58 73 27)(21 59 74 28)

G:=sub<Sym(84)| (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)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84), (1,68,8,75,15,82)(2,69,9,76,16,83)(3,70,10,77,17,84)(4,71,11,78,18,64)(5,72,12,79,19,65)(6,73,13,80,20,66)(7,74,14,81,21,67)(22,60,36,53,29,46)(23,61,37,54,30,47)(24,62,38,55,31,48)(25,63,39,56,32,49)(26,43,40,57,33,50)(27,44,41,58,34,51)(28,45,42,59,35,52), (1,60,75,29)(2,61,76,30)(3,62,77,31)(4,63,78,32)(5,43,79,33)(6,44,80,34)(7,45,81,35)(8,46,82,36)(9,47,83,37)(10,48,84,38)(11,49,64,39)(12,50,65,40)(13,51,66,41)(14,52,67,42)(15,53,68,22)(16,54,69,23)(17,55,70,24)(18,56,71,25)(19,57,72,26)(20,58,73,27)(21,59,74,28)>;

G:=Group( (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)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84), (1,68,8,75,15,82)(2,69,9,76,16,83)(3,70,10,77,17,84)(4,71,11,78,18,64)(5,72,12,79,19,65)(6,73,13,80,20,66)(7,74,14,81,21,67)(22,60,36,53,29,46)(23,61,37,54,30,47)(24,62,38,55,31,48)(25,63,39,56,32,49)(26,43,40,57,33,50)(27,44,41,58,34,51)(28,45,42,59,35,52), (1,60,75,29)(2,61,76,30)(3,62,77,31)(4,63,78,32)(5,43,79,33)(6,44,80,34)(7,45,81,35)(8,46,82,36)(9,47,83,37)(10,48,84,38)(11,49,64,39)(12,50,65,40)(13,51,66,41)(14,52,67,42)(15,53,68,22)(16,54,69,23)(17,55,70,24)(18,56,71,25)(19,57,72,26)(20,58,73,27)(21,59,74,28) );

G=PermutationGroup([[(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),(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)], [(1,68,8,75,15,82),(2,69,9,76,16,83),(3,70,10,77,17,84),(4,71,11,78,18,64),(5,72,12,79,19,65),(6,73,13,80,20,66),(7,74,14,81,21,67),(22,60,36,53,29,46),(23,61,37,54,30,47),(24,62,38,55,31,48),(25,63,39,56,32,49),(26,43,40,57,33,50),(27,44,41,58,34,51),(28,45,42,59,35,52)], [(1,60,75,29),(2,61,76,30),(3,62,77,31),(4,63,78,32),(5,43,79,33),(6,44,80,34),(7,45,81,35),(8,46,82,36),(9,47,83,37),(10,48,84,38),(11,49,64,39),(12,50,65,40),(13,51,66,41),(14,52,67,42),(15,53,68,22),(16,54,69,23),(17,55,70,24),(18,56,71,25),(19,57,72,26),(20,58,73,27),(21,59,74,28)]])

126 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 7A ··· 7F 12A 12B 12C 12D 14A ··· 14F 21A ··· 21L 21M ··· 21AD 28A ··· 28L 42A ··· 42L 42M ··· 42AD 84A ··· 84X order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 7 ··· 7 12 12 12 12 14 ··· 14 21 ··· 21 21 ··· 21 28 ··· 28 42 ··· 42 42 ··· 42 84 ··· 84 size 1 1 1 1 2 2 2 3 3 1 1 2 2 2 1 ··· 1 3 3 3 3 1 ··· 1 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C7 C12 C14 C21 C28 C42 C84 S3 Dic3 C3×S3 C3×Dic3 S3×C7 C7×Dic3 S3×C21 Dic3×C21 kernel Dic3×C21 C3×C42 C7×Dic3 C3×C21 C42 C3×Dic3 C21 C3×C6 Dic3 C32 C6 C3 C42 C21 C14 C7 C6 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 12 12 12 24 1 1 2 2 6 6 12 12

Matrix representation of Dic3×C21 in GL2(𝔽43) generated by

 38 0 0 38
,
 37 0 0 7
,
 0 2 21 0
G:=sub<GL(2,GF(43))| [38,0,0,38],[37,0,0,7],[0,21,2,0] >;

Dic3×C21 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{21}
% in TeX

G:=Group("Dic3xC21");
// GroupNames label

G:=SmallGroup(252,21);
// by ID

G=gap.SmallGroup(252,21);
# by ID

G:=PCGroup([5,-2,-3,-7,-2,-3,210,4204]);
// Polycyclic

G:=Group<a,b,c|a^21=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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