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## G = C3×Dic21order 252 = 22·32·7

### Direct product of C3 and Dic21

Aliases: C3×Dic21, C215C12, C42.9C6, C42.4S3, C6.4D21, C212Dic3, C322Dic7, C6.(C3×D7), (C3×C21)⋊4C4, C3⋊(C3×Dic7), C2.(C3×D21), (C3×C6).1D7, C73(C3×Dic3), C14.3(C3×S3), (C3×C42).2C2, SmallGroup(252,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C3×Dic21
 Chief series C1 — C7 — C21 — C42 — C3×C42 — C3×Dic21
 Lower central C21 — C3×Dic21
 Upper central C1 — C6

Generators and relations for C3×Dic21
G = < a,b,c | a3=b42=1, c2=b21, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×Dic21
On 84 points
Generators in S84
(1 15 29)(2 16 30)(3 17 31)(4 18 32)(5 19 33)(6 20 34)(7 21 35)(8 22 36)(9 23 37)(10 24 38)(11 25 39)(12 26 40)(13 27 41)(14 28 42)(43 71 57)(44 72 58)(45 73 59)(46 74 60)(47 75 61)(48 76 62)(49 77 63)(50 78 64)(51 79 65)(52 80 66)(53 81 67)(54 82 68)(55 83 69)(56 84 70)
(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 64 22 43)(2 63 23 84)(3 62 24 83)(4 61 25 82)(5 60 26 81)(6 59 27 80)(7 58 28 79)(8 57 29 78)(9 56 30 77)(10 55 31 76)(11 54 32 75)(12 53 33 74)(13 52 34 73)(14 51 35 72)(15 50 36 71)(16 49 37 70)(17 48 38 69)(18 47 39 68)(19 46 40 67)(20 45 41 66)(21 44 42 65)

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

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

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

72 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 7A 7B 7C 12A 12B 12C 12D 14A 14B 14C 21A ··· 21X 42A ··· 42X order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 7 7 7 12 12 12 12 14 14 14 21 ··· 21 42 ··· 42 size 1 1 1 1 2 2 2 21 21 1 1 2 2 2 2 2 2 21 21 21 21 2 2 2 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + - + - + - image C1 C2 C3 C4 C6 C12 S3 Dic3 D7 C3×S3 Dic7 C3×Dic3 C3×D7 D21 C3×Dic7 Dic21 C3×D21 C3×Dic21 kernel C3×Dic21 C3×C42 Dic21 C3×C21 C42 C21 C42 C21 C3×C6 C14 C32 C7 C6 C6 C3 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 3 2 3 2 6 6 6 6 12 12

Matrix representation of C3×Dic21 in GL2(𝔽43) generated by

 6 0 0 6
,
 28 0 0 20
,
 0 42 1 0
G:=sub<GL(2,GF(43))| [6,0,0,6],[28,0,0,20],[0,1,42,0] >;

C3×Dic21 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{21}
% in TeX

G:=Group("C3xDic21");
// GroupNames label

G:=SmallGroup(252,22);
// by ID

G=gap.SmallGroup(252,22);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-7,30,483,5404]);
// Polycyclic

G:=Group<a,b,c|a^3=b^42=1,c^2=b^21,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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