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G = C3×Dic21order 252 = 22·32·7

Direct product of C3 and Dic21

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

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

C1C21 — C3×Dic21
C1C7C21C42C3×C42 — C3×Dic21
C21 — C3×Dic21
C1C6

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

2C3
21C4
2C6
2C21
7Dic3
21C12
3Dic7
2C42
7C3×Dic3
3C3×Dic7

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 3A3B3C3D3E4A4B6A6B6C6D6E7A7B7C12A12B12C12D14A14B14C21A···21X42A···42X
order123333344666667771212121214141421···2142···42
size1111222212111222222212121212222···22···2

72 irreducible representations

dim111111222222222222
type+++-+-+-
imageC1C2C3C4C6C12S3Dic3D7C3×S3Dic7C3×Dic3C3×D7D21C3×Dic7Dic21C3×D21C3×Dic21
kernelC3×Dic21C3×C42Dic21C3×C21C42C21C42C21C3×C6C14C32C7C6C6C3C3C2C1
# reps11222411323266661212

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

60
06
,
280
020
,
042
10
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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Subgroup lattice of C3×Dic21 in TeX

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