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G = C6xD21order 252 = 22·32·7

Direct product of C6 and D21

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

Aliases: C6xD21, C42:5C6, C42:2S3, C21:7D6, C32:5D14, C6:(C3xD7), C7:5(S3xC6), (C3xC6):1D7, C3:2(C6xD7), C14:3(C3xS3), C21:7(C2xC6), (C3xC42):2C2, (C3xC21):7C22, SmallGroup(252,43)

Series: Derived Chief Lower central Upper central

C1C21 — C6xD21
C1C7C21C3xC21C3xD21 — C6xD21
C21 — C6xD21
C1C6

Generators and relations for C6xD21
 G = < a,b,c | a6=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 216 in 44 conjugacy classes, 22 normal (18 characteristic)
Quotients: C1, C2, C3, C22, S3, C6, D6, C2xC6, D7, C3xS3, D14, S3xC6, C3xD7, D21, C6xD7, D42, C3xD21, C6xD21
21C2
21C2
2C3
21C22
2C6
7S3
7S3
21C6
21C6
3D7
3D7
2C21
7D6
21C2xC6
7C3xS3
7C3xS3
3D14
2C42
3C3xD7
3C3xD7
7S3xC6
3C6xD7

Smallest permutation representation of C6xD21
On 84 points
Generators in S84
(1 28 15 42 8 35)(2 29 16 22 9 36)(3 30 17 23 10 37)(4 31 18 24 11 38)(5 32 19 25 12 39)(6 33 20 26 13 40)(7 34 21 27 14 41)(43 78 50 64 57 71)(44 79 51 65 58 72)(45 80 52 66 59 73)(46 81 53 67 60 74)(47 82 54 68 61 75)(48 83 55 69 62 76)(49 84 56 70 63 77)
(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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 84)(17 83)(18 82)(19 81)(20 80)(21 79)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 63)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)

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

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

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

72 conjugacy classes

class 1 2A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I7A7B7C14A14B14C21A···21X42A···42X
order12223333366666666677714141421···2142···42
size1121211122211222212121212222222···22···2

72 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C3C6C6S3D6D7C3xS3D14S3xC6C3xD7D21C6xD7D42C3xD21C6xD21
kernelC6xD21C3xD21C3xC42D42D21C42C42C21C3xC6C14C32C7C6C6C3C3C2C1
# reps12124211323266661212

Matrix representation of C6xD21 in GL2(F43) generated by

370
037
,
140
040
,
018
120
G:=sub<GL(2,GF(43))| [37,0,0,37],[14,0,0,40],[0,12,18,0] >;

C6xD21 in GAP, Magma, Sage, TeX

C_6\times D_{21}
% in TeX

G:=Group("C6xD21");
// GroupNames label

G:=SmallGroup(252,43);
// by ID

G=gap.SmallGroup(252,43);
# by ID

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

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

Export

Subgroup lattice of C6xD21 in TeX

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