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G = C6×D21order 252 = 22·32·7

Direct product of C6 and D21

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

Aliases: C6×D21, C425C6, C422S3, C217D6, C325D14, C6⋊(C3×D7), C75(S3×C6), (C3×C6)⋊1D7, C32(C6×D7), C143(C3×S3), C217(C2×C6), (C3×C42)⋊2C2, (C3×C21)⋊7C22, SmallGroup(252,43)

Series: Derived Chief Lower central Upper central

C1C21 — C6×D21
C1C7C21C3×C21C3×D21 — C6×D21
C21 — C6×D21
C1C6

Generators and relations for C6×D21
 G = < a,b,c | a6=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >

21C2
21C2
2C3
21C22
2C6
7S3
7S3
21C6
21C6
3D7
3D7
2C21
7D6
21C2×C6
7C3×S3
7C3×S3
3D14
2C42
3C3×D7
3C3×D7
7S3×C6
3C6×D7

Smallest permutation representation of C6×D21
On 84 points
Generators in S84
(1 26 15 40 8 33)(2 27 16 41 9 34)(3 28 17 42 10 35)(4 29 18 22 11 36)(5 30 19 23 12 37)(6 31 20 24 13 38)(7 32 21 25 14 39)(43 65 50 72 57 79)(44 66 51 73 58 80)(45 67 52 74 59 81)(46 68 53 75 60 82)(47 69 54 76 61 83)(48 70 55 77 62 84)(49 71 56 78 63 64)
(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 82)(2 81)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 72)(12 71)(13 70)(14 69)(15 68)(16 67)(17 66)(18 65)(19 64)(20 84)(21 83)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)

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

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

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

72 conjugacy classes

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

72 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C3C6C6S3D6D7C3×S3D14S3×C6C3×D7D21C6×D7D42C3×D21C6×D21
kernelC6×D21C3×D21C3×C42D42D21C42C42C21C3×C6C14C32C7C6C6C3C3C2C1
# reps12124211323266661212

Matrix representation of C6×D21 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] >;

C6×D21 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 C6×D21 in TeX

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