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G = C9×D21order 378 = 2·33·7

Direct product of C9 and D21

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

Aliases: C9×D21, C633S3, C215C18, C3⋊(C9×D7), C73(S3×C9), (C3×C9)⋊1D7, (C3×C63)⋊7C2, (C3×C21).9C6, C3.4(C3×D21), C21.14(C3×S3), (C3×D21).2C3, C32.2(C3×D7), SmallGroup(378,37)

Series: Derived Chief Lower central Upper central

C1C21 — C9×D21
C1C7C21C3×C21C3×C63 — C9×D21
C21 — C9×D21
C1C9

Generators and relations for C9×D21
 G = < a,b,c | a9=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >

21C2
2C3
7S3
21C6
2C9
3D7
2C21
7C3×S3
21C18
3C3×D7
2C63
7S3×C9
3C9×D7

Smallest permutation representation of C9×D21
On 126 points
Generators in S126
(1 60 29 15 53 22 8 46 36)(2 61 30 16 54 23 9 47 37)(3 62 31 17 55 24 10 48 38)(4 63 32 18 56 25 11 49 39)(5 43 33 19 57 26 12 50 40)(6 44 34 20 58 27 13 51 41)(7 45 35 21 59 28 14 52 42)(64 107 96 71 114 103 78 121 89)(65 108 97 72 115 104 79 122 90)(66 109 98 73 116 105 80 123 91)(67 110 99 74 117 85 81 124 92)(68 111 100 75 118 86 82 125 93)(69 112 101 76 119 87 83 126 94)(70 113 102 77 120 88 84 106 95)
(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)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)
(1 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 84)(19 83)(20 82)(21 81)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 105)(37 104)(38 103)(39 102)(40 101)(41 100)(42 99)(43 119)(44 118)(45 117)(46 116)(47 115)(48 114)(49 113)(50 112)(51 111)(52 110)(53 109)(54 108)(55 107)(56 106)(57 126)(58 125)(59 124)(60 123)(61 122)(62 121)(63 120)

G:=sub<Sym(126)| (1,60,29,15,53,22,8,46,36)(2,61,30,16,54,23,9,47,37)(3,62,31,17,55,24,10,48,38)(4,63,32,18,56,25,11,49,39)(5,43,33,19,57,26,12,50,40)(6,44,34,20,58,27,13,51,41)(7,45,35,21,59,28,14,52,42)(64,107,96,71,114,103,78,121,89)(65,108,97,72,115,104,79,122,90)(66,109,98,73,116,105,80,123,91)(67,110,99,74,117,85,81,124,92)(68,111,100,75,118,86,82,125,93)(69,112,101,76,119,87,83,126,94)(70,113,102,77,120,88,84,106,95), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,84)(19,83)(20,82)(21,81)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,113)(50,112)(51,111)(52,110)(53,109)(54,108)(55,107)(56,106)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120)>;

G:=Group( (1,60,29,15,53,22,8,46,36)(2,61,30,16,54,23,9,47,37)(3,62,31,17,55,24,10,48,38)(4,63,32,18,56,25,11,49,39)(5,43,33,19,57,26,12,50,40)(6,44,34,20,58,27,13,51,41)(7,45,35,21,59,28,14,52,42)(64,107,96,71,114,103,78,121,89)(65,108,97,72,115,104,79,122,90)(66,109,98,73,116,105,80,123,91)(67,110,99,74,117,85,81,124,92)(68,111,100,75,118,86,82,125,93)(69,112,101,76,119,87,83,126,94)(70,113,102,77,120,88,84,106,95), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,84)(19,83)(20,82)(21,81)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,113)(50,112)(51,111)(52,110)(53,109)(54,108)(55,107)(56,106)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120) );

G=PermutationGroup([(1,60,29,15,53,22,8,46,36),(2,61,30,16,54,23,9,47,37),(3,62,31,17,55,24,10,48,38),(4,63,32,18,56,25,11,49,39),(5,43,33,19,57,26,12,50,40),(6,44,34,20,58,27,13,51,41),(7,45,35,21,59,28,14,52,42),(64,107,96,71,114,103,78,121,89),(65,108,97,72,115,104,79,122,90),(66,109,98,73,116,105,80,123,91),(67,110,99,74,117,85,81,124,92),(68,111,100,75,118,86,82,125,93),(69,112,101,76,119,87,83,126,94),(70,113,102,77,120,88,84,106,95)], [(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)], [(1,80),(2,79),(3,78),(4,77),(5,76),(6,75),(7,74),(8,73),(9,72),(10,71),(11,70),(12,69),(13,68),(14,67),(15,66),(16,65),(17,64),(18,84),(19,83),(20,82),(21,81),(22,98),(23,97),(24,96),(25,95),(26,94),(27,93),(28,92),(29,91),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,105),(37,104),(38,103),(39,102),(40,101),(41,100),(42,99),(43,119),(44,118),(45,117),(46,116),(47,115),(48,114),(49,113),(50,112),(51,111),(52,110),(53,109),(54,108),(55,107),(56,106),(57,126),(58,125),(59,124),(60,123),(61,122),(62,121),(63,120)])

108 conjugacy classes

class 1  2 3A3B3C3D3E6A6B7A7B7C9A···9F9G···9L18A···18F21A···21X63A···63BB
order1233333667779···99···918···1821···2163···63
size1211122221212221···12···221···212···22···2

108 irreducible representations

dim111111222222222
type+++++
imageC1C2C3C6C9C18S3D7C3×S3D21C3×D7S3×C9C9×D7C3×D21C9×D21
kernelC9×D21C3×C63C3×D21C3×C21D21C21C63C3×C9C21C9C32C7C3C3C1
# reps112266132666181236

Matrix representation of C9×D21 in GL2(𝔽127) generated by

520
052
,
940
050
,
050
940
G:=sub<GL(2,GF(127))| [52,0,0,52],[94,0,0,50],[0,94,50,0] >;

C9×D21 in GAP, Magma, Sage, TeX

C_9\times D_{21}
% in TeX

G:=Group("C9xD21");
// GroupNames label

G:=SmallGroup(378,37);
// by ID

G=gap.SmallGroup(378,37);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,36,723,8104]);
// Polycyclic

G:=Group<a,b,c|a^9=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C9×D21 in TeX

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