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G = D9×C21order 378 = 2·33·7

Direct product of C21 and D9

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

Aliases: D9×C21, C93C42, C6318C6, (C3×C9)⋊2C14, (C3×C63)⋊4C2, C3.1(S3×C21), (C3×C21).5S3, C21.15(C3×S3), C32.2(S3×C7), SmallGroup(378,32)

Series: Derived Chief Lower central Upper central

C1C9 — D9×C21
C1C3C9C63C3×C63 — D9×C21
C9 — D9×C21
C1C21

Generators and relations for D9×C21
 G = < a,b,c | a21=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
2C3
3S3
9C6
2C9
9C14
2C21
3C3×S3
3S3×C7
9C42
2C63
3S3×C21

Smallest permutation representation of D9×C21
On 126 points
Generators in S126
(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 68 114 8 75 121 15 82 107)(2 69 115 9 76 122 16 83 108)(3 70 116 10 77 123 17 84 109)(4 71 117 11 78 124 18 64 110)(5 72 118 12 79 125 19 65 111)(6 73 119 13 80 126 20 66 112)(7 74 120 14 81 106 21 67 113)(22 58 92 36 51 85 29 44 99)(23 59 93 37 52 86 30 45 100)(24 60 94 38 53 87 31 46 101)(25 61 95 39 54 88 32 47 102)(26 62 96 40 55 89 33 48 103)(27 63 97 41 56 90 34 49 104)(28 43 98 42 57 91 35 50 105)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 121)(23 122)(24 123)(25 124)(26 125)(27 126)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 81)(44 82)(45 83)(46 84)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)

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

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

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

126 conjugacy classes

class 1  2 3A3B3C3D3E6A6B7A···7F9A···9I14A···14F21A···21L21M···21AD42A···42L63A···63BB
order1233333667···79···914···1421···2121···2142···4263···63
size1911222991···12···29···91···12···29···92···2

126 irreducible representations

dim1111111122222222
type++++
imageC1C2C3C6C7C14C21C42S3D9C3×S3S3×C7C3×D9C7×D9S3×C21D9×C21
kernelD9×C21C3×C63C7×D9C63C3×D9C3×C9D9C9C3×C21C21C21C32C7C3C3C1
# reps112266121213266181236

Matrix representation of D9×C21 in GL2(𝔽127) generated by

1000
0100
,
520
5322
,
10546
5322
G:=sub<GL(2,GF(127))| [100,0,0,100],[52,53,0,22],[105,53,46,22] >;

D9×C21 in GAP, Magma, Sage, TeX

D_9\times C_{21}
% in TeX

G:=Group("D9xC21");
// GroupNames label

G:=SmallGroup(378,32);
// by ID

G=gap.SmallGroup(378,32);
# by ID

G:=PCGroup([5,-2,-3,-7,-3,-3,4203,138,6304]);
// Polycyclic

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

Export

Subgroup lattice of D9×C21 in TeX

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