direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D9×C21, C9⋊3C42, C63⋊18C6, (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
| C9 — D9×C21 |
Generators and relations for D9×C21
G = < a,b,c | a21=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
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)
(1 112 65 8 119 72 15 126 79)(2 113 66 9 120 73 16 106 80)(3 114 67 10 121 74 17 107 81)(4 115 68 11 122 75 18 108 82)(5 116 69 12 123 76 19 109 83)(6 117 70 13 124 77 20 110 84)(7 118 71 14 125 78 21 111 64)(22 85 50 36 99 43 29 92 57)(23 86 51 37 100 44 30 93 58)(24 87 52 38 101 45 31 94 59)(25 88 53 39 102 46 32 95 60)(26 89 54 40 103 47 33 96 61)(27 90 55 41 104 48 34 97 62)(28 91 56 42 105 49 35 98 63)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(43 119)(44 120)(45 121)(46 122)(47 123)(48 124)(49 125)(50 126)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)
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), (1,112,65,8,119,72,15,126,79)(2,113,66,9,120,73,16,106,80)(3,114,67,10,121,74,17,107,81)(4,115,68,11,122,75,18,108,82)(5,116,69,12,123,76,19,109,83)(6,117,70,13,124,77,20,110,84)(7,118,71,14,125,78,21,111,64)(22,85,50,36,99,43,29,92,57)(23,86,51,37,100,44,30,93,58)(24,87,52,38,101,45,31,94,59)(25,88,53,39,102,46,32,95,60)(26,89,54,40,103,47,33,96,61)(27,90,55,41,104,48,34,97,62)(28,91,56,42,105,49,35,98,63), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(49,125)(50,126)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)>;
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), (1,112,65,8,119,72,15,126,79)(2,113,66,9,120,73,16,106,80)(3,114,67,10,121,74,17,107,81)(4,115,68,11,122,75,18,108,82)(5,116,69,12,123,76,19,109,83)(6,117,70,13,124,77,20,110,84)(7,118,71,14,125,78,21,111,64)(22,85,50,36,99,43,29,92,57)(23,86,51,37,100,44,30,93,58)(24,87,52,38,101,45,31,94,59)(25,88,53,39,102,46,32,95,60)(26,89,54,40,103,47,33,96,61)(27,90,55,41,104,48,34,97,62)(28,91,56,42,105,49,35,98,63), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(49,125)(50,126)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90) );
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)], [(1,112,65,8,119,72,15,126,79),(2,113,66,9,120,73,16,106,80),(3,114,67,10,121,74,17,107,81),(4,115,68,11,122,75,18,108,82),(5,116,69,12,123,76,19,109,83),(6,117,70,13,124,77,20,110,84),(7,118,71,14,125,78,21,111,64),(22,85,50,36,99,43,29,92,57),(23,86,51,37,100,44,30,93,58),(24,87,52,38,101,45,31,94,59),(25,88,53,39,102,46,32,95,60),(26,89,54,40,103,47,33,96,61),(27,90,55,41,104,48,34,97,62),(28,91,56,42,105,49,35,98,63)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(43,119),(44,120),(45,121),(46,122),(47,123),(48,124),(49,125),(50,126),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(57,112),(58,113),(59,114),(60,115),(61,116),(62,117),(63,118),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99),(73,100),(74,101),(75,102),(76,103),(77,104),(78,105),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90)]])
126 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 7A | ··· | 7F | 9A | ··· | 9I | 14A | ··· | 14F | 21A | ··· | 21L | 21M | ··· | 21AD | 42A | ··· | 42L | 63A | ··· | 63BB |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 9 | ··· | 9 | 14 | ··· | 14 | 21 | ··· | 21 | 21 | ··· | 21 | 42 | ··· | 42 | 63 | ··· | 63 |
| size | 1 | 9 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 | S3 | D9 | C3×S3 | S3×C7 | C3×D9 | C7×D9 | S3×C21 | D9×C21 |
| kernel | D9×C21 | C3×C63 | C7×D9 | C63 | C3×D9 | C3×C9 | D9 | C9 | C3×C21 | C21 | C21 | C32 | C7 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 1 | 3 | 2 | 6 | 6 | 18 | 12 | 36 |
Matrix representation of D9×C21 ►in GL2(𝔽127) generated by
| 100 | 0 |
| 0 | 100 |
| 52 | 0 |
| 53 | 22 |
| 105 | 46 |
| 53 | 22 |
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