direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C9×D21, C63⋊3S3, C21⋊5C18, C3⋊(C9×D7), C7⋊3(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
| C21 — C9×D21 |
Generators and relations for C9×D21
G = < a,b,c | a9=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C9×D21
►On 126 points
Generators in S126
(1 54 24 15 47 38 8 61 31)(2 55 25 16 48 39 9 62 32)(3 56 26 17 49 40 10 63 33)(4 57 27 18 50 41 11 43 34)(5 58 28 19 51 42 12 44 35)(6 59 29 20 52 22 13 45 36)(7 60 30 21 53 23 14 46 37)(64 119 100 71 126 86 78 112 93)(65 120 101 72 106 87 79 113 94)(66 121 102 73 107 88 80 114 95)(67 122 103 74 108 89 81 115 96)(68 123 104 75 109 90 82 116 97)(69 124 105 76 110 91 83 117 98)(70 125 85 77 111 92 84 118 99)
(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 74)(2 73)(3 72)(4 71)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 84)(13 83)(14 82)(15 81)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 91)(23 90)(24 89)(25 88)(26 87)(27 86)(28 85)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 93)(42 92)(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)| (1,54,24,15,47,38,8,61,31)(2,55,25,16,48,39,9,62,32)(3,56,26,17,49,40,10,63,33)(4,57,27,18,50,41,11,43,34)(5,58,28,19,51,42,12,44,35)(6,59,29,20,52,22,13,45,36)(7,60,30,21,53,23,14,46,37)(64,119,100,71,126,86,78,112,93)(65,120,101,72,106,87,79,113,94)(66,121,102,73,107,88,80,114,95)(67,122,103,74,108,89,81,115,96)(68,123,104,75,109,90,82,116,97)(69,124,105,76,110,91,83,117,98)(70,125,85,77,111,92,84,118,99), (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), (1,74)(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,84)(13,83)(14,82)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(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( (1,54,24,15,47,38,8,61,31)(2,55,25,16,48,39,9,62,32)(3,56,26,17,49,40,10,63,33)(4,57,27,18,50,41,11,43,34)(5,58,28,19,51,42,12,44,35)(6,59,29,20,52,22,13,45,36)(7,60,30,21,53,23,14,46,37)(64,119,100,71,126,86,78,112,93)(65,120,101,72,106,87,79,113,94)(66,121,102,73,107,88,80,114,95)(67,122,103,74,108,89,81,115,96)(68,123,104,75,109,90,82,116,97)(69,124,105,76,110,91,83,117,98)(70,125,85,77,111,92,84,118,99), (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), (1,74)(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,84)(13,83)(14,82)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(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([[(1,54,24,15,47,38,8,61,31),(2,55,25,16,48,39,9,62,32),(3,56,26,17,49,40,10,63,33),(4,57,27,18,50,41,11,43,34),(5,58,28,19,51,42,12,44,35),(6,59,29,20,52,22,13,45,36),(7,60,30,21,53,23,14,46,37),(64,119,100,71,126,86,78,112,93),(65,120,101,72,106,87,79,113,94),(66,121,102,73,107,88,80,114,95),(67,122,103,74,108,89,81,115,96),(68,123,104,75,109,90,82,116,97),(69,124,105,76,110,91,83,117,98),(70,125,85,77,111,92,84,118,99)], [(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)], [(1,74),(2,73),(3,72),(4,71),(5,70),(6,69),(7,68),(8,67),(9,66),(10,65),(11,64),(12,84),(13,83),(14,82),(15,81),(16,80),(17,79),(18,78),(19,77),(20,76),(21,75),(22,91),(23,90),(24,89),(25,88),(26,87),(27,86),(28,85),(29,105),(30,104),(31,103),(32,102),(33,101),(34,100),(35,99),(36,98),(37,97),(38,96),(39,95),(40,94),(41,93),(42,92),(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 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 7A | 7B | 7C | 9A | ··· | 9F | 9G | ··· | 9L | 18A | ··· | 18F | 21A | ··· | 21X | 63A | ··· | 63BB |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 7 | 7 | 7 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 21 | ··· | 21 | 63 | ··· | 63 |
| size | 1 | 21 | 1 | 1 | 2 | 2 | 2 | 21 | 21 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 21 | ··· | 21 | 2 | ··· | 2 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | S3 | D7 | C3×S3 | D21 | C3×D7 | S3×C9 | C9×D7 | C3×D21 | C9×D21 |
| kernel | C9×D21 | C3×C63 | C3×D21 | C3×C21 | D21 | C21 | C63 | C3×C9 | C21 | C9 | C32 | C7 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 3 | 2 | 6 | 6 | 6 | 18 | 12 | 36 |
Matrix representation of C9×D21 ►in GL2(𝔽127) generated by
| 52 | 0 |
| 0 | 52 |
| 94 | 0 |
| 0 | 50 |
| 0 | 50 |
| 94 | 0 |
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
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