metacyclic, supersoluble, monomial, A-group
Aliases: D21⋊C9, C21⋊1C18, C32.2F7, C7⋊C9⋊S3, C7⋊(S3×C9), C3⋊(C7⋊C18), (C3×D21).C3, C21.5(C3×S3), (C3×C21).1C6, C3.5(C3⋊F7), (C3×C7⋊C9)⋊1C2, SmallGroup(378,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — D21⋊C9 |
Generators and relations for D21⋊C9
G = < a,b,c | a21=b2=c9=1, bab=a-1, cac-1=a4, cbc-1=a3b >
Smallest permutation representation of D21⋊C9
►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 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 85)(23 105)(24 104)(25 103)(26 102)(27 101)(28 100)(29 99)(30 98)(31 97)(32 96)(33 95)(34 94)(35 93)(36 92)(37 91)(38 90)(39 89)(40 88)(41 87)(42 86)(43 106)(44 126)(45 125)(46 124)(47 123)(48 122)(49 121)(50 120)(51 119)(52 118)(53 117)(54 116)(55 115)(56 114)(57 113)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)
(1 57 40 15 50 33 8 43 26)(2 52 23 16 45 37 9 59 30)(3 47 27 17 61 41 10 54 34)(4 63 31 18 56 24 11 49 38)(5 58 35 19 51 28 12 44 42)(6 53 39 20 46 32 13 60 25)(7 48 22 21 62 36 14 55 29)(64 115 105 71 122 91 78 108 98)(65 110 88 72 117 95 79 124 102)(66 126 92 73 112 99 80 119 85)(67 121 96 74 107 103 81 114 89)(68 116 100 75 123 86 82 109 93)(69 111 104 76 118 90 83 125 97)(70 106 87 77 113 94 84 120 101)
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), (1,80)(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)(22,85)(23,105)(24,104)(25,103)(26,102)(27,101)(28,100)(29,99)(30,98)(31,97)(32,96)(33,95)(34,94)(35,93)(36,92)(37,91)(38,90)(39,89)(40,88)(41,87)(42,86)(43,106)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107), (1,57,40,15,50,33,8,43,26)(2,52,23,16,45,37,9,59,30)(3,47,27,17,61,41,10,54,34)(4,63,31,18,56,24,11,49,38)(5,58,35,19,51,28,12,44,42)(6,53,39,20,46,32,13,60,25)(7,48,22,21,62,36,14,55,29)(64,115,105,71,122,91,78,108,98)(65,110,88,72,117,95,79,124,102)(66,126,92,73,112,99,80,119,85)(67,121,96,74,107,103,81,114,89)(68,116,100,75,123,86,82,109,93)(69,111,104,76,118,90,83,125,97)(70,106,87,77,113,94,84,120,101)>;
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), (1,80)(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)(22,85)(23,105)(24,104)(25,103)(26,102)(27,101)(28,100)(29,99)(30,98)(31,97)(32,96)(33,95)(34,94)(35,93)(36,92)(37,91)(38,90)(39,89)(40,88)(41,87)(42,86)(43,106)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107), (1,57,40,15,50,33,8,43,26)(2,52,23,16,45,37,9,59,30)(3,47,27,17,61,41,10,54,34)(4,63,31,18,56,24,11,49,38)(5,58,35,19,51,28,12,44,42)(6,53,39,20,46,32,13,60,25)(7,48,22,21,62,36,14,55,29)(64,115,105,71,122,91,78,108,98)(65,110,88,72,117,95,79,124,102)(66,126,92,73,112,99,80,119,85)(67,121,96,74,107,103,81,114,89)(68,116,100,75,123,86,82,109,93)(69,111,104,76,118,90,83,125,97)(70,106,87,77,113,94,84,120,101) );
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)], [(1,80),(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),(22,85),(23,105),(24,104),(25,103),(26,102),(27,101),(28,100),(29,99),(30,98),(31,97),(32,96),(33,95),(34,94),(35,93),(36,92),(37,91),(38,90),(39,89),(40,88),(41,87),(42,86),(43,106),(44,126),(45,125),(46,124),(47,123),(48,122),(49,121),(50,120),(51,119),(52,118),(53,117),(54,116),(55,115),(56,114),(57,113),(58,112),(59,111),(60,110),(61,109),(62,108),(63,107)], [(1,57,40,15,50,33,8,43,26),(2,52,23,16,45,37,9,59,30),(3,47,27,17,61,41,10,54,34),(4,63,31,18,56,24,11,49,38),(5,58,35,19,51,28,12,44,42),(6,53,39,20,46,32,13,60,25),(7,48,22,21,62,36,14,55,29),(64,115,105,71,122,91,78,108,98),(65,110,88,72,117,95,79,124,102),(66,126,92,73,112,99,80,119,85),(67,121,96,74,107,103,81,114,89),(68,116,100,75,123,86,82,109,93),(69,111,104,76,118,90,83,125,97),(70,106,87,77,113,94,84,120,101)]])
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 7 | 9A | ··· | 9F | 9G | ··· | 9L | 18A | ··· | 18F | 21A | ··· | 21H |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 7 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 21 | ··· | 21 |
| size | 1 | 21 | 1 | 1 | 2 | 2 | 2 | 21 | 21 | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 6 | ··· | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | S3 | C3×S3 | S3×C9 | F7 | C7⋊C18 | C3⋊F7 | D21⋊C9 |
| kernel | D21⋊C9 | C3×C7⋊C9 | C3×D21 | C3×C21 | D21 | C21 | C7⋊C9 | C21 | C7 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 6 | 1 | 2 | 2 | 4 |
Matrix representation of D21⋊C9 ►in GL6(𝔽127)
| 0 | 68 | 79 | 0 | 0 | 0 |
| 20 | 20 | 59 | 0 | 0 | 0 |
| 0 | 20 | 59 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 108 | 19 |
| 0 | 0 | 0 | 90 | 0 | 56 |
| 0 | 0 | 0 | 19 | 0 | 37 |
| 0 | 0 | 0 | 0 | 108 | 19 |
| 0 | 0 | 0 | 90 | 0 | 56 |
| 0 | 0 | 0 | 19 | 0 | 37 |
| 0 | 68 | 79 | 0 | 0 | 0 |
| 20 | 20 | 59 | 0 | 0 | 0 |
| 0 | 20 | 59 | 0 | 0 | 0 |
| 115 | 45 | 120 | 0 | 0 | 0 |
| 85 | 26 | 102 | 0 | 0 | 0 |
| 86 | 7 | 113 | 0 | 0 | 0 |
| 0 | 0 | 0 | 115 | 45 | 120 |
| 0 | 0 | 0 | 85 | 26 | 102 |
| 0 | 0 | 0 | 86 | 7 | 113 |
G:=sub<GL(6,GF(127))| [0,20,0,0,0,0,68,20,20,0,0,0,79,59,59,0,0,0,0,0,0,0,90,19,0,0,0,108,0,0,0,0,0,19,56,37],[0,0,0,0,20,0,0,0,0,68,20,20,0,0,0,79,59,59,0,90,19,0,0,0,108,0,0,0,0,0,19,56,37,0,0,0],[115,85,86,0,0,0,45,26,7,0,0,0,120,102,113,0,0,0,0,0,0,115,85,86,0,0,0,45,26,7,0,0,0,120,102,113] >;
D21⋊C9 in GAP, Magma, Sage, TeX
D_{21}\rtimes C_9 % in TeX
G:=Group("D21:C9"); // GroupNames label
G:=SmallGroup(378,21);
// by ID
G=gap.SmallGroup(378,21);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,36,723,8104,1359]);
// Polycyclic
G:=Group<a,b,c|a^21=b^2=c^9=1,b*a*b=a^-1,c*a*c^-1=a^4,c*b*c^-1=a^3*b>;
// generators/relations
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