direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C9×D16, C144⋊5C2, C16⋊1C18, C48.2C6, D8⋊1C18, C36.37D4, C18.15D8, C72.24C22, C3.(C3×D16), (C9×D8)⋊5C2, (C3×D16).C3, C4.1(D4×C9), C2.3(C9×D8), C8.2(C2×C18), (C3×D8).2C6, C6.15(C3×D8), C24.21(C2×C6), C12.36(C3×D4), SmallGroup(288,61)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×D16
G = < a,b,c | a9=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C9×D16
►On 144 points
Generators in S144
(1 56 111 21 122 72 41 137 84)(2 57 112 22 123 73 42 138 85)(3 58 97 23 124 74 43 139 86)(4 59 98 24 125 75 44 140 87)(5 60 99 25 126 76 45 141 88)(6 61 100 26 127 77 46 142 89)(7 62 101 27 128 78 47 143 90)(8 63 102 28 113 79 48 144 91)(9 64 103 29 114 80 33 129 92)(10 49 104 30 115 65 34 130 93)(11 50 105 31 116 66 35 131 94)(12 51 106 32 117 67 36 132 95)(13 52 107 17 118 68 37 133 96)(14 53 108 18 119 69 38 134 81)(15 54 109 19 120 70 39 135 82)(16 55 110 20 121 71 40 136 83)
(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)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 24)(18 23)(19 22)(20 21)(25 32)(26 31)(27 30)(28 29)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(63 64)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 80)(81 86)(82 85)(83 84)(87 96)(88 95)(89 94)(90 93)(91 92)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(109 112)(110 111)(113 114)(115 128)(116 127)(117 126)(118 125)(119 124)(120 123)(121 122)(129 144)(130 143)(131 142)(132 141)(133 140)(134 139)(135 138)(136 137)
G:=sub<Sym(144)| (1,56,111,21,122,72,41,137,84)(2,57,112,22,123,73,42,138,85)(3,58,97,23,124,74,43,139,86)(4,59,98,24,125,75,44,140,87)(5,60,99,25,126,76,45,141,88)(6,61,100,26,127,77,46,142,89)(7,62,101,27,128,78,47,143,90)(8,63,102,28,113,79,48,144,91)(9,64,103,29,114,80,33,129,92)(10,49,104,30,115,65,34,130,93)(11,50,105,31,116,66,35,131,94)(12,51,106,32,117,67,36,132,95)(13,52,107,17,118,68,37,133,96)(14,53,108,18,119,69,38,134,81)(15,54,109,19,120,70,39,135,82)(16,55,110,20,121,71,40,136,83), (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,24)(18,23)(19,22)(20,21)(25,32)(26,31)(27,30)(28,29)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(63,64)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,80)(81,86)(82,85)(83,84)(87,96)(88,95)(89,94)(90,93)(91,92)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,112)(110,111)(113,114)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(121,122)(129,144)(130,143)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)>;
G:=Group( (1,56,111,21,122,72,41,137,84)(2,57,112,22,123,73,42,138,85)(3,58,97,23,124,74,43,139,86)(4,59,98,24,125,75,44,140,87)(5,60,99,25,126,76,45,141,88)(6,61,100,26,127,77,46,142,89)(7,62,101,27,128,78,47,143,90)(8,63,102,28,113,79,48,144,91)(9,64,103,29,114,80,33,129,92)(10,49,104,30,115,65,34,130,93)(11,50,105,31,116,66,35,131,94)(12,51,106,32,117,67,36,132,95)(13,52,107,17,118,68,37,133,96)(14,53,108,18,119,69,38,134,81)(15,54,109,19,120,70,39,135,82)(16,55,110,20,121,71,40,136,83), (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,24)(18,23)(19,22)(20,21)(25,32)(26,31)(27,30)(28,29)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(63,64)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,80)(81,86)(82,85)(83,84)(87,96)(88,95)(89,94)(90,93)(91,92)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,112)(110,111)(113,114)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(121,122)(129,144)(130,143)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137) );
G=PermutationGroup([[(1,56,111,21,122,72,41,137,84),(2,57,112,22,123,73,42,138,85),(3,58,97,23,124,74,43,139,86),(4,59,98,24,125,75,44,140,87),(5,60,99,25,126,76,45,141,88),(6,61,100,26,127,77,46,142,89),(7,62,101,27,128,78,47,143,90),(8,63,102,28,113,79,48,144,91),(9,64,103,29,114,80,33,129,92),(10,49,104,30,115,65,34,130,93),(11,50,105,31,116,66,35,131,94),(12,51,106,32,117,67,36,132,95),(13,52,107,17,118,68,37,133,96),(14,53,108,18,119,69,38,134,81),(15,54,109,19,120,70,39,135,82),(16,55,110,20,121,71,40,136,83)], [(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),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,24),(18,23),(19,22),(20,21),(25,32),(26,31),(27,30),(28,29),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,56),(63,64),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(79,80),(81,86),(82,85),(83,84),(87,96),(88,95),(89,94),(90,93),(91,92),(97,108),(98,107),(99,106),(100,105),(101,104),(102,103),(109,112),(110,111),(113,114),(115,128),(116,127),(117,126),(118,125),(119,124),(120,123),(121,122),(129,144),(130,143),(131,142),(132,141),(133,140),(134,139),(135,138),(136,137)]])
99 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 9A | ··· | 9F | 12A | 12B | 16A | 16B | 16C | 16D | 18A | ··· | 18F | 18G | ··· | 18R | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 48A | ··· | 48H | 72A | ··· | 72L | 144A | ··· | 144X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 16 | 16 | 16 | 16 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 | 144 | ··· | 144 |
| size | 1 | 1 | 8 | 8 | 1 | 1 | 2 | 1 | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | D4 | D8 | C3×D4 | D16 | C3×D8 | D4×C9 | C3×D16 | C9×D8 | C9×D16 |
| kernel | C9×D16 | C144 | C9×D8 | C3×D16 | C48 | C3×D8 | D16 | C16 | D8 | C36 | C18 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 1 | 2 | 2 | 4 | 4 | 6 | 8 | 12 | 24 |
Matrix representation of C9×D16 ►in GL2(𝔽433) generated by
| 296 | 0 |
| 0 | 296 |
| 260 | 41 |
| 392 | 260 |
| 260 | 41 |
| 41 | 173 |
G:=sub<GL(2,GF(433))| [296,0,0,296],[260,392,41,260],[260,41,41,173] >;
C9×D16 in GAP, Magma, Sage, TeX
C_9\times D_{16} % in TeX
G:=Group("C9xD16"); // GroupNames label
G:=SmallGroup(288,61);
// by ID
G=gap.SmallGroup(288,61);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-2,197,142,2355,1186,528,9077,4548,124]);
// Polycyclic
G:=Group<a,b,c|a^9=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export