metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C9⋊2D16, D8⋊1D9, D72⋊3C2, C8.4D18, C24.6D6, C36.3D4, C18.8D8, C72.2C22, C9⋊C16⋊1C2, (C9×D8)⋊1C2, C3.(C3⋊D16), (C3×D8).1S3, C2.4(D4⋊D9), C4.1(C9⋊D4), C6.15(D4⋊S3), C12.1(C3⋊D4), SmallGroup(288,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9⋊D16
G = < a,b,c | a9=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C9⋊D16
►On 144 points
Generators in S144
(1 132 55 122 90 44 102 23 77)(2 78 24 103 45 91 123 56 133)(3 134 57 124 92 46 104 25 79)(4 80 26 105 47 93 125 58 135)(5 136 59 126 94 48 106 27 65)(6 66 28 107 33 95 127 60 137)(7 138 61 128 96 34 108 29 67)(8 68 30 109 35 81 113 62 139)(9 140 63 114 82 36 110 31 69)(10 70 32 111 37 83 115 64 141)(11 142 49 116 84 38 112 17 71)(12 72 18 97 39 85 117 50 143)(13 144 51 118 86 40 98 19 73)(14 74 20 99 41 87 119 52 129)(15 130 53 120 88 42 100 21 75)(16 76 22 101 43 89 121 54 131)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 61)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 64)(31 63)(32 62)(33 85)(34 84)(35 83)(36 82)(37 81)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)(65 144)(66 143)(67 142)(68 141)(69 140)(70 139)(71 138)(72 137)(73 136)(74 135)(75 134)(76 133)(77 132)(78 131)(79 130)(80 129)(97 127)(98 126)(99 125)(100 124)(101 123)(102 122)(103 121)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(112 128)
G:=sub<Sym(144)| (1,132,55,122,90,44,102,23,77)(2,78,24,103,45,91,123,56,133)(3,134,57,124,92,46,104,25,79)(4,80,26,105,47,93,125,58,135)(5,136,59,126,94,48,106,27,65)(6,66,28,107,33,95,127,60,137)(7,138,61,128,96,34,108,29,67)(8,68,30,109,35,81,113,62,139)(9,140,63,114,82,36,110,31,69)(10,70,32,111,37,83,115,64,141)(11,142,49,116,84,38,112,17,71)(12,72,18,97,39,85,117,50,143)(13,144,51,118,86,40,98,19,73)(14,74,20,99,41,87,119,52,129)(15,130,53,120,88,42,100,21,75)(16,76,22,101,43,89,121,54,131), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,64)(31,63)(32,62)(33,85)(34,84)(35,83)(36,82)(37,81)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(65,144)(66,143)(67,142)(68,141)(69,140)(70,139)(71,138)(72,137)(73,136)(74,135)(75,134)(76,133)(77,132)(78,131)(79,130)(80,129)(97,127)(98,126)(99,125)(100,124)(101,123)(102,122)(103,121)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(112,128)>;
G:=Group( (1,132,55,122,90,44,102,23,77)(2,78,24,103,45,91,123,56,133)(3,134,57,124,92,46,104,25,79)(4,80,26,105,47,93,125,58,135)(5,136,59,126,94,48,106,27,65)(6,66,28,107,33,95,127,60,137)(7,138,61,128,96,34,108,29,67)(8,68,30,109,35,81,113,62,139)(9,140,63,114,82,36,110,31,69)(10,70,32,111,37,83,115,64,141)(11,142,49,116,84,38,112,17,71)(12,72,18,97,39,85,117,50,143)(13,144,51,118,86,40,98,19,73)(14,74,20,99,41,87,119,52,129)(15,130,53,120,88,42,100,21,75)(16,76,22,101,43,89,121,54,131), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,64)(31,63)(32,62)(33,85)(34,84)(35,83)(36,82)(37,81)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(65,144)(66,143)(67,142)(68,141)(69,140)(70,139)(71,138)(72,137)(73,136)(74,135)(75,134)(76,133)(77,132)(78,131)(79,130)(80,129)(97,127)(98,126)(99,125)(100,124)(101,123)(102,122)(103,121)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(112,128) );
G=PermutationGroup([[(1,132,55,122,90,44,102,23,77),(2,78,24,103,45,91,123,56,133),(3,134,57,124,92,46,104,25,79),(4,80,26,105,47,93,125,58,135),(5,136,59,126,94,48,106,27,65),(6,66,28,107,33,95,127,60,137),(7,138,61,128,96,34,108,29,67),(8,68,30,109,35,81,113,62,139),(9,140,63,114,82,36,110,31,69),(10,70,32,111,37,83,115,64,141),(11,142,49,116,84,38,112,17,71),(12,72,18,97,39,85,117,50,143),(13,144,51,118,86,40,98,19,73),(14,74,20,99,41,87,119,52,129),(15,130,53,120,88,42,100,21,75),(16,76,22,101,43,89,121,54,131)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,61),(18,60),(19,59),(20,58),(21,57),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(29,49),(30,64),(31,63),(32,62),(33,85),(34,84),(35,83),(36,82),(37,81),(38,96),(39,95),(40,94),(41,93),(42,92),(43,91),(44,90),(45,89),(46,88),(47,87),(48,86),(65,144),(66,143),(67,142),(68,141),(69,140),(70,139),(71,138),(72,137),(73,136),(74,135),(75,134),(76,133),(77,132),(78,131),(79,130),(80,129),(97,127),(98,126),(99,125),(100,124),(101,123),(102,122),(103,121),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113),(112,128)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 12 | 16A | 16B | 16C | 16D | 18A | 18B | 18C | 18D | ··· | 18I | 24A | 24B | 36A | 36B | 36C | 72A | ··· | 72F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 16 | 16 | 16 | 16 | 18 | 18 | 18 | 18 | ··· | 18 | 24 | 24 | 36 | 36 | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 8 | 72 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D8 | D9 | C3⋊D4 | D16 | D18 | C9⋊D4 | D4⋊S3 | C3⋊D16 | D4⋊D9 | C9⋊D16 |
| kernel | C9⋊D16 | C9⋊C16 | D72 | C9×D8 | C3×D8 | C36 | C24 | C18 | D8 | C12 | C9 | C8 | C4 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 4 | 3 | 6 | 1 | 2 | 3 | 6 |
Matrix representation of C9⋊D16 ►in GL4(𝔽433) generated by
| 350 | 36 | 0 | 0 |
| 397 | 386 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 214 | 82 |
| 0 | 0 | 392 | 132 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 432 | 432 |
G:=sub<GL(4,GF(433))| [350,397,0,0,36,386,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,214,392,0,0,82,132],[0,1,0,0,1,0,0,0,0,0,1,432,0,0,0,432] >;
C9⋊D16 in GAP, Magma, Sage, TeX
C_9\rtimes D_{16} % in TeX
G:=Group("C9:D16"); // GroupNames label
G:=SmallGroup(288,33);
// by ID
G=gap.SmallGroup(288,33);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,254,135,142,675,346,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^9=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export