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 70 55 86 27 119 136 47 108)(2 109 48 137 120 28 87 56 71)(3 72 57 88 29 121 138 33 110)(4 111 34 139 122 30 89 58 73)(5 74 59 90 31 123 140 35 112)(6 97 36 141 124 32 91 60 75)(7 76 61 92 17 125 142 37 98)(8 99 38 143 126 18 93 62 77)(9 78 63 94 19 127 144 39 100)(10 101 40 129 128 20 95 64 79)(11 80 49 96 21 113 130 41 102)(12 103 42 131 114 22 81 50 65)(13 66 51 82 23 115 132 43 104)(14 105 44 133 116 24 83 52 67)(15 68 53 84 25 117 134 45 106)(16 107 46 135 118 26 85 54 69)
(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)(17 113)(18 128)(19 127)(20 126)(21 125)(22 124)(23 123)(24 122)(25 121)(26 120)(27 119)(28 118)(29 117)(30 116)(31 115)(32 114)(33 53)(34 52)(35 51)(36 50)(37 49)(38 64)(39 63)(40 62)(41 61)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(65 97)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 141)(82 140)(83 139)(84 138)(85 137)(86 136)(87 135)(88 134)(89 133)(90 132)(91 131)(92 130)(93 129)(94 144)(95 143)(96 142)
G:=sub<Sym(144)| (1,70,55,86,27,119,136,47,108)(2,109,48,137,120,28,87,56,71)(3,72,57,88,29,121,138,33,110)(4,111,34,139,122,30,89,58,73)(5,74,59,90,31,123,140,35,112)(6,97,36,141,124,32,91,60,75)(7,76,61,92,17,125,142,37,98)(8,99,38,143,126,18,93,62,77)(9,78,63,94,19,127,144,39,100)(10,101,40,129,128,20,95,64,79)(11,80,49,96,21,113,130,41,102)(12,103,42,131,114,22,81,50,65)(13,66,51,82,23,115,132,43,104)(14,105,44,133,116,24,83,52,67)(15,68,53,84,25,117,134,45,106)(16,107,46,135,118,26,85,54,69), (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)(17,113)(18,128)(19,127)(20,126)(21,125)(22,124)(23,123)(24,122)(25,121)(26,120)(27,119)(28,118)(29,117)(30,116)(31,115)(32,114)(33,53)(34,52)(35,51)(36,50)(37,49)(38,64)(39,63)(40,62)(41,61)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(65,97)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,141)(82,140)(83,139)(84,138)(85,137)(86,136)(87,135)(88,134)(89,133)(90,132)(91,131)(92,130)(93,129)(94,144)(95,143)(96,142)>;
G:=Group( (1,70,55,86,27,119,136,47,108)(2,109,48,137,120,28,87,56,71)(3,72,57,88,29,121,138,33,110)(4,111,34,139,122,30,89,58,73)(5,74,59,90,31,123,140,35,112)(6,97,36,141,124,32,91,60,75)(7,76,61,92,17,125,142,37,98)(8,99,38,143,126,18,93,62,77)(9,78,63,94,19,127,144,39,100)(10,101,40,129,128,20,95,64,79)(11,80,49,96,21,113,130,41,102)(12,103,42,131,114,22,81,50,65)(13,66,51,82,23,115,132,43,104)(14,105,44,133,116,24,83,52,67)(15,68,53,84,25,117,134,45,106)(16,107,46,135,118,26,85,54,69), (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)(17,113)(18,128)(19,127)(20,126)(21,125)(22,124)(23,123)(24,122)(25,121)(26,120)(27,119)(28,118)(29,117)(30,116)(31,115)(32,114)(33,53)(34,52)(35,51)(36,50)(37,49)(38,64)(39,63)(40,62)(41,61)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(65,97)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,141)(82,140)(83,139)(84,138)(85,137)(86,136)(87,135)(88,134)(89,133)(90,132)(91,131)(92,130)(93,129)(94,144)(95,143)(96,142) );
G=PermutationGroup([(1,70,55,86,27,119,136,47,108),(2,109,48,137,120,28,87,56,71),(3,72,57,88,29,121,138,33,110),(4,111,34,139,122,30,89,58,73),(5,74,59,90,31,123,140,35,112),(6,97,36,141,124,32,91,60,75),(7,76,61,92,17,125,142,37,98),(8,99,38,143,126,18,93,62,77),(9,78,63,94,19,127,144,39,100),(10,101,40,129,128,20,95,64,79),(11,80,49,96,21,113,130,41,102),(12,103,42,131,114,22,81,50,65),(13,66,51,82,23,115,132,43,104),(14,105,44,133,116,24,83,52,67),(15,68,53,84,25,117,134,45,106),(16,107,46,135,118,26,85,54,69)], [(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),(17,113),(18,128),(19,127),(20,126),(21,125),(22,124),(23,123),(24,122),(25,121),(26,120),(27,119),(28,118),(29,117),(30,116),(31,115),(32,114),(33,53),(34,52),(35,51),(36,50),(37,49),(38,64),(39,63),(40,62),(41,61),(42,60),(43,59),(44,58),(45,57),(46,56),(47,55),(48,54),(65,97),(66,112),(67,111),(68,110),(69,109),(70,108),(71,107),(72,106),(73,105),(74,104),(75,103),(76,102),(77,101),(78,100),(79,99),(80,98),(81,141),(82,140),(83,139),(84,138),(85,137),(86,136),(87,135),(88,134),(89,133),(90,132),(91,131),(92,130),(93,129),(94,144),(95,143),(96,142)])
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