direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C16×D9, C144⋊3C2, C48.7S3, D18.2C8, C24.84D6, C8.18D18, Dic9.2C8, C72.19C22, C9⋊C16⋊6C2, C9⋊1(C2×C16), C9⋊C8.3C4, C3.(S3×C16), C6.6(S3×C8), C2.1(C8×D9), C18.1(C2×C8), (C8×D9).3C2, (C4×D9).4C4, C4.16(C4×D9), C12.65(C4×S3), C36.21(C2×C4), SmallGroup(288,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C16×D9 |
Generators and relations for C16×D9
G = < a,b,c | a16=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C16×D9
►On 144 points
Generators in S144
(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 60 82 98 74 31 119 44 133)(2 61 83 99 75 32 120 45 134)(3 62 84 100 76 17 121 46 135)(4 63 85 101 77 18 122 47 136)(5 64 86 102 78 19 123 48 137)(6 49 87 103 79 20 124 33 138)(7 50 88 104 80 21 125 34 139)(8 51 89 105 65 22 126 35 140)(9 52 90 106 66 23 127 36 141)(10 53 91 107 67 24 128 37 142)(11 54 92 108 68 25 113 38 143)(12 55 93 109 69 26 114 39 144)(13 56 94 110 70 27 115 40 129)(14 57 95 111 71 28 116 41 130)(15 58 96 112 72 29 117 42 131)(16 59 81 97 73 30 118 43 132)
(1 141)(2 142)(3 143)(4 144)(5 129)(6 130)(7 131)(8 132)(9 133)(10 134)(11 135)(12 136)(13 137)(14 138)(15 139)(16 140)(17 108)(18 109)(19 110)(20 111)(21 112)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(81 126)(82 127)(83 128)(84 113)(85 114)(86 115)(87 116)(88 117)(89 118)(90 119)(91 120)(92 121)(93 122)(94 123)(95 124)(96 125)
G:=sub<Sym(144)| (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), (1,60,82,98,74,31,119,44,133)(2,61,83,99,75,32,120,45,134)(3,62,84,100,76,17,121,46,135)(4,63,85,101,77,18,122,47,136)(5,64,86,102,78,19,123,48,137)(6,49,87,103,79,20,124,33,138)(7,50,88,104,80,21,125,34,139)(8,51,89,105,65,22,126,35,140)(9,52,90,106,66,23,127,36,141)(10,53,91,107,67,24,128,37,142)(11,54,92,108,68,25,113,38,143)(12,55,93,109,69,26,114,39,144)(13,56,94,110,70,27,115,40,129)(14,57,95,111,71,28,116,41,130)(15,58,96,112,72,29,117,42,131)(16,59,81,97,73,30,118,43,132), (1,141)(2,142)(3,143)(4,144)(5,129)(6,130)(7,131)(8,132)(9,133)(10,134)(11,135)(12,136)(13,137)(14,138)(15,139)(16,140)(17,108)(18,109)(19,110)(20,111)(21,112)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(81,126)(82,127)(83,128)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119)(91,120)(92,121)(93,122)(94,123)(95,124)(96,125)>;
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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,60,82,98,74,31,119,44,133)(2,61,83,99,75,32,120,45,134)(3,62,84,100,76,17,121,46,135)(4,63,85,101,77,18,122,47,136)(5,64,86,102,78,19,123,48,137)(6,49,87,103,79,20,124,33,138)(7,50,88,104,80,21,125,34,139)(8,51,89,105,65,22,126,35,140)(9,52,90,106,66,23,127,36,141)(10,53,91,107,67,24,128,37,142)(11,54,92,108,68,25,113,38,143)(12,55,93,109,69,26,114,39,144)(13,56,94,110,70,27,115,40,129)(14,57,95,111,71,28,116,41,130)(15,58,96,112,72,29,117,42,131)(16,59,81,97,73,30,118,43,132), (1,141)(2,142)(3,143)(4,144)(5,129)(6,130)(7,131)(8,132)(9,133)(10,134)(11,135)(12,136)(13,137)(14,138)(15,139)(16,140)(17,108)(18,109)(19,110)(20,111)(21,112)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(81,126)(82,127)(83,128)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119)(91,120)(92,121)(93,122)(94,123)(95,124)(96,125) );
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),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,60,82,98,74,31,119,44,133),(2,61,83,99,75,32,120,45,134),(3,62,84,100,76,17,121,46,135),(4,63,85,101,77,18,122,47,136),(5,64,86,102,78,19,123,48,137),(6,49,87,103,79,20,124,33,138),(7,50,88,104,80,21,125,34,139),(8,51,89,105,65,22,126,35,140),(9,52,90,106,66,23,127,36,141),(10,53,91,107,67,24,128,37,142),(11,54,92,108,68,25,113,38,143),(12,55,93,109,69,26,114,39,144),(13,56,94,110,70,27,115,40,129),(14,57,95,111,71,28,116,41,130),(15,58,96,112,72,29,117,42,131),(16,59,81,97,73,30,118,43,132)], [(1,141),(2,142),(3,143),(4,144),(5,129),(6,130),(7,131),(8,132),(9,133),(10,134),(11,135),(12,136),(13,137),(14,138),(15,139),(16,140),(17,108),(18,109),(19,110),(20,111),(21,112),(22,97),(23,98),(24,99),(25,100),(26,101),(27,102),(28,103),(29,104),(30,105),(31,106),(32,107),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80),(81,126),(82,127),(83,128),(84,113),(85,114),(86,115),(87,116),(88,117),(89,118),(90,119),(91,120),(92,121),(93,122),(94,123),(95,124),(96,125)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 9A | 9B | 9C | 12A | 12B | 16A | ··· | 16H | 16I | ··· | 16P | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 48A | ··· | 48H | 72A | ··· | 72L | 144A | ··· | 144X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 16 | ··· | 16 | 16 | ··· | 16 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 | 144 | ··· | 144 |
| size | 1 | 1 | 9 | 9 | 2 | 1 | 1 | 9 | 9 | 2 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | D6 | D9 | C4×S3 | D18 | S3×C8 | C4×D9 | S3×C16 | C8×D9 | C16×D9 |
| kernel | C16×D9 | C9⋊C16 | C144 | C8×D9 | C9⋊C8 | C4×D9 | Dic9 | D18 | D9 | C48 | C24 | C16 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 8 | 12 | 24 |
Matrix representation of C16×D9 ►in GL2(𝔽17) generated by
| 6 | 0 |
| 0 | 6 |
| 14 | 6 |
| 14 | 0 |
| 0 | 6 |
| 3 | 0 |
G:=sub<GL(2,GF(17))| [6,0,0,6],[14,14,6,0],[0,3,6,0] >;
C16×D9 in GAP, Magma, Sage, TeX
C_{16}\times D_9 % in TeX
G:=Group("C16xD9"); // GroupNames label
G:=SmallGroup(288,4);
// by ID
G=gap.SmallGroup(288,4);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,36,58,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^16=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export