Copied to
clipboard

## G = C16×D9order 288 = 25·32

### Direct product of C16 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C16×D9
 Chief series C1 — C3 — C9 — C18 — C36 — C72 — C8×D9 — C16×D9
 Lower central C9 — C16×D9
 Upper central C1 — C16

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

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

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

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

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

׿
×
𝔽