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G = C16×D9order 288 = 25·32

Direct product of C16 and D9

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C16×D9, C1443C2, C48.7S3, D18.2C8, C24.84D6, C8.18D18, Dic9.2C8, C72.19C22, C9⋊C166C2, C91(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

C1C9 — C16×D9
C1C3C9C18C36C72C8×D9 — C16×D9
C9 — C16×D9
C1C16

Generators and relations for C16×D9
 G = < a,b,c | a16=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
9C2
9C4
9C22
3S3
3S3
9C2×C4
9C8
3D6
3Dic3
9C2×C8
9C16
3C3⋊C8
3C4×S3
9C2×C16
3C3⋊C16
3S3×C8
3S3×C16

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 2A2B2C 3 4A4B4C4D 6 8A8B8C8D8E8F8G8H9A9B9C12A12B16A···16H16I···16P18A18B18C24A24B24C24D36A···36F48A···48H72A···72L144A···144X
order122234444688888888999121216···1616···161818182424242436···3648···4872···72144···144
size119921199211119999222221···19···922222222···22···22···22···2

96 irreducible representations

dim1111111112222222222
type++++++++
imageC1C2C2C2C4C4C8C8C16S3D6D9C4×S3D18S3×C8C4×D9S3×C16C8×D9C16×D9
kernelC16×D9C9⋊C16C144C8×D9C9⋊C8C4×D9Dic9D18D9C48C24C16C12C8C6C4C3C2C1
# reps1111224416113234681224

Matrix representation of C16×D9 in GL2(𝔽17) generated by

60
06
,
146
140
,
06
30
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

Subgroup lattice of C16×D9 in TeX

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