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G = C16⋊D9order 288 = 25·32

3rd semidirect product of C16 and D9 acting via D9/C9=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C163D9, D18.C8, C1444C2, C48.8S3, Dic9.C8, C91M5(2), C24.85D6, C8.19D18, C72.20C22, C9⋊C164C2, C9⋊C8.2C4, C6.7(S3×C8), C2.3(C8×D9), C18.2(C2×C8), (C8×D9).2C2, (C4×D9).2C4, C4.17(C4×D9), C3.(D6.C8), C12.66(C4×S3), C36.22(C2×C4), SmallGroup(288,5)

Series: Derived Chief Lower central Upper central

C1C18 — C16⋊D9
C1C3C9C18C36C72C8×D9 — C16⋊D9
C9C18 — C16⋊D9
C1C8C16

Generators and relations for C16⋊D9
 G = < a,b,c | a16=b9=c2=1, ab=ba, cac=a9, cbc=b-1 >

18C2
9C22
9C4
6S3
9C8
9C2×C4
3Dic3
3D6
2D9
9C16
9C2×C8
3C3⋊C8
3C4×S3
9M5(2)
3S3×C8
3C3⋊C16
3D6.C8

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

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

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

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

84 conjugacy classes

class 1 2A2B 3 4A4B4C 6 8A8B8C8D8E8F9A9B9C12A12B16A16B16C16D16E16F16G16H18A18B18C24A24B24C24D36A···36F48A···48H72A···72L144A···144X
order12234446888888999121216161616161616161818182424242436···3648···4872···72144···144
size1118211182111118182222222221818181822222222···22···22···22···2

84 irreducible representations

dim1111111122222222222
type++++++++
imageC1C2C2C2C4C4C8C8S3D6D9C4×S3M5(2)D18S3×C8C4×D9D6.C8C8×D9C16⋊D9
kernelC16⋊D9C9⋊C16C144C8×D9C9⋊C8C4×D9Dic9D18C48C24C16C12C9C8C6C4C3C2C1
# reps111122441132434681224

Matrix representation of C16⋊D9 in GL4(𝔽433) generated by

2588300
35017500
004320
000432
,
0100
43243200
0047350
0083397
,
0100
1000
0083397
0047350
G:=sub<GL(4,GF(433))| [258,350,0,0,83,175,0,0,0,0,432,0,0,0,0,432],[0,432,0,0,1,432,0,0,0,0,47,83,0,0,350,397],[0,1,0,0,1,0,0,0,0,0,83,47,0,0,397,350] >;

C16⋊D9 in GAP, Magma, Sage, TeX

C_{16}\rtimes D_9
% in TeX

G:=Group("C16:D9");
// GroupNames label

G:=SmallGroup(288,5);
// by ID

G=gap.SmallGroup(288,5);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,36,58,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^16=b^9=c^2=1,a*b=b*a,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C16⋊D9 in TeX

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