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G = Q16⋊D9order 288 = 25·32

2nd semidirect product of Q16 and D9 acting via D9/C9=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q162D9, C8.3D18, D18.9D4, C24.34D6, Q8.9D18, C36.9C23, C72.10C22, Dic9.11D4, D36.4C22, Dic18.5C22, (Q8×D9)⋊3C2, C72⋊C24C2, C8⋊D94C2, (C9×Q16)⋊4C2, C9⋊Q164C2, C2.23(D4×D9), C6.97(S3×D4), C9⋊C8.2C22, Q83D9.C2, Q82D93C2, C18.35(C2×D4), C93(C8.C22), C3.(Q16⋊S3), (C3×Q16).6S3, (C3×Q8).29D6, C4.9(C22×D9), (C4×D9).4C22, (Q8×C9).4C22, C12.48(C22×S3), SmallGroup(288,128)

Series: Derived Chief Lower central Upper central

C1C36 — Q16⋊D9
C1C3C9C18C36C4×D9Q8×D9 — Q16⋊D9
C9C18C36 — Q16⋊D9
C1C2C4Q16

Generators and relations for Q16⋊D9
 G = < a,b,c,d | a8=c9=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 436 in 90 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×2], C6, C8, C8, C2×C4 [×3], D4 [×2], Q8 [×2], Q8 [×2], C9, Dic3 [×2], C12, C12 [×2], D6 [×2], M4(2), SD16 [×2], Q16, Q16, C2×Q8, C4○D4, D9 [×2], C18, C3⋊C8, C24, Dic6 [×2], C4×S3 [×3], D12 [×2], C3×Q8 [×2], C8.C22, Dic9, Dic9, C36, C36 [×2], D18, D18, C8⋊S3, C24⋊C2, Q82S3, C3⋊Q16, C3×Q16, S3×Q8, Q83S3, C9⋊C8, C72, Dic18, Dic18, C4×D9, C4×D9 [×2], D36, D36, Q8×C9 [×2], Q16⋊S3, C8⋊D9, C72⋊C2, C9⋊Q16, Q82D9, C9×Q16, Q8×D9, Q83D9, Q16⋊D9
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C22×S3, C8.C22, D18 [×3], S3×D4, C22×D9, Q16⋊S3, D4×D9, Q16⋊D9

Smallest permutation representation of Q16⋊D9
On 144 points
Generators in S144
(1 131 23 122 14 140 32 113)(2 132 24 123 15 141 33 114)(3 133 25 124 16 142 34 115)(4 134 26 125 17 143 35 116)(5 135 27 126 18 144 36 117)(6 127 19 118 10 136 28 109)(7 128 20 119 11 137 29 110)(8 129 21 120 12 138 30 111)(9 130 22 121 13 139 31 112)(37 91 64 73 46 100 55 82)(38 92 65 74 47 101 56 83)(39 93 66 75 48 102 57 84)(40 94 67 76 49 103 58 85)(41 95 68 77 50 104 59 86)(42 96 69 78 51 105 60 87)(43 97 70 79 52 106 61 88)(44 98 71 80 53 107 62 89)(45 99 72 81 54 108 63 90)
(1 50 14 41)(2 51 15 42)(3 52 16 43)(4 53 17 44)(5 54 18 45)(6 46 10 37)(7 47 11 38)(8 48 12 39)(9 49 13 40)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)(73 136 82 127)(74 137 83 128)(75 138 84 129)(76 139 85 130)(77 140 86 131)(78 141 87 132)(79 142 88 133)(80 143 89 134)(81 144 90 135)(91 109 100 118)(92 110 101 119)(93 111 102 120)(94 112 103 121)(95 113 104 122)(96 114 105 123)(97 115 106 124)(98 116 107 125)(99 117 108 126)
(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 40)(2 39)(3 38)(4 37)(5 45)(6 44)(7 43)(8 42)(9 41)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 54)(19 71)(20 70)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 72)(28 62)(29 61)(30 60)(31 59)(32 58)(33 57)(34 56)(35 55)(36 63)(73 116)(74 115)(75 114)(76 113)(77 112)(78 111)(79 110)(80 109)(81 117)(82 125)(83 124)(84 123)(85 122)(86 121)(87 120)(88 119)(89 118)(90 126)(91 143)(92 142)(93 141)(94 140)(95 139)(96 138)(97 137)(98 136)(99 144)(100 134)(101 133)(102 132)(103 131)(104 130)(105 129)(106 128)(107 127)(108 135)

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

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

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

39 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E 6 8A8B9A9B9C12A12B12C18A18B18C24A24B36A36B36C36D···36I72A···72F
order1222344444688999121212181818242436363636···3672···72
size111836224418362436222488222444448···84···4

39 irreducible representations

dim111111112222222244444
type++++++++++++++++-++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D9D18D18C8.C22S3×D4Q16⋊S3D4×D9Q16⋊D9
kernelQ16⋊D9C8⋊D9C72⋊C2C9⋊Q16Q82D9C9×Q16Q8×D9Q83D9C3×Q16Dic9D18C24C3×Q8Q16C8Q8C9C6C3C2C1
# reps111111111111233611236

Matrix representation of Q16⋊D9 in GL4(𝔽73) generated by

001122
005162
31621122
11425162
,
71400
596600
7146659
5966147
,
33100
424500
00331
004245
,
254400
194800
002544
001948
G:=sub<GL(4,GF(73))| [0,0,31,11,0,0,62,42,11,51,11,51,22,62,22,62],[7,59,7,59,14,66,14,66,0,0,66,14,0,0,59,7],[3,42,0,0,31,45,0,0,0,0,3,42,0,0,31,45],[25,19,0,0,44,48,0,0,0,0,25,19,0,0,44,48] >;

Q16⋊D9 in GAP, Magma, Sage, TeX

Q_{16}\rtimes D_9
% in TeX

G:=Group("Q16:D9");
// GroupNames label

G:=SmallGroup(288,128);
// by ID

G=gap.SmallGroup(288,128);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,100,346,185,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^9=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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