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## G = SD16⋊D9order 288 = 25·32

### 2nd semidirect product of SD16 and D9 acting via D9/C9=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — SD16⋊D9
 Chief series C1 — C3 — C9 — C18 — C36 — C4×D9 — Q8×D9 — SD16⋊D9
 Lower central C9 — C18 — C36 — SD16⋊D9
 Upper central C1 — C2 — C4 — SD16

Generators and relations for SD16⋊D9
G = < a,b,c,d | a8=b2=c9=d2=1, bab=a3, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 396 in 90 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C9, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, SD16, Q16, C2×Q8, C4○D4, D9, C18, C18, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C8.C22, Dic9, Dic9, C36, C36, D18, C2×C18, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D42S3, S3×Q8, C9⋊C8, C72, Dic18, Dic18, C4×D9, C4×D9, C2×Dic9, C9⋊D4, D4×C9, Q8×C9, D4.D6, Dic36, C8⋊D9, D4.D9, C9⋊Q16, C9×SD16, D42D9, Q8×D9, SD16⋊D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C22×S3, C8.C22, D18, S3×D4, C22×D9, D4.D6, D4×D9, SD16⋊D9

Smallest permutation representation of SD16⋊D9
On 144 points
Generators in S144
(1 140 32 122 14 131 23 113)(2 141 33 123 15 132 24 114)(3 142 34 124 16 133 25 115)(4 143 35 125 17 134 26 116)(5 144 36 126 18 135 27 117)(6 136 28 118 10 127 19 109)(7 137 29 119 11 128 20 110)(8 138 30 120 12 129 21 111)(9 139 31 121 13 130 22 112)(37 91 55 82 46 100 64 73)(38 92 56 83 47 101 65 74)(39 93 57 84 48 102 66 75)(40 94 58 85 49 103 67 76)(41 95 59 86 50 104 68 77)(42 96 60 87 51 105 69 78)(43 97 61 88 52 106 70 79)(44 98 62 89 53 107 71 80)(45 99 63 90 54 108 72 81)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 37)(7 38)(8 39)(9 40)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(73 127)(74 128)(75 129)(76 130)(77 131)(78 132)(79 133)(80 134)(81 135)(82 136)(83 137)(84 138)(85 139)(86 140)(87 141)(88 142)(89 143)(90 144)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 109)(101 110)(102 111)(103 112)(104 113)(105 114)(106 115)(107 116)(108 117)
(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 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(45 54)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 72)(73 80)(74 79)(75 78)(76 77)(82 89)(83 88)(84 87)(85 86)(91 98)(92 97)(93 96)(94 95)(100 107)(101 106)(102 105)(103 104)(109 125)(110 124)(111 123)(112 122)(113 121)(114 120)(115 119)(116 118)(117 126)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(135 144)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 8A 8B 9A 9B 9C 12A 12B 18A 18B 18C 18D 18E 18F 24A 24B 36A 36B 36C 36D 36E 36F 72A ··· 72F order 1 2 2 2 3 4 4 4 4 4 6 6 8 8 9 9 9 12 12 18 18 18 18 18 18 24 24 36 36 36 36 36 36 72 ··· 72 size 1 1 4 18 2 2 4 18 36 36 2 8 4 36 2 2 2 4 8 2 2 2 8 8 8 4 4 4 4 4 8 8 8 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D9 D18 D18 D18 C8.C22 S3×D4 D4.D6 D4×D9 SD16⋊D9 kernel SD16⋊D9 Dic36 C8⋊D9 D4.D9 C9⋊Q16 C9×SD16 D4⋊2D9 Q8×D9 C3×SD16 Dic9 D18 C24 C3×D4 C3×Q8 SD16 C8 D4 Q8 C9 C6 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 1 1 2 3 6

Matrix representation of SD16⋊D9 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 66 13 0 0 0 0 13 7 0 0 13 7 0 0 0 0 7 60 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 42 45 0 0 0 0 28 70 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 42 70 0 0 0 0 28 31 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,7,0,0,0,0,7,60,0,0,66,13,0,0,0,0,13,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[42,28,0,0,0,0,45,70,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,28,0,0,0,0,70,31,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

SD16⋊D9 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes D_9
% in TeX

G:=Group("SD16:D9");
// GroupNames label

G:=SmallGroup(288,125);
// by ID

G=gap.SmallGroup(288,125);
# by ID

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

G:=Group<a,b,c,d|a^8=b^2=c^9=d^2=1,b*a*b=a^3,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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