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## G = SD16⋊3D9order 288 = 25·32

### The semidirect product of SD16 and D9 acting through Inn(SD16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — SD16⋊3D9
 Chief series C1 — C3 — C9 — C18 — C36 — C4×D9 — D4⋊2D9 — SD16⋊3D9
 Lower central C9 — C18 — C36 — SD16⋊3D9
 Upper central C1 — C2 — C4 — SD16

Generators and relations for SD163D9
G = < a,b,c,d | a8=b2=c9=d2=1, bab=a3, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 448 in 93 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], S3 [×2], C6, C6, C8, C8, C2×C4 [×3], D4, D4 [×3], Q8, Q8, C9, Dic3 [×2], C12, C12, D6 [×2], C2×C6, C2×C8, D8, SD16, SD16, Q16, C4○D4 [×2], D9 [×2], C18, C18, C3⋊C8, C24, Dic6, C4×S3 [×2], D12 [×2], C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C4○D8, Dic9, Dic9, C36, C36, D18, D18, C2×C18, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D42S3, Q83S3, C9⋊C8, C72, Dic18, C4×D9, C4×D9, D36, D36, C2×Dic9, C9⋊D4, D4×C9, Q8×C9, Q8.7D6, C8×D9, C72⋊C2, D4⋊D9, C9⋊Q16, C9×SD16, D42D9, Q83D9, SD163D9
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C22×S3, C4○D8, D18 [×3], S3×D4, C22×D9, Q8.7D6, D4×D9, SD163D9

Smallest permutation representation of SD163D9
On 144 points
Generators in S144
(1 122 23 140 14 113 32 131)(2 123 24 141 15 114 33 132)(3 124 25 142 16 115 34 133)(4 125 26 143 17 116 35 134)(5 126 27 144 18 117 36 135)(6 118 19 136 10 109 28 127)(7 119 20 137 11 110 29 128)(8 120 21 138 12 111 30 129)(9 121 22 139 13 112 31 130)(37 73 64 100 46 82 55 91)(38 74 65 101 47 83 56 92)(39 75 66 102 48 84 57 93)(40 76 67 103 49 85 58 94)(41 77 68 104 50 86 59 95)(42 78 69 105 51 87 60 96)(43 79 70 106 52 88 61 97)(44 80 71 107 53 89 62 98)(45 81 72 108 54 90 63 99)
(1 77)(2 78)(3 79)(4 80)(5 81)(6 73)(7 74)(8 75)(9 76)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(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 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)(73 125)(74 124)(75 123)(76 122)(77 121)(78 120)(79 119)(80 118)(81 126)(82 116)(83 115)(84 114)(85 113)(86 112)(87 111)(88 110)(89 109)(90 117)(91 134)(92 133)(93 132)(94 131)(95 130)(96 129)(97 128)(98 127)(99 135)(100 143)(101 142)(102 141)(103 140)(104 139)(105 138)(106 137)(107 136)(108 144)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 8A 8B 8C 8D 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 2 3 4 4 4 4 4 6 6 8 8 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 36 2 2 4 9 9 36 2 8 2 2 18 18 2 2 2 4 8 2 2 2 8 8 8 4 4 4 4 4 8 8 8 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D9 C4○D8 D18 D18 D18 S3×D4 Q8.7D6 D4×D9 SD16⋊3D9 kernel SD16⋊3D9 C8×D9 C72⋊C2 D4⋊D9 C9⋊Q16 C9×SD16 D4⋊2D9 Q8⋊3D9 C3×SD16 Dic9 D18 C24 C3×D4 C3×Q8 SD16 C9 C8 D4 Q8 C6 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 4 3 3 3 1 2 3 6

Matrix representation of SD163D9 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 6 6 0 0 67 6
,
 1 0 0 0 0 1 0 0 0 0 16 16 0 0 16 57
,
 3 31 0 0 42 45 0 0 0 0 1 0 0 0 0 1
,
 31 3 0 0 45 42 0 0 0 0 0 46 0 0 27 0
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,6,67,0,0,6,6],[1,0,0,0,0,1,0,0,0,0,16,16,0,0,16,57],[3,42,0,0,31,45,0,0,0,0,1,0,0,0,0,1],[31,45,0,0,3,42,0,0,0,0,0,27,0,0,46,0] >;

SD163D9 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_3D_9
% in TeX

G:=Group("SD16:3D9");
// GroupNames label

G:=SmallGroup(288,126);
// by ID

G=gap.SmallGroup(288,126);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,135,100,346,185,80,6725,292,9414]);
// 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,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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