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G = SD163D9order 288 = 25·32

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: SD163D9, D4.5D18, D18.2D4, C8.11D18, C24.37D6, Q8.7D18, C36.7C23, C72.11C22, Dic9.13D4, D36.3C22, Dic18.3C22, D4⋊D94C2, (C8×D9)⋊5C2, C93(C4○D8), C72⋊C26C2, C9⋊Q162C2, (C3×D4).7D6, C6.95(S3×D4), C2.21(D4×D9), C9⋊C8.6C22, D42D93C2, Q83D92C2, (C9×SD16)⋊4C2, C18.33(C2×D4), (C3×Q8).27D6, C4.7(C22×D9), C3.(Q8.7D6), (C3×SD16).4S3, (D4×C9).5C22, (Q8×C9).2C22, C12.46(C22×S3), (C4×D9).10C22, SmallGroup(288,126)

Series: Derived Chief Lower central Upper central

C1C36 — SD163D9
C1C3C9C18C36C4×D9D42D9 — SD163D9
C9C18C36 — SD163D9
C1C2C4SD16

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 2A2B2C2D 3 4A4B4C4D4E6A6B8A8B8C8D9A9B9C12A12B18A18B18C18D18E18F24A24B36A36B36C36D36E36F72A···72F
order122223444446688889991212181818181818242436363636363672···72
size114183622499362822181822248222888444448884···4

42 irreducible representations

dim11111111222222222224444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6D9C4○D8D18D18D18S3×D4Q8.7D6D4×D9SD163D9
kernelSD163D9C8×D9C72⋊C2D4⋊D9C9⋊Q16C9×SD16D42D9Q83D9C3×SD16Dic9D18C24C3×D4C3×Q8SD16C9C8D4Q8C6C3C2C1
# reps11111111111111343331236

Matrix representation of SD163D9 in GL4(𝔽73) generated by

72000
07200
0066
00676
,
1000
0100
001616
001657
,
33100
424500
0010
0001
,
31300
454200
00046
00270
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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