metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD16⋊3D9, D4.5D18, D18.2D4, C8.11D18, C24.37D6, Q8.7D18, C36.7C23, C72.11C22, Dic9.13D4, D36.3C22, Dic18.3C22, D4⋊D9⋊4C2, (C8×D9)⋊5C2, C9⋊3(C4○D8), C72⋊C2⋊6C2, C9⋊Q16⋊2C2, (C3×D4).7D6, C6.95(S3×D4), C2.21(D4×D9), C9⋊C8.6C22, D4⋊2D9⋊3C2, Q8⋊3D9⋊2C2, (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
Generators and relations for SD16⋊3D9
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, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C9, Dic3, C12, C12, D6, C2×C6, C2×C8, D8, SD16, SD16, Q16, C4○D4, D9, C18, C18, C3⋊C8, C24, Dic6, C4×S3, D12, 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, D4⋊2S3, Q8⋊3S3, 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, D4⋊2D9, Q8⋊3D9, SD16⋊3D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C22×S3, C4○D8, D18, S3×D4, C22×D9, Q8.7D6, D4×D9, SD16⋊3D9
►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 SD16⋊3D9 ►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] >;
SD16⋊3D9 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