metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16⋊2D9, C8.3D18, D18.9D4, C24.34D6, Q8.9D18, C36.9C23, C72.10C22, Dic9.11D4, D36.4C22, Dic18.5C22, (Q8×D9)⋊3C2, C72⋊C2⋊4C2, C8⋊D9⋊4C2, (C9×Q16)⋊4C2, C9⋊Q16⋊4C2, C2.23(D4×D9), C6.97(S3×D4), C9⋊C8.2C22, Q8⋊3D9.C2, Q8⋊2D9⋊3C2, C18.35(C2×D4), C9⋊3(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
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, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, Q8, C9, Dic3, C12, C12, D6, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, D9, C18, C3⋊C8, C24, Dic6, C4×S3, D12, C3×Q8, C8.C22, Dic9, Dic9, C36, C36, D18, D18, C8⋊S3, C24⋊C2, Q8⋊2S3, C3⋊Q16, C3×Q16, S3×Q8, Q8⋊3S3, C9⋊C8, C72, Dic18, Dic18, C4×D9, C4×D9, D36, D36, Q8×C9, Q16⋊S3, C8⋊D9, C72⋊C2, C9⋊Q16, Q8⋊2D9, C9×Q16, Q8×D9, Q8⋊3D9, Q16⋊D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C22×S3, C8.C22, D18, S3×D4, C22×D9, Q16⋊S3, D4×D9, 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 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6 | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 24A | 24B | 36A | 36B | 36C | 36D | ··· | 36I | 72A | ··· | 72F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 18 | 18 | 18 | 24 | 24 | 36 | 36 | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 18 | 36 | 2 | 2 | 4 | 4 | 18 | 36 | 2 | 4 | 36 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | D9 | D18 | D18 | C8.C22 | S3×D4 | Q16⋊S3 | D4×D9 | Q16⋊D9 |
| kernel | Q16⋊D9 | C8⋊D9 | C72⋊C2 | C9⋊Q16 | Q8⋊2D9 | C9×Q16 | Q8×D9 | Q8⋊3D9 | C3×Q16 | Dic9 | D18 | C24 | C3×Q8 | Q16 | C8 | Q8 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 1 | 1 | 2 | 3 | 6 |
Matrix representation of Q16⋊D9 ►in GL4(𝔽73) generated by
| 0 | 0 | 11 | 22 |
| 0 | 0 | 51 | 62 |
| 31 | 62 | 11 | 22 |
| 11 | 42 | 51 | 62 |
| 7 | 14 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 7 | 14 | 66 | 59 |
| 59 | 66 | 14 | 7 |
| 3 | 31 | 0 | 0 |
| 42 | 45 | 0 | 0 |
| 0 | 0 | 3 | 31 |
| 0 | 0 | 42 | 45 |
| 25 | 44 | 0 | 0 |
| 19 | 48 | 0 | 0 |
| 0 | 0 | 25 | 44 |
| 0 | 0 | 19 | 48 |
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