direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16×D9, C8.9D18, C24.12D6, Q8.8D18, Dic36⋊5C2, D18.14D4, C36.8C23, C72.7C22, Dic9.5D4, Dic18.4C22, C9⋊2(C2×Q16), (Q8×D9).C2, C3.(S3×Q16), (C9×Q16)⋊2C2, C9⋊Q16⋊3C2, (C8×D9).1C2, C6.96(S3×D4), C2.22(D4×D9), C9⋊C8.7C22, C18.34(C2×D4), (C3×Q16).3S3, (C3×Q8).28D6, C4.8(C22×D9), (Q8×C9).3C22, C12.47(C22×S3), (C4×D9).11C22, SmallGroup(288,127)
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, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 384 in 90 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, Q8, Q8, C9, Dic3, C12, C12, D6, C2×C8, Q16, Q16, C2×Q8, D9, C18, C3⋊C8, C24, Dic6, C4×S3, C3×Q8, C2×Q16, Dic9, Dic9, C36, C36, D18, S3×C8, Dic12, C3⋊Q16, C3×Q16, S3×Q8, C9⋊C8, C72, Dic18, Dic18, C4×D9, C4×D9, Q8×C9, S3×Q16, Dic36, C8×D9, C9⋊Q16, C9×Q16, Q8×D9, Q16×D9
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, D9, C22×S3, C2×Q16, D18, S3×D4, C22×D9, S3×Q16, 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 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)(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 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(81 90)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(99 108)(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)| (1,131,23,122,14,140,32,113)(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), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(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)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(81,90)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(99,108)(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( (1,131,23,122,14,140,32,113)(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), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(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)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(81,90)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(99,108)(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([[(1,131,23,122,14,140,32,113),(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)], [(1,13),(2,12),(3,11),(4,10),(5,18),(6,17),(7,16),(8,15),(9,14),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(27,36),(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),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(81,90),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100),(99,108),(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)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6 | 8A | 8B | 8C | 8D | 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 | 4 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 18 | 18 | 18 | 24 | 24 | 36 | 36 | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 4 | 18 | 36 | 36 | 2 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | Q16 | D9 | D18 | D18 | S3×D4 | S3×Q16 | D4×D9 | Q16×D9 |
| kernel | Q16×D9 | Dic36 | C8×D9 | C9⋊Q16 | C9×Q16 | Q8×D9 | C3×Q16 | Dic9 | D18 | C24 | C3×Q8 | D9 | Q16 | C8 | Q8 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 3 | 3 | 6 | 1 | 2 | 3 | 6 |
Matrix representation of Q16×D9 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 57 | 57 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 16 | 20 |
| 0 | 0 | 20 | 57 |
| 70 | 45 | 0 | 0 |
| 28 | 42 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 42 | 28 | 0 | 0 |
| 70 | 31 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,57,57,0,0,16,57],[72,0,0,0,0,72,0,0,0,0,16,20,0,0,20,57],[70,28,0,0,45,42,0,0,0,0,1,0,0,0,0,1],[42,70,0,0,28,31,0,0,0,0,72,0,0,0,0,72] >;
Q16×D9 in GAP, Magma, Sage, TeX
Q_{16}\times D_9 % in TeX
G:=Group("Q16xD9"); // GroupNames label
G:=SmallGroup(288,127);
// by ID
G=gap.SmallGroup(288,127);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,135,100,346,185,80,6725,292,9414]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations