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## G = Q16×D9order 288 = 25·32

### Direct product of Q16 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — Q16×D9
 Chief series C1 — C3 — C9 — C18 — C36 — C4×D9 — Q8×D9 — Q16×D9
 Lower central C9 — C18 — C36 — Q16×D9
 Upper central C1 — C2 — C4 — Q16

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

Smallest permutation representation of 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

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