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G = D83D9order 288 = 25·32

The semidirect product of D8 and D9 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D83D9, C8.8D18, D18.1D4, D4.1D18, C24.11D6, Dic364C2, C72.6C22, C36.3C23, Dic9.12D4, Dic18.1C22, (C8×D9)⋊2C2, (C9×D8)⋊3C2, C92(C4○D8), D4.D92C2, (C3×D8).4S3, (C3×D4).3D6, C6.91(S3×D4), C2.17(D4×D9), C9⋊C8.5C22, D42D92C2, C3.(D83S3), C18.29(C2×D4), C4.3(C22×D9), (C4×D9).8C22, (D4×C9).1C22, C12.42(C22×S3), SmallGroup(288,122)

Series: Derived Chief Lower central Upper central

C1C36 — D83D9
C1C3C9C18C36C4×D9D42D9 — D83D9
C9C18C36 — D83D9
C1C2C4D8

Generators and relations for D83D9
 G = < a,b,c,d | a8=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 408 in 93 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], S3, C6, C6 [×2], C8, C8, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], C9, Dic3 [×3], C12, D6, C2×C6 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], D9, C18, C18 [×2], C3⋊C8, C24, Dic6 [×2], C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C3×D4 [×2], C4○D8, Dic9, Dic9 [×2], C36, D18, C2×C18 [×2], S3×C8, Dic12, D4.S3 [×2], C3×D8, D42S3 [×2], C9⋊C8, C72, Dic18 [×2], C4×D9, C2×Dic9 [×2], C9⋊D4 [×2], D4×C9 [×2], D83S3, Dic36, C8×D9, D4.D9 [×2], C9×D8, D42D9 [×2], D83D9
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, D83S3, D4×D9, D83D9

Smallest permutation representation of D83D9
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)(37 100 64 82 46 91 55 73)(38 101 65 83 47 92 56 74)(39 102 66 84 48 93 57 75)(40 103 67 85 49 94 58 76)(41 104 68 86 50 95 59 77)(42 105 69 87 51 96 60 78)(43 106 70 88 52 97 61 79)(44 107 71 89 53 98 62 80)(45 108 72 90 54 99 63 81)
(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 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(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)(109 116)(110 115)(111 114)(112 113)(118 125)(119 124)(120 123)(121 122)(127 134)(128 133)(129 132)(130 131)(136 143)(137 142)(138 141)(139 140)

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)(37,100,64,82,46,91,55,73)(38,101,65,83,47,92,56,74)(39,102,66,84,48,93,57,75)(40,103,67,85,49,94,58,76)(41,104,68,86,50,95,59,77)(42,105,69,87,51,96,60,78)(43,106,70,88,52,97,61,79)(44,107,71,89,53,98,62,80)(45,108,72,90,54,99,63,81), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(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)(109,116)(110,115)(111,114)(112,113)(118,125)(119,124)(120,123)(121,122)(127,134)(128,133)(129,132)(130,131)(136,143)(137,142)(138,141)(139,140)>;

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)(37,100,64,82,46,91,55,73)(38,101,65,83,47,92,56,74)(39,102,66,84,48,93,57,75)(40,103,67,85,49,94,58,76)(41,104,68,86,50,95,59,77)(42,105,69,87,51,96,60,78)(43,106,70,88,52,97,61,79)(44,107,71,89,53,98,62,80)(45,108,72,90,54,99,63,81), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(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)(109,116)(110,115)(111,114)(112,113)(118,125)(119,124)(120,123)(121,122)(127,134)(128,133)(129,132)(130,131)(136,143)(137,142)(138,141)(139,140) );

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),(37,100,64,82,46,91,55,73),(38,101,65,83,47,92,56,74),(39,102,66,84,48,93,57,75),(40,103,67,85,49,94,58,76),(41,104,68,86,50,95,59,77),(42,105,69,87,51,96,60,78),(43,106,70,88,52,97,61,79),(44,107,71,89,53,98,62,80),(45,108,72,90,54,99,63,81)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,26),(20,25),(21,24),(22,23),(28,35),(29,34),(30,33),(31,32),(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),(109,116),(110,115),(111,114),(112,113),(118,125),(119,124),(120,123),(121,122),(127,134),(128,133),(129,132),(130,131),(136,143),(137,142),(138,141),(139,140)])

42 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B6C8A8B8C8D9A9B9C 12 18A18B18C18D···18I24A24B36A36B36C72A···72F
order1222234444466688889991218181818···18242436363672···72
size1144182299363628822181822242228···8444444···4

42 irreducible representations

dim1111112222222224444
type+++++++++++++++-+-
imageC1C2C2C2C2C2S3D4D4D6D6D9C4○D8D18D18S3×D4D83S3D4×D9D83D9
kernelD83D9Dic36C8×D9D4.D9C9×D8D42D9C3×D8Dic9D18C24C3×D4D8C9C8D4C6C3C2C1
# reps1112121111234361236

Matrix representation of D83D9 in GL4(𝔽73) generated by

22000
231000
0010
0001
,
523000
342100
0010
0001
,
1000
0100
007045
002842
,
1000
167200
00720
00721
G:=sub<GL(4,GF(73))| [22,23,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[52,34,0,0,30,21,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,70,28,0,0,45,42],[1,16,0,0,0,72,0,0,0,0,72,72,0,0,0,1] >;

D83D9 in GAP, Magma, Sage, TeX

D_8\rtimes_3D_9
% in TeX

G:=Group("D8:3D9");
// GroupNames label

G:=SmallGroup(288,122);
// by ID

G=gap.SmallGroup(288,122);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,135,346,185,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^9=d^2=1,b*a*b=a^-1,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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