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## G = Dic9⋊D4order 288 = 25·32

### 2nd semidirect product of Dic9 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — Dic9⋊D4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C2×C9⋊D4 — Dic9⋊D4
 Lower central C9 — C2×C18 — Dic9⋊D4
 Upper central C1 — C22 — C2×D4

Generators and relations for Dic9⋊D4
G = < a,b,c,d | a18=c4=d2=1, b2=a9, bab-1=a-1, ac=ca, ad=da, cbc-1=a9b, bd=db, dcd=c-1 >

Subgroups: 612 in 141 conjugacy classes, 46 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, D9, C18, C18, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4⋊D4, Dic9, Dic9, C36, D18, C2×C18, C2×C18, C2×C18, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C6×D4, C2×Dic9, C2×Dic9, C9⋊D4, C2×C36, D4×C9, C22×D9, C22×C18, C23.14D6, Dic9⋊C4, D18⋊C4, C18.D4, C22×Dic9, C2×C9⋊D4, D4×C18, Dic9⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, C3⋊D4, C22×S3, C4⋊D4, D18, S3×D4, D42S3, C2×C3⋊D4, C9⋊D4, C22×D9, C23.14D6, D4×D9, D42D9, C2×C9⋊D4, Dic9⋊D4

Smallest permutation representation of Dic9⋊D4
On 144 points
Generators in S144
(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 70 10 61)(2 69 11 60)(3 68 12 59)(4 67 13 58)(5 66 14 57)(6 65 15 56)(7 64 16 55)(8 63 17 72)(9 62 18 71)(19 50 28 41)(20 49 29 40)(21 48 30 39)(22 47 31 38)(23 46 32 37)(24 45 33 54)(25 44 34 53)(26 43 35 52)(27 42 36 51)(73 134 82 143)(74 133 83 142)(75 132 84 141)(76 131 85 140)(77 130 86 139)(78 129 87 138)(79 128 88 137)(80 127 89 136)(81 144 90 135)(91 111 100 120)(92 110 101 119)(93 109 102 118)(94 126 103 117)(95 125 104 116)(96 124 105 115)(97 123 106 114)(98 122 107 113)(99 121 108 112)
(1 126 28 84)(2 109 29 85)(3 110 30 86)(4 111 31 87)(5 112 32 88)(6 113 33 89)(7 114 34 90)(8 115 35 73)(9 116 36 74)(10 117 19 75)(11 118 20 76)(12 119 21 77)(13 120 22 78)(14 121 23 79)(15 122 24 80)(16 123 25 81)(17 124 26 82)(18 125 27 83)(37 128 66 108)(38 129 67 91)(39 130 68 92)(40 131 69 93)(41 132 70 94)(42 133 71 95)(43 134 72 96)(44 135 55 97)(45 136 56 98)(46 137 57 99)(47 138 58 100)(48 139 59 101)(49 140 60 102)(50 141 61 103)(51 142 62 104)(52 143 63 105)(53 144 64 106)(54 127 65 107)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 91)(17 92)(18 93)(19 141)(20 142)(21 143)(22 144)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 109)(72 110)

G:=sub<Sym(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), (1,70,10,61)(2,69,11,60)(3,68,12,59)(4,67,13,58)(5,66,14,57)(6,65,15,56)(7,64,16,55)(8,63,17,72)(9,62,18,71)(19,50,28,41)(20,49,29,40)(21,48,30,39)(22,47,31,38)(23,46,32,37)(24,45,33,54)(25,44,34,53)(26,43,35,52)(27,42,36,51)(73,134,82,143)(74,133,83,142)(75,132,84,141)(76,131,85,140)(77,130,86,139)(78,129,87,138)(79,128,88,137)(80,127,89,136)(81,144,90,135)(91,111,100,120)(92,110,101,119)(93,109,102,118)(94,126,103,117)(95,125,104,116)(96,124,105,115)(97,123,106,114)(98,122,107,113)(99,121,108,112), (1,126,28,84)(2,109,29,85)(3,110,30,86)(4,111,31,87)(5,112,32,88)(6,113,33,89)(7,114,34,90)(8,115,35,73)(9,116,36,74)(10,117,19,75)(11,118,20,76)(12,119,21,77)(13,120,22,78)(14,121,23,79)(15,122,24,80)(16,123,25,81)(17,124,26,82)(18,125,27,83)(37,128,66,108)(38,129,67,91)(39,130,68,92)(40,131,69,93)(41,132,70,94)(42,133,71,95)(43,134,72,96)(44,135,55,97)(45,136,56,98)(46,137,57,99)(47,138,58,100)(48,139,59,101)(49,140,60,102)(50,141,61,103)(51,142,62,104)(52,143,63,105)(53,144,64,106)(54,127,65,107), (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,91)(17,92)(18,93)(19,141)(20,142)(21,143)(22,144)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,109)(72,110)>;

G:=Group( (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), (1,70,10,61)(2,69,11,60)(3,68,12,59)(4,67,13,58)(5,66,14,57)(6,65,15,56)(7,64,16,55)(8,63,17,72)(9,62,18,71)(19,50,28,41)(20,49,29,40)(21,48,30,39)(22,47,31,38)(23,46,32,37)(24,45,33,54)(25,44,34,53)(26,43,35,52)(27,42,36,51)(73,134,82,143)(74,133,83,142)(75,132,84,141)(76,131,85,140)(77,130,86,139)(78,129,87,138)(79,128,88,137)(80,127,89,136)(81,144,90,135)(91,111,100,120)(92,110,101,119)(93,109,102,118)(94,126,103,117)(95,125,104,116)(96,124,105,115)(97,123,106,114)(98,122,107,113)(99,121,108,112), (1,126,28,84)(2,109,29,85)(3,110,30,86)(4,111,31,87)(5,112,32,88)(6,113,33,89)(7,114,34,90)(8,115,35,73)(9,116,36,74)(10,117,19,75)(11,118,20,76)(12,119,21,77)(13,120,22,78)(14,121,23,79)(15,122,24,80)(16,123,25,81)(17,124,26,82)(18,125,27,83)(37,128,66,108)(38,129,67,91)(39,130,68,92)(40,131,69,93)(41,132,70,94)(42,133,71,95)(43,134,72,96)(44,135,55,97)(45,136,56,98)(46,137,57,99)(47,138,58,100)(48,139,59,101)(49,140,60,102)(50,141,61,103)(51,142,62,104)(52,143,63,105)(53,144,64,106)(54,127,65,107), (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,91)(17,92)(18,93)(19,141)(20,142)(21,143)(22,144)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,109)(72,110) );

G=PermutationGroup([[(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)], [(1,70,10,61),(2,69,11,60),(3,68,12,59),(4,67,13,58),(5,66,14,57),(6,65,15,56),(7,64,16,55),(8,63,17,72),(9,62,18,71),(19,50,28,41),(20,49,29,40),(21,48,30,39),(22,47,31,38),(23,46,32,37),(24,45,33,54),(25,44,34,53),(26,43,35,52),(27,42,36,51),(73,134,82,143),(74,133,83,142),(75,132,84,141),(76,131,85,140),(77,130,86,139),(78,129,87,138),(79,128,88,137),(80,127,89,136),(81,144,90,135),(91,111,100,120),(92,110,101,119),(93,109,102,118),(94,126,103,117),(95,125,104,116),(96,124,105,115),(97,123,106,114),(98,122,107,113),(99,121,108,112)], [(1,126,28,84),(2,109,29,85),(3,110,30,86),(4,111,31,87),(5,112,32,88),(6,113,33,89),(7,114,34,90),(8,115,35,73),(9,116,36,74),(10,117,19,75),(11,118,20,76),(12,119,21,77),(13,120,22,78),(14,121,23,79),(15,122,24,80),(16,123,25,81),(17,124,26,82),(18,125,27,83),(37,128,66,108),(38,129,67,91),(39,130,68,92),(40,131,69,93),(41,132,70,94),(42,133,71,95),(43,134,72,96),(44,135,55,97),(45,136,56,98),(46,137,57,99),(47,138,58,100),(48,139,59,101),(49,140,60,102),(50,141,61,103),(51,142,62,104),(52,143,63,105),(53,144,64,106),(54,127,65,107)], [(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,100),(8,101),(9,102),(10,103),(11,104),(12,105),(13,106),(14,107),(15,108),(16,91),(17,92),(18,93),(19,141),(20,142),(21,143),(22,144),(23,127),(24,128),(25,129),(26,130),(27,131),(28,132),(29,133),(30,134),(31,135),(32,136),(33,137),(34,138),(35,139),(36,140),(37,80),(38,81),(39,82),(40,83),(41,84),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,73),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,111),(56,112),(57,113),(58,114),(59,115),(60,116),(61,117),(62,118),(63,119),(64,120),(65,121),(66,122),(67,123),(68,124),(69,125),(70,126),(71,109),(72,110)]])

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 12A 12B 18A ··· 18I 18J ··· 18U 36A ··· 36F order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 6 6 6 6 6 6 6 9 9 9 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 4 36 2 4 18 18 18 18 36 2 2 2 4 4 4 4 2 2 2 4 4 2 ··· 2 4 ··· 4 4 ··· 4

54 irreducible representations

 dim 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 S3 D4 D4 D6 D6 C4○D4 D9 C3⋊D4 D18 D18 C9⋊D4 S3×D4 D4⋊2S3 D4×D9 D4⋊2D9 kernel Dic9⋊D4 Dic9⋊C4 D18⋊C4 C18.D4 C22×Dic9 C2×C9⋊D4 D4×C18 C6×D4 Dic9 C2×C18 C2×C12 C22×C6 C18 C2×D4 C2×C6 C2×C4 C23 C22 C6 C6 C2 C2 # reps 1 1 1 1 1 2 1 1 2 2 1 2 2 3 4 3 6 12 1 1 3 3

Matrix representation of Dic9⋊D4 in GL4(𝔽37) generated by

 17 31 0 0 6 11 0 0 0 0 1 0 0 0 0 1
,
 2 11 0 0 13 35 0 0 0 0 36 0 0 0 0 36
,
 7 23 0 0 14 30 0 0 0 0 36 35 0 0 1 1
,
 36 0 0 0 0 36 0 0 0 0 36 35 0 0 0 1
G:=sub<GL(4,GF(37))| [17,6,0,0,31,11,0,0,0,0,1,0,0,0,0,1],[2,13,0,0,11,35,0,0,0,0,36,0,0,0,0,36],[7,14,0,0,23,30,0,0,0,0,36,1,0,0,35,1],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,35,1] >;

Dic9⋊D4 in GAP, Magma, Sage, TeX

{\rm Dic}_9\rtimes D_4
% in TeX

G:=Group("Dic9:D4");
// GroupNames label

G:=SmallGroup(288,149);
// by ID

G=gap.SmallGroup(288,149);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,219,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^18=c^4=d^2=1,b^2=a^9,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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