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## G = D18⋊D4order 288 = 25·32

### 1st semidirect product of D18 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — D18⋊D4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C2×C4×D9 — D18⋊D4
 Lower central C9 — C2×C18 — D18⋊D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for D18⋊D4
G = < a,b,c,d | a18=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a16b, dbd=a7b, dcd=c-1 >

Subgroups: 732 in 141 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], S3 [×3], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C9, Dic3 [×3], C12 [×2], D6 [×7], C2×C6, C2×C6 [×3], C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4 [×3], D9 [×3], C18 [×3], C18, C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C22×S3 [×2], C22×C6, C4⋊D4, Dic9 [×2], Dic9, C36 [×2], D18 [×2], D18 [×5], C2×C18, C2×C18 [×3], Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4 [×2], C4×D9 [×2], D36 [×2], C2×Dic9 [×2], C9⋊D4 [×4], C2×C36 [×2], C22×D9 [×2], C22×C18, Dic3⋊D4, Dic9⋊C4, D18⋊C4, C9×C22⋊C4, C2×C4×D9, C2×D36, C2×C9⋊D4 [×2], D18⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D9, C22×S3, C4⋊D4, D18 [×3], C4○D12, S3×D4 [×2], C22×D9, Dic3⋊D4, D365C2, D4×D9 [×2], D18⋊D4

Smallest permutation representation of D18⋊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)
(1 134)(2 133)(3 132)(4 131)(5 130)(6 129)(7 128)(8 127)(9 144)(10 143)(11 142)(12 141)(13 140)(14 139)(15 138)(16 137)(17 136)(18 135)(19 102)(20 101)(21 100)(22 99)(23 98)(24 97)(25 96)(26 95)(27 94)(28 93)(29 92)(30 91)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 113)(38 112)(39 111)(40 110)(41 109)(42 126)(43 125)(44 124)(45 123)(46 122)(47 121)(48 120)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 90)(72 89)
(1 43 74 36)(2 42 75 35)(3 41 76 34)(4 40 77 33)(5 39 78 32)(6 38 79 31)(7 37 80 30)(8 54 81 29)(9 53 82 28)(10 52 83 27)(11 51 84 26)(12 50 85 25)(13 49 86 24)(14 48 87 23)(15 47 88 22)(16 46 89 21)(17 45 90 20)(18 44 73 19)(55 101 138 123)(56 100 139 122)(57 99 140 121)(58 98 141 120)(59 97 142 119)(60 96 143 118)(61 95 144 117)(62 94 127 116)(63 93 128 115)(64 92 129 114)(65 91 130 113)(66 108 131 112)(67 107 132 111)(68 106 133 110)(69 105 134 109)(70 104 135 126)(71 103 136 125)(72 102 137 124)
(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(55 58)(56 57)(59 72)(60 71)(61 70)(62 69)(63 68)(64 67)(65 66)(73 75)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 112)(92 111)(93 110)(94 109)(95 126)(96 125)(97 124)(98 123)(99 122)(100 121)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(127 134)(128 133)(129 132)(130 131)(135 144)(136 143)(137 142)(138 141)(139 140)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 18J ··· 18O 36A ··· 36L order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 6 6 6 6 6 9 9 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 4 18 18 36 2 2 2 4 18 18 36 2 2 2 4 4 2 2 2 4 4 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 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 D9 D18 D18 C4○D12 D36⋊5C2 S3×D4 D4×D9 kernel D18⋊D4 Dic9⋊C4 D18⋊C4 C9×C22⋊C4 C2×C4×D9 C2×D36 C2×C9⋊D4 C3×C22⋊C4 Dic9 D18 C2×C12 C22×C6 C18 C22⋊C4 C2×C4 C23 C6 C2 C6 C2 # reps 1 1 1 1 1 1 2 1 2 2 2 1 2 3 6 3 4 12 2 6

Matrix representation of D18⋊D4 in GL4(𝔽37) generated by

 36 0 0 0 0 36 0 0 0 0 6 17 0 0 20 26
,
 36 3 0 0 0 1 0 0 0 0 4 12 0 0 8 33
,
 36 0 0 0 0 36 0 0 0 0 2 13 0 0 11 35
,
 1 0 0 0 13 36 0 0 0 0 20 26 0 0 6 17
`G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,6,20,0,0,17,26],[36,0,0,0,3,1,0,0,0,0,4,8,0,0,12,33],[36,0,0,0,0,36,0,0,0,0,2,11,0,0,13,35],[1,13,0,0,0,36,0,0,0,0,20,6,0,0,26,17] >;`

D18⋊D4 in GAP, Magma, Sage, TeX

`D_{18}\rtimes D_4`
`% in TeX`

`G:=Group("D18:D4");`
`// GroupNames label`

`G:=SmallGroup(288,94);`
`// by ID`

`G=gap.SmallGroup(288,94);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^18=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^16*b,d*b*d=a^7*b,d*c*d=c^-1>;`
`// generators/relations`

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