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## G = D18.D4order 288 = 25·32

### 2nd non-split extension by D18 of D4 acting via D4/C22=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 — C4⋊C4

Generators and relations for D18.D4
G = < a,b,c,d | a18=b2=c4=1, d2=a9, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a9b, dcd-1=c-1 >

Subgroups: 604 in 117 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, C9, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C22.D4, Dic9, C36, D18, D18, C2×C18, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C4×D9, D36, C2×Dic9, C2×C36, C22×D9, D6.D4, Dic9⋊C4, D18⋊C4, C9×C4⋊C4, C2×C4×D9, C2×D36, D18.D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, C22×S3, C22.D4, D18, C4○D12, S3×D4, Q83S3, C22×D9, D6.D4, D365C2, D4×D9, Q83D9, 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)
(1 131)(2 130)(3 129)(4 128)(5 127)(6 144)(7 143)(8 142)(9 141)(10 140)(11 139)(12 138)(13 137)(14 136)(15 135)(16 134)(17 133)(18 132)(19 67)(20 66)(21 65)(22 64)(23 63)(24 62)(25 61)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 72)(33 71)(34 70)(35 69)(36 68)(37 85)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 90)(51 89)(52 88)(53 87)(54 86)(91 121)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 126)(105 125)(106 124)(107 123)(108 122)
(1 53 105 60)(2 54 106 61)(3 37 107 62)(4 38 108 63)(5 39 91 64)(6 40 92 65)(7 41 93 66)(8 42 94 67)(9 43 95 68)(10 44 96 69)(11 45 97 70)(12 46 98 71)(13 47 99 72)(14 48 100 55)(15 49 101 56)(16 50 102 57)(17 51 103 58)(18 52 104 59)(19 133 80 109)(20 134 81 110)(21 135 82 111)(22 136 83 112)(23 137 84 113)(24 138 85 114)(25 139 86 115)(26 140 87 116)(27 141 88 117)(28 142 89 118)(29 143 90 119)(30 144 73 120)(31 127 74 121)(32 128 75 122)(33 129 76 123)(34 130 77 124)(35 131 78 125)(36 132 79 126)
(1 117 10 126)(2 118 11 109)(3 119 12 110)(4 120 13 111)(5 121 14 112)(6 122 15 113)(7 123 16 114)(8 124 17 115)(9 125 18 116)(19 61 28 70)(20 62 29 71)(21 63 30 72)(22 64 31 55)(23 65 32 56)(24 66 33 57)(25 67 34 58)(26 68 35 59)(27 69 36 60)(37 90 46 81)(38 73 47 82)(39 74 48 83)(40 75 49 84)(41 76 50 85)(42 77 51 86)(43 78 52 87)(44 79 53 88)(45 80 54 89)(91 127 100 136)(92 128 101 137)(93 129 102 138)(94 130 103 139)(95 131 104 140)(96 132 105 141)(97 133 106 142)(98 134 107 143)(99 135 108 144)```

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D6 C4○D4 D9 D18 C4○D12 D36⋊5C2 S3×D4 Q8⋊3S3 D4×D9 Q8⋊3D9 kernel D18.D4 Dic9⋊C4 D18⋊C4 C9×C4⋊C4 C2×C4×D9 C2×D36 C3×C4⋊C4 D18 C2×C12 C18 C4⋊C4 C2×C4 C6 C2 C6 C6 C2 C2 # reps 1 1 3 1 1 1 1 2 3 4 3 9 4 12 1 1 3 3

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

 31 26 0 0 11 20 0 0 0 0 36 0 0 0 0 36
,
 4 25 0 0 29 33 0 0 0 0 26 36 0 0 9 11
,
 7 23 0 0 14 30 0 0 0 0 6 0 0 0 16 31
,
 32 10 0 0 27 5 0 0 0 0 26 36 0 0 11 11
`G:=sub<GL(4,GF(37))| [31,11,0,0,26,20,0,0,0,0,36,0,0,0,0,36],[4,29,0,0,25,33,0,0,0,0,26,9,0,0,36,11],[7,14,0,0,23,30,0,0,0,0,6,16,0,0,0,31],[32,27,0,0,10,5,0,0,0,0,26,11,0,0,36,11] >;`

D18.D4 in GAP, Magma, Sage, TeX

`D_{18}.D_4`
`% in TeX`

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

`G:=SmallGroup(288,104);`
`// by ID`

`G=gap.SmallGroup(288,104);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,100,6725,292,9414]);`
`// Polycyclic`

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

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