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## G = D18⋊3Q8order 288 = 25·32

### 3rd semidirect product of D18 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — D18⋊3Q8
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C2×C4×D9 — D18⋊3Q8
 Lower central C9 — C2×C18 — D18⋊3Q8
 Upper central C1 — C22 — C2×Q8

Generators and relations for D183Q8
G = < a,b,c,d | a18=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a9b, bd=db, dcd-1=c-1 >

Subgroups: 460 in 111 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, C9, Dic3 [×3], C12 [×2], C12 [×2], D6 [×4], C2×C6, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, D9 [×2], C18 [×3], C4×S3 [×2], C2×Dic3 [×3], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, C22⋊Q8, Dic9 [×3], C36 [×2], C36 [×2], D18 [×2], D18 [×2], C2×C18, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], S3×C2×C4, C6×Q8, C4×D9 [×2], C2×Dic9, C2×Dic9 [×2], C2×C36, C2×C36 [×2], Q8×C9 [×2], C22×D9, D63Q8, Dic9⋊C4 [×2], C4⋊Dic9, D18⋊C4 [×2], C2×C4×D9, Q8×C18, D183Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, D9, C3⋊D4 [×2], C22×S3, C22⋊Q8, D18 [×3], S3×Q8, Q83S3, C2×C3⋊D4, C9⋊D4 [×2], C22×D9, D63Q8, Q8×D9, Q83D9, C2×C9⋊D4, D183Q8

Smallest permutation representation of D183Q8
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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(35 36)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(73 75)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 92)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)(109 117)(110 116)(111 115)(112 114)(118 126)(119 125)(120 124)(121 123)(127 131)(128 130)(132 144)(133 143)(134 142)(135 141)(136 140)(137 139)
(1 134 101 88)(2 135 102 89)(3 136 103 90)(4 137 104 73)(5 138 105 74)(6 139 106 75)(7 140 107 76)(8 141 108 77)(9 142 91 78)(10 143 92 79)(11 144 93 80)(12 127 94 81)(13 128 95 82)(14 129 96 83)(15 130 97 84)(16 131 98 85)(17 132 99 86)(18 133 100 87)(19 61 47 110)(20 62 48 111)(21 63 49 112)(22 64 50 113)(23 65 51 114)(24 66 52 115)(25 67 53 116)(26 68 54 117)(27 69 37 118)(28 70 38 119)(29 71 39 120)(30 72 40 121)(31 55 41 122)(32 56 42 123)(33 57 43 124)(34 58 44 125)(35 59 45 126)(36 60 46 109)
(1 27 101 37)(2 28 102 38)(3 29 103 39)(4 30 104 40)(5 31 105 41)(6 32 106 42)(7 33 107 43)(8 34 108 44)(9 35 91 45)(10 36 92 46)(11 19 93 47)(12 20 94 48)(13 21 95 49)(14 22 96 50)(15 23 97 51)(16 24 98 52)(17 25 99 53)(18 26 100 54)(55 138 122 74)(56 139 123 75)(57 140 124 76)(58 141 125 77)(59 142 126 78)(60 143 109 79)(61 144 110 80)(62 127 111 81)(63 128 112 82)(64 129 113 83)(65 130 114 84)(66 131 115 85)(67 132 116 86)(68 133 117 87)(69 134 118 88)(70 135 119 89)(71 136 120 90)(72 137 121 73)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 2 2 2 3 4 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 2 2 2 4 4 18 18 36 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 2 4 4 4 4 type + + + + + + + + - + + + - + - + image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D6 C4○D4 D9 C3⋊D4 D18 C9⋊D4 S3×Q8 Q8⋊3S3 Q8×D9 Q8⋊3D9 kernel D18⋊3Q8 Dic9⋊C4 C4⋊Dic9 D18⋊C4 C2×C4×D9 Q8×C18 C6×Q8 C36 D18 C2×C12 C18 C2×Q8 C12 C2×C4 C4 C6 C6 C2 C2 # reps 1 2 1 2 1 1 1 2 2 3 2 3 4 9 12 1 1 3 3

Matrix representation of D183Q8 in GL4(𝔽37) generated by

 6 17 0 0 20 26 0 0 0 0 36 0 0 0 0 36
,
 6 17 0 0 11 31 0 0 0 0 36 0 0 0 36 1
,
 7 14 0 0 23 30 0 0 0 0 1 35 0 0 1 36
,
 1 0 0 0 0 1 0 0 0 0 6 0 0 0 6 31
`G:=sub<GL(4,GF(37))| [6,20,0,0,17,26,0,0,0,0,36,0,0,0,0,36],[6,11,0,0,17,31,0,0,0,0,36,36,0,0,0,1],[7,23,0,0,14,30,0,0,0,0,1,1,0,0,35,36],[1,0,0,0,0,1,0,0,0,0,6,6,0,0,0,31] >;`

D183Q8 in GAP, Magma, Sage, TeX

`D_{18}\rtimes_3Q_8`
`% in TeX`

`G:=Group("D18:3Q8");`
`// GroupNames label`

`G:=SmallGroup(288,156);`
`// by ID`

`G=gap.SmallGroup(288,156);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,100,6725,292,9414]);`
`// Polycyclic`

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

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