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## G = D18⋊Dic3order 432 = 24·33

### The semidirect product of D18 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — D18⋊Dic3
 Chief series C1 — C3 — C9 — C3×C9 — C3×C18 — C6×C18 — C2×C6×D9 — D18⋊Dic3
 Lower central C3×C9 — C3×C18 — D18⋊Dic3
 Upper central C1 — C22

Generators and relations for D18⋊Dic3
G = < a,b,c,d | a18=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a9b, dcd-1=c-1 >

Subgroups: 652 in 118 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, D9, C18, C18, C3×S3, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×C9, Dic9, C36, D18, D18, C2×C18, C2×C18, C3×Dic3, C3⋊Dic3, S3×C6, C62, D6⋊C4, C6.D4, C3×D9, C3×C18, C2×Dic9, C2×C36, C22×D9, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C9×Dic3, C9⋊Dic3, C6×D9, C6×D9, C6×C18, D18⋊C4, D6⋊Dic3, Dic3×C18, C2×C9⋊Dic3, C2×C6×D9, D18⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D9, C4×S3, D12, C2×Dic3, C3⋊D4, D18, S32, D6⋊C4, C6.D4, C4×D9, D36, C9⋊D4, S3×Dic3, D6⋊S3, C3⋊D12, S3×D9, D18⋊C4, D6⋊Dic3, Dic3×D9, C3⋊D36, D6⋊D9, D18⋊Dic3

Smallest permutation representation of D18⋊Dic3
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)
(1 108)(2 107)(3 106)(4 105)(5 104)(6 103)(7 102)(8 101)(9 100)(10 99)(11 98)(12 97)(13 96)(14 95)(15 94)(16 93)(17 92)(18 91)(19 78)(20 77)(21 76)(22 75)(23 74)(24 73)(25 90)(26 89)(27 88)(28 87)(29 86)(30 85)(31 84)(32 83)(33 82)(34 81)(35 80)(36 79)(37 128)(38 127)(39 144)(40 143)(41 142)(42 141)(43 140)(44 139)(45 138)(46 137)(47 136)(48 135)(49 134)(50 133)(51 132)(52 131)(53 130)(54 129)(55 126)(56 125)(57 124)(58 123)(59 122)(60 121)(61 120)(62 119)(63 118)(64 117)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)
(1 20 7 26 13 32)(2 21 8 27 14 33)(3 22 9 28 15 34)(4 23 10 29 16 35)(5 24 11 30 17 36)(6 25 12 31 18 19)(37 57 49 69 43 63)(38 58 50 70 44 64)(39 59 51 71 45 65)(40 60 52 72 46 66)(41 61 53 55 47 67)(42 62 54 56 48 68)(73 98 85 92 79 104)(74 99 86 93 80 105)(75 100 87 94 81 106)(76 101 88 95 82 107)(77 102 89 96 83 108)(78 103 90 97 84 91)(109 137 115 143 121 131)(110 138 116 144 122 132)(111 139 117 127 123 133)(112 140 118 128 124 134)(113 141 119 129 125 135)(114 142 120 130 126 136)
(1 65 26 51)(2 66 27 52)(3 67 28 53)(4 68 29 54)(5 69 30 37)(6 70 31 38)(7 71 32 39)(8 72 33 40)(9 55 34 41)(10 56 35 42)(11 57 36 43)(12 58 19 44)(13 59 20 45)(14 60 21 46)(15 61 22 47)(16 62 23 48)(17 63 24 49)(18 64 25 50)(73 143 92 109)(74 144 93 110)(75 127 94 111)(76 128 95 112)(77 129 96 113)(78 130 97 114)(79 131 98 115)(80 132 99 116)(81 133 100 117)(82 134 101 118)(83 135 102 119)(84 136 103 120)(85 137 104 121)(86 138 105 122)(87 139 106 123)(88 140 107 124)(89 141 108 125)(90 142 91 126)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 18A ··· 18I 18J ··· 18R 36A ··· 36L order 1 2 2 2 2 2 3 3 3 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 9 9 9 9 9 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 18 18 2 2 4 6 6 54 54 2 ··· 2 4 4 4 18 18 18 18 2 2 2 4 4 4 6 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + - + + + + + + + - - + + - + - image C1 C2 C2 C2 C4 S3 S3 D4 Dic3 D6 D6 D9 C3⋊D4 C4×S3 D12 C3⋊D4 D18 C4×D9 D36 C9⋊D4 S32 S3×Dic3 D6⋊S3 C3⋊D12 S3×D9 Dic3×D9 C3⋊D36 D6⋊D9 kernel D18⋊Dic3 Dic3×C18 C2×C9⋊Dic3 C2×C6×D9 C6×D9 C22×D9 C6×Dic3 C3×C18 D18 C2×C18 C62 C2×Dic3 C18 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 1 2 2 1 1 3 4 2 2 2 3 6 6 6 1 1 1 1 3 3 3 3

Matrix representation of D18⋊Dic3 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 34
,
 28 25 0 0 0 0 19 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 25 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 36 1 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 36 1 0 0 0 0 0 1 0 0 0 0 0 0 0 31 0 0 0 0 31 0 0 0 0 0 0 0 6 0 0 0 0 0 0 6

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,34],[28,19,0,0,0,0,25,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,25,0,0,0,0,3,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,1,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,1,1,0,0,0,0,0,0,0,31,0,0,0,0,31,0,0,0,0,0,0,0,6,0,0,0,0,0,0,6] >;

D18⋊Dic3 in GAP, Magma, Sage, TeX

D_{18}\rtimes {\rm Dic}_3
% in TeX

G:=Group("D18:Dic3");
// GroupNames label

G:=SmallGroup(432,91);
// by ID

G=gap.SmallGroup(432,91);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,571,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^18=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=c^-1>;
// generators/relations

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