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## G = D6⋊Dic9order 432 = 24·33

### The semidirect product of D6 and Dic9 acting via Dic9/C18=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — D6⋊Dic9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C6×C18 — S3×C2×C18 — D6⋊Dic9
 Lower central C3×C9 — C3×C18 — D6⋊Dic9
 Upper central C1 — C22

Generators and relations for D6⋊Dic9
G = < a,b,c,d | a6=b2=c18=1, d2=c9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 540 in 118 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C18, C18, C3×S3, C3×C6, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×C9, Dic9, C2×C18, C2×C18, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, D6⋊C4, C6.D4, S3×C9, C3×C18, C2×Dic9, C2×Dic9, C22×C18, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C3×Dic9, C9⋊Dic3, S3×C18, S3×C18, C6×C18, C18.D4, D6⋊Dic3, C6×Dic9, C2×C9⋊Dic3, S3×C2×C18, D6⋊Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D9, C4×S3, D12, C2×Dic3, C3⋊D4, Dic9, D18, S32, D6⋊C4, C6.D4, C2×Dic9, C9⋊D4, S3×Dic3, D6⋊S3, C3⋊D12, S3×D9, C18.D4, D6⋊Dic3, S3×Dic9, D6⋊D9, C9⋊D12, D6⋊Dic9

Smallest permutation representation of D6⋊Dic9
On 144 points
Generators in S144
(1 64 13 58 7 70)(2 65 14 59 8 71)(3 66 15 60 9 72)(4 67 16 61 10 55)(5 68 17 62 11 56)(6 69 18 63 12 57)(19 96 31 108 25 102)(20 97 32 91 26 103)(21 98 33 92 27 104)(22 99 34 93 28 105)(23 100 35 94 29 106)(24 101 36 95 30 107)(37 87 43 75 49 81)(38 88 44 76 50 82)(39 89 45 77 51 83)(40 90 46 78 52 84)(41 73 47 79 53 85)(42 74 48 80 54 86)(109 130 115 136 121 142)(110 131 116 137 122 143)(111 132 117 138 123 144)(112 133 118 139 124 127)(113 134 119 140 125 128)(114 135 120 141 126 129)
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 121)(12 122)(13 123)(14 124)(15 125)(16 126)(17 109)(18 110)(19 77)(20 78)(21 79)(22 80)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 73)(34 74)(35 75)(36 76)(37 106)(38 107)(39 108)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 97)(47 98)(48 99)(49 100)(50 101)(51 102)(52 103)(53 104)(54 105)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 127)(66 128)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)
(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 79 10 88)(2 78 11 87)(3 77 12 86)(4 76 13 85)(5 75 14 84)(6 74 15 83)(7 73 16 82)(8 90 17 81)(9 89 18 80)(19 131 28 140)(20 130 29 139)(21 129 30 138)(22 128 31 137)(23 127 32 136)(24 144 33 135)(25 143 34 134)(26 142 35 133)(27 141 36 132)(37 71 46 62)(38 70 47 61)(39 69 48 60)(40 68 49 59)(41 67 50 58)(42 66 51 57)(43 65 52 56)(44 64 53 55)(45 63 54 72)(91 121 100 112)(92 120 101 111)(93 119 102 110)(94 118 103 109)(95 117 104 126)(96 116 105 125)(97 115 106 124)(98 114 107 123)(99 113 108 122)

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

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

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

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 18S ··· 18AD 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 18 ··· 18 size 1 1 1 1 6 6 2 2 4 18 18 54 54 2 ··· 2 4 4 4 6 6 6 6 2 2 2 4 4 4 18 18 18 18 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 4 4 4 4 4 4 4 4 type + + + + + + + + - + + + - + + - - + + - - + image C1 C2 C2 C2 C4 S3 S3 D4 D6 Dic3 D6 D9 C4×S3 D12 C3⋊D4 C3⋊D4 Dic9 D18 C9⋊D4 S32 S3×Dic3 D6⋊S3 C3⋊D12 S3×D9 S3×Dic9 D6⋊D9 C9⋊D12 kernel D6⋊Dic9 C6×Dic9 C2×C9⋊Dic3 S3×C2×C18 S3×C18 C2×Dic9 S3×C2×C6 C3×C18 C2×C18 S3×C6 C62 C22×S3 C18 C18 C18 C3×C6 D6 C2×C6 C6 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 1 2 1 2 1 3 2 2 2 4 6 3 12 1 1 1 1 3 3 3 3

Matrix representation of D6⋊Dic9 in GL6(𝔽37)

 1 36 0 0 0 0 1 0 0 0 0 0 0 0 1 36 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 32 10 0 0 0 0 5 5 0 0 0 0 0 0 5 27 0 0 0 0 32 32 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 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 20 34 0 0 0 0 12 26
,
 32 10 0 0 0 0 27 5 0 0 0 0 0 0 30 14 0 0 0 0 23 7 0 0 0 0 0 0 1 0 0 0 0 0 35 36

G:=sub<GL(6,GF(37))| [1,1,0,0,0,0,36,0,0,0,0,0,0,0,1,1,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,5,0,0,0,0,10,5,0,0,0,0,0,0,5,32,0,0,0,0,27,32,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,12,0,0,0,0,34,26],[32,27,0,0,0,0,10,5,0,0,0,0,0,0,30,23,0,0,0,0,14,7,0,0,0,0,0,0,1,35,0,0,0,0,0,36] >;

D6⋊Dic9 in GAP, Magma, Sage, TeX

D_6\rtimes {\rm Dic}_9
% in TeX

G:=Group("D6:Dic9");
// GroupNames label

G:=SmallGroup(432,93);
// by ID

G=gap.SmallGroup(432,93);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,3091,662,4037,7069]);
// Polycyclic

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

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