metabelian, supersoluble, monomial
Aliases: D6⋊Dic9, C18.18D12, C62.59D6, (S3×C18)⋊2C4, C9⋊3(D6⋊C4), (C22×S3).D9, C18.20(C4×S3), (C6×Dic9)⋊2C2, (C2×Dic9)⋊2S3, (C2×C18).14D6, (C3×C18).18D4, (C2×C6).14D18, C6.7(C9⋊D4), C2.5(S3×Dic9), C6.5(C2×Dic9), (C6×C18).8C22, (S3×C6).3Dic3, C6.26(S3×Dic3), C22.10(S3×D9), C2.3(D6⋊D9), C2.3(C9⋊D12), C18.13(C3⋊D4), C3.3(D6⋊Dic3), C6.18(D6⋊S3), C6.20(C3⋊D12), C3⋊1(C18.D4), C32.2(C6.D4), (C2×C6).20S32, (S3×C2×C6).2S3, (S3×C2×C18).2C2, (C3×C9)⋊3(C22⋊C4), (C2×C9⋊Dic3)⋊2C2, (C3×C18).13(C2×C4), (C3×C6).53(C3⋊D4), (C3×C6).33(C2×Dic3), SmallGroup(432,93)
Series: Derived ►Chief ►Lower central ►Upper central
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
►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