metabelian, supersoluble, monomial
Aliases: D18⋊Dic3, C6.17D36, C62.57D6, (C6×D9)⋊1C4, C6.20(C4×D9), C3⋊3(D18⋊C4), (C22×D9).S3, (C2×Dic3)⋊1D9, (C2×C6).12D18, (C2×C18).12D6, (C3×C6).34D12, (C3×C18).16D4, C2.5(Dic3×D9), C6.4(S3×Dic3), C22.8(S3×D9), (Dic3×C18)⋊1C2, C6.13(C9⋊D4), (C6×C18).6C22, (C6×Dic3).3S3, C18.5(C2×Dic3), C2.2(D6⋊D9), C6.6(C3⋊D12), C2.2(C3⋊D36), C9⋊1(C6.D4), C18.12(C3⋊D4), C3.1(D6⋊Dic3), C32.2(D6⋊C4), C6.17(D6⋊S3), (C2×C6).18S32, (C2×C6×D9).1C2, (C3×C9)⋊1(C22⋊C4), (C3×C6).40(C4×S3), (C2×C9⋊Dic3)⋊1C2, (C3×C18).11(C2×C4), (C3×C6).51(C3⋊D4), SmallGroup(432,91)
Series: Derived ►Chief ►Lower central ►Upper central
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
►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