metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D18.5D4, C4⋊C4⋊2D9, C6.86(S3×D4), (C2×C12).8D6, C2.12(D4×D9), D18⋊C4⋊13C2, Dic9⋊C4⋊6C2, (C2×D36).3C2, C18.25(C2×D4), (C2×C4).33D18, C3.(D6.D4), C18.12(C4○D4), C6.82(C4○D12), (C2×C18).35C23, (C2×C36).10C22, C2.5(Q8⋊3D9), C9⋊3(C22.D4), C6.40(Q8⋊3S3), C2.14(D36⋊5C2), (C2×Dic9).9C22, (C22×D9).7C22, C22.49(C22×D9), (C9×C4⋊C4)⋊5C2, (C2×C4×D9)⋊13C2, (C3×C4⋊C4).12S3, (C2×C6).192(C22×S3), SmallGroup(288,104)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D18.D4
G = < a,b,c,d | a18=b2=c4=1, d2=a9, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a9b, dcd-1=c-1 >
Subgroups: 604 in 117 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, C9, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C22.D4, Dic9, C36, D18, D18, C2×C18, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C4×D9, D36, C2×Dic9, C2×C36, C22×D9, D6.D4, Dic9⋊C4, D18⋊C4, C9×C4⋊C4, C2×C4×D9, C2×D36, D18.D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, C22×S3, C22.D4, D18, C4○D12, S3×D4, Q8⋊3S3, C22×D9, D6.D4, D36⋊5C2, D4×D9, Q8⋊3D9, D18.D4
►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)
(1 131)(2 130)(3 129)(4 128)(5 127)(6 144)(7 143)(8 142)(9 141)(10 140)(11 139)(12 138)(13 137)(14 136)(15 135)(16 134)(17 133)(18 132)(19 67)(20 66)(21 65)(22 64)(23 63)(24 62)(25 61)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 72)(33 71)(34 70)(35 69)(36 68)(37 85)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 90)(51 89)(52 88)(53 87)(54 86)(91 121)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 126)(105 125)(106 124)(107 123)(108 122)
(1 53 105 60)(2 54 106 61)(3 37 107 62)(4 38 108 63)(5 39 91 64)(6 40 92 65)(7 41 93 66)(8 42 94 67)(9 43 95 68)(10 44 96 69)(11 45 97 70)(12 46 98 71)(13 47 99 72)(14 48 100 55)(15 49 101 56)(16 50 102 57)(17 51 103 58)(18 52 104 59)(19 133 80 109)(20 134 81 110)(21 135 82 111)(22 136 83 112)(23 137 84 113)(24 138 85 114)(25 139 86 115)(26 140 87 116)(27 141 88 117)(28 142 89 118)(29 143 90 119)(30 144 73 120)(31 127 74 121)(32 128 75 122)(33 129 76 123)(34 130 77 124)(35 131 78 125)(36 132 79 126)
(1 117 10 126)(2 118 11 109)(3 119 12 110)(4 120 13 111)(5 121 14 112)(6 122 15 113)(7 123 16 114)(8 124 17 115)(9 125 18 116)(19 61 28 70)(20 62 29 71)(21 63 30 72)(22 64 31 55)(23 65 32 56)(24 66 33 57)(25 67 34 58)(26 68 35 59)(27 69 36 60)(37 90 46 81)(38 73 47 82)(39 74 48 83)(40 75 49 84)(41 76 50 85)(42 77 51 86)(43 78 52 87)(44 79 53 88)(45 80 54 89)(91 127 100 136)(92 128 101 137)(93 129 102 138)(94 130 103 139)(95 131 104 140)(96 132 105 141)(97 133 106 142)(98 134 107 143)(99 135 108 144)
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), (1,131)(2,130)(3,129)(4,128)(5,127)(6,144)(7,143)(8,142)(9,141)(10,140)(11,139)(12,138)(13,137)(14,136)(15,135)(16,134)(17,133)(18,132)(19,67)(20,66)(21,65)(22,64)(23,63)(24,62)(25,61)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,72)(33,71)(34,70)(35,69)(36,68)(37,85)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,90)(51,89)(52,88)(53,87)(54,86)(91,121)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,126)(105,125)(106,124)(107,123)(108,122), (1,53,105,60)(2,54,106,61)(3,37,107,62)(4,38,108,63)(5,39,91,64)(6,40,92,65)(7,41,93,66)(8,42,94,67)(9,43,95,68)(10,44,96,69)(11,45,97,70)(12,46,98,71)(13,47,99,72)(14,48,100,55)(15,49,101,56)(16,50,102,57)(17,51,103,58)(18,52,104,59)(19,133,80,109)(20,134,81,110)(21,135,82,111)(22,136,83,112)(23,137,84,113)(24,138,85,114)(25,139,86,115)(26,140,87,116)(27,141,88,117)(28,142,89,118)(29,143,90,119)(30,144,73,120)(31,127,74,121)(32,128,75,122)(33,129,76,123)(34,130,77,124)(35,131,78,125)(36,132,79,126), (1,117,10,126)(2,118,11,109)(3,119,12,110)(4,120,13,111)(5,121,14,112)(6,122,15,113)(7,123,16,114)(8,124,17,115)(9,125,18,116)(19,61,28,70)(20,62,29,71)(21,63,30,72)(22,64,31,55)(23,65,32,56)(24,66,33,57)(25,67,34,58)(26,68,35,59)(27,69,36,60)(37,90,46,81)(38,73,47,82)(39,74,48,83)(40,75,49,84)(41,76,50,85)(42,77,51,86)(43,78,52,87)(44,79,53,88)(45,80,54,89)(91,127,100,136)(92,128,101,137)(93,129,102,138)(94,130,103,139)(95,131,104,140)(96,132,105,141)(97,133,106,142)(98,134,107,143)(99,135,108,144)>;
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), (1,131)(2,130)(3,129)(4,128)(5,127)(6,144)(7,143)(8,142)(9,141)(10,140)(11,139)(12,138)(13,137)(14,136)(15,135)(16,134)(17,133)(18,132)(19,67)(20,66)(21,65)(22,64)(23,63)(24,62)(25,61)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,72)(33,71)(34,70)(35,69)(36,68)(37,85)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,90)(51,89)(52,88)(53,87)(54,86)(91,121)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,126)(105,125)(106,124)(107,123)(108,122), (1,53,105,60)(2,54,106,61)(3,37,107,62)(4,38,108,63)(5,39,91,64)(6,40,92,65)(7,41,93,66)(8,42,94,67)(9,43,95,68)(10,44,96,69)(11,45,97,70)(12,46,98,71)(13,47,99,72)(14,48,100,55)(15,49,101,56)(16,50,102,57)(17,51,103,58)(18,52,104,59)(19,133,80,109)(20,134,81,110)(21,135,82,111)(22,136,83,112)(23,137,84,113)(24,138,85,114)(25,139,86,115)(26,140,87,116)(27,141,88,117)(28,142,89,118)(29,143,90,119)(30,144,73,120)(31,127,74,121)(32,128,75,122)(33,129,76,123)(34,130,77,124)(35,131,78,125)(36,132,79,126), (1,117,10,126)(2,118,11,109)(3,119,12,110)(4,120,13,111)(5,121,14,112)(6,122,15,113)(7,123,16,114)(8,124,17,115)(9,125,18,116)(19,61,28,70)(20,62,29,71)(21,63,30,72)(22,64,31,55)(23,65,32,56)(24,66,33,57)(25,67,34,58)(26,68,35,59)(27,69,36,60)(37,90,46,81)(38,73,47,82)(39,74,48,83)(40,75,49,84)(41,76,50,85)(42,77,51,86)(43,78,52,87)(44,79,53,88)(45,80,54,89)(91,127,100,136)(92,128,101,137)(93,129,102,138)(94,130,103,139)(95,131,104,140)(96,132,105,141)(97,133,106,142)(98,134,107,143)(99,135,108,144) );
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)], [(1,131),(2,130),(3,129),(4,128),(5,127),(6,144),(7,143),(8,142),(9,141),(10,140),(11,139),(12,138),(13,137),(14,136),(15,135),(16,134),(17,133),(18,132),(19,67),(20,66),(21,65),(22,64),(23,63),(24,62),(25,61),(26,60),(27,59),(28,58),(29,57),(30,56),(31,55),(32,72),(33,71),(34,70),(35,69),(36,68),(37,85),(38,84),(39,83),(40,82),(41,81),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,73),(50,90),(51,89),(52,88),(53,87),(54,86),(91,121),(92,120),(93,119),(94,118),(95,117),(96,116),(97,115),(98,114),(99,113),(100,112),(101,111),(102,110),(103,109),(104,126),(105,125),(106,124),(107,123),(108,122)], [(1,53,105,60),(2,54,106,61),(3,37,107,62),(4,38,108,63),(5,39,91,64),(6,40,92,65),(7,41,93,66),(8,42,94,67),(9,43,95,68),(10,44,96,69),(11,45,97,70),(12,46,98,71),(13,47,99,72),(14,48,100,55),(15,49,101,56),(16,50,102,57),(17,51,103,58),(18,52,104,59),(19,133,80,109),(20,134,81,110),(21,135,82,111),(22,136,83,112),(23,137,84,113),(24,138,85,114),(25,139,86,115),(26,140,87,116),(27,141,88,117),(28,142,89,118),(29,143,90,119),(30,144,73,120),(31,127,74,121),(32,128,75,122),(33,129,76,123),(34,130,77,124),(35,131,78,125),(36,132,79,126)], [(1,117,10,126),(2,118,11,109),(3,119,12,110),(4,120,13,111),(5,121,14,112),(6,122,15,113),(7,123,16,114),(8,124,17,115),(9,125,18,116),(19,61,28,70),(20,62,29,71),(21,63,30,72),(22,64,31,55),(23,65,32,56),(24,66,33,57),(25,67,34,58),(26,68,35,59),(27,69,36,60),(37,90,46,81),(38,73,47,82),(39,74,48,83),(40,75,49,84),(41,76,50,85),(42,77,51,86),(43,78,52,87),(44,79,53,88),(45,80,54,89),(91,127,100,136),(92,128,101,137),(93,129,102,138),(94,130,103,139),(95,131,104,140),(96,132,105,141),(97,133,106,142),(98,134,107,143),(99,135,108,144)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 36 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | C4○D4 | D9 | D18 | C4○D12 | D36⋊5C2 | S3×D4 | Q8⋊3S3 | D4×D9 | Q8⋊3D9 |
| kernel | D18.D4 | Dic9⋊C4 | D18⋊C4 | C9×C4⋊C4 | C2×C4×D9 | C2×D36 | C3×C4⋊C4 | D18 | C2×C12 | C18 | C4⋊C4 | C2×C4 | C6 | C2 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 3 | 9 | 4 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of D18.D4 ►in GL4(𝔽37) generated by
| 31 | 26 | 0 | 0 |
| 11 | 20 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 4 | 25 | 0 | 0 |
| 29 | 33 | 0 | 0 |
| 0 | 0 | 26 | 36 |
| 0 | 0 | 9 | 11 |
| 7 | 23 | 0 | 0 |
| 14 | 30 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 16 | 31 |
| 32 | 10 | 0 | 0 |
| 27 | 5 | 0 | 0 |
| 0 | 0 | 26 | 36 |
| 0 | 0 | 11 | 11 |
G:=sub<GL(4,GF(37))| [31,11,0,0,26,20,0,0,0,0,36,0,0,0,0,36],[4,29,0,0,25,33,0,0,0,0,26,9,0,0,36,11],[7,14,0,0,23,30,0,0,0,0,6,16,0,0,0,31],[32,27,0,0,10,5,0,0,0,0,26,11,0,0,36,11] >;
D18.D4 in GAP, Magma, Sage, TeX
D_{18}.D_4 % in TeX
G:=Group("D18.D4"); // GroupNames label
G:=SmallGroup(288,104);
// by ID
G=gap.SmallGroup(288,104);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^18=b^2=c^4=1,d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^9*b,d*c*d^-1=c^-1>;
// generators/relations