metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D36⋊3C4, C18.7D8, C36.2D4, C4.10D36, C12.2D12, C18.7SD16, C4⋊C4⋊1D9, C4.2(C4×D9), C12.4(C4×S3), C36.4(C2×C4), C9⋊1(D4⋊C4), C2.2(D4⋊D9), (C2×D36).5C2, (C2×C12).40D6, (C2×C18).32D4, (C2×C4).38D18, C3.(C6.D8), C6.11(D6⋊C4), C2.6(D18⋊C4), C6.14(D4⋊S3), C18.4(C22⋊C4), (C2×C36).18C22, C6.8(Q8⋊2S3), C2.2(Q8⋊2D9), C22.15(C9⋊D4), (C2×C9⋊C8)⋊1C2, (C9×C4⋊C4)⋊1C2, (C3×C4⋊C4).4S3, (C2×C6).70(C3⋊D4), SmallGroup(288,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C18.D8
G = < a,b,c | a18=b8=c2=1, bab-1=cac=a-1, cbc=a9b-1 >
Subgroups: 448 in 75 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, C9, C12, C12, D6, C2×C6, C4⋊C4, C2×C8, C2×D4, D9, C18, C3⋊C8, D12, C2×C12, C2×C12, C22×S3, D4⋊C4, C36, C36, D18, C2×C18, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C9⋊C8, D36, D36, C2×C36, C2×C36, C22×D9, C6.D8, C2×C9⋊C8, C9×C4⋊C4, C2×D36, C18.D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, D9, C4×S3, D12, C3⋊D4, D4⋊C4, D18, D6⋊C4, D4⋊S3, Q8⋊2S3, C4×D9, D36, C9⋊D4, C6.D8, D18⋊C4, D4⋊D9, Q8⋊2D9, C18.D8
►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)
(1 107 28 119 89 52 58 135)(2 106 29 118 90 51 59 134)(3 105 30 117 73 50 60 133)(4 104 31 116 74 49 61 132)(5 103 32 115 75 48 62 131)(6 102 33 114 76 47 63 130)(7 101 34 113 77 46 64 129)(8 100 35 112 78 45 65 128)(9 99 36 111 79 44 66 127)(10 98 19 110 80 43 67 144)(11 97 20 109 81 42 68 143)(12 96 21 126 82 41 69 142)(13 95 22 125 83 40 70 141)(14 94 23 124 84 39 71 140)(15 93 24 123 85 38 72 139)(16 92 25 122 86 37 55 138)(17 91 26 121 87 54 56 137)(18 108 27 120 88 53 57 136)
(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(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)(37 125)(38 124)(39 123)(40 122)(41 121)(42 120)(43 119)(44 118)(45 117)(46 116)(47 115)(48 114)(49 113)(50 112)(51 111)(52 110)(53 109)(54 126)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)(79 81)(88 90)(91 142)(92 141)(93 140)(94 139)(95 138)(96 137)(97 136)(98 135)(99 134)(100 133)(101 132)(102 131)(103 130)(104 129)(105 128)(106 127)(107 144)(108 143)
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), (1,107,28,119,89,52,58,135)(2,106,29,118,90,51,59,134)(3,105,30,117,73,50,60,133)(4,104,31,116,74,49,61,132)(5,103,32,115,75,48,62,131)(6,102,33,114,76,47,63,130)(7,101,34,113,77,46,64,129)(8,100,35,112,78,45,65,128)(9,99,36,111,79,44,66,127)(10,98,19,110,80,43,67,144)(11,97,20,109,81,42,68,143)(12,96,21,126,82,41,69,142)(13,95,22,125,83,40,70,141)(14,94,23,124,84,39,71,140)(15,93,24,123,85,38,72,139)(16,92,25,122,86,37,55,138)(17,91,26,121,87,54,56,137)(18,108,27,120,88,53,57,136), (2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(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)(37,125)(38,124)(39,123)(40,122)(41,121)(42,120)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,113)(50,112)(51,111)(52,110)(53,109)(54,126)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)(88,90)(91,142)(92,141)(93,140)(94,139)(95,138)(96,137)(97,136)(98,135)(99,134)(100,133)(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,144)(108,143)>;
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), (1,107,28,119,89,52,58,135)(2,106,29,118,90,51,59,134)(3,105,30,117,73,50,60,133)(4,104,31,116,74,49,61,132)(5,103,32,115,75,48,62,131)(6,102,33,114,76,47,63,130)(7,101,34,113,77,46,64,129)(8,100,35,112,78,45,65,128)(9,99,36,111,79,44,66,127)(10,98,19,110,80,43,67,144)(11,97,20,109,81,42,68,143)(12,96,21,126,82,41,69,142)(13,95,22,125,83,40,70,141)(14,94,23,124,84,39,71,140)(15,93,24,123,85,38,72,139)(16,92,25,122,86,37,55,138)(17,91,26,121,87,54,56,137)(18,108,27,120,88,53,57,136), (2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(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)(37,125)(38,124)(39,123)(40,122)(41,121)(42,120)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,113)(50,112)(51,111)(52,110)(53,109)(54,126)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)(88,90)(91,142)(92,141)(93,140)(94,139)(95,138)(96,137)(97,136)(98,135)(99,134)(100,133)(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,144)(108,143) );
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)], [(1,107,28,119,89,52,58,135),(2,106,29,118,90,51,59,134),(3,105,30,117,73,50,60,133),(4,104,31,116,74,49,61,132),(5,103,32,115,75,48,62,131),(6,102,33,114,76,47,63,130),(7,101,34,113,77,46,64,129),(8,100,35,112,78,45,65,128),(9,99,36,111,79,44,66,127),(10,98,19,110,80,43,67,144),(11,97,20,109,81,42,68,143),(12,96,21,126,82,41,69,142),(13,95,22,125,83,40,70,141),(14,94,23,124,84,39,71,140),(15,93,24,123,85,38,72,139),(16,92,25,122,86,37,55,138),(17,91,26,121,87,54,56,137),(18,108,27,120,88,53,57,136)], [(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(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),(37,125),(38,124),(39,123),(40,122),(41,121),(42,120),(43,119),(44,118),(45,117),(46,116),(47,115),(48,114),(49,113),(50,112),(51,111),(52,110),(53,109),(54,126),(73,87),(74,86),(75,85),(76,84),(77,83),(78,82),(79,81),(88,90),(91,142),(92,141),(93,140),(94,139),(95,138),(96,137),(97,136),(98,135),(99,134),(100,133),(101,132),(102,131),(103,130),(104,129),(105,128),(106,127),(107,144),(108,143)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
54 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D8 | SD16 | D9 | C4×S3 | D12 | C3⋊D4 | D18 | C4×D9 | D36 | C9⋊D4 | D4⋊S3 | Q8⋊2S3 | D4⋊D9 | Q8⋊2D9 |
| kernel | C18.D8 | C2×C9⋊C8 | C9×C4⋊C4 | C2×D36 | D36 | C3×C4⋊C4 | C36 | C2×C18 | C2×C12 | C18 | C18 | C4⋊C4 | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 1 | 1 | 3 | 3 |
Matrix representation of C18.D8 ►in GL4(𝔽73) generated by
| 3 | 45 | 0 | 0 |
| 28 | 31 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 34 | 0 | 0 |
| 26 | 65 | 0 | 0 |
| 0 | 0 | 0 | 48 |
| 0 | 0 | 38 | 41 |
| 70 | 42 | 0 | 0 |
| 45 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 48 | 72 |
G:=sub<GL(4,GF(73))| [3,28,0,0,45,31,0,0,0,0,1,0,0,0,0,1],[8,26,0,0,34,65,0,0,0,0,0,38,0,0,48,41],[70,45,0,0,42,3,0,0,0,0,1,48,0,0,0,72] >;
C18.D8 in GAP, Magma, Sage, TeX
C_{18}.D_8 % in TeX
G:=Group("C18.D8"); // GroupNames label
G:=SmallGroup(288,17);
// by ID
G=gap.SmallGroup(288,17);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,675,346,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^18=b^8=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^9*b^-1>;
// generators/relations