metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D18⋊C8, C36.35D4, C4.19D36, C12.54D12, C18.3M4(2), (C2×C8)⋊1D9, (C2×C72)⋊1C2, C3.(D6⋊C8), C2.5(C8×D9), C6.10(S3×C8), C9⋊1(C22⋊C8), C18.5(C2×C8), (C2×C24).3S3, (C2×C4).93D18, C6.6(C8⋊S3), C2.1(D18⋊C4), C6.13(D6⋊C4), (C2×C12).407D6, C2.3(C8⋊D9), C4.27(C9⋊D4), (C2×Dic9).4C4, (C22×D9).2C4, C22.11(C4×D9), C18.6(C22⋊C4), C12.122(C3⋊D4), (C2×C36).105C22, (C2×C9⋊C8)⋊9C2, (C2×C4×D9).6C2, (C2×C6).37(C4×S3), (C2×C18).12(C2×C4), SmallGroup(288,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D18⋊C8
G = < a,b,c | a18=b2=c8=1, bab=a-1, ac=ca, cbc-1=a9b >
Subgroups: 332 in 75 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, C9, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, C22×C4, D9, C18, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C22⋊C8, Dic9, C36, D18, D18, C2×C18, C2×C3⋊C8, C2×C24, S3×C2×C4, C9⋊C8, C72, C4×D9, C2×Dic9, C2×C36, C22×D9, D6⋊C8, C2×C9⋊C8, C2×C72, C2×C4×D9, D18⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D6, C22⋊C4, C2×C8, M4(2), D9, C4×S3, D12, C3⋊D4, C22⋊C8, D18, S3×C8, C8⋊S3, D6⋊C4, C4×D9, D36, C9⋊D4, D6⋊C8, C8×D9, C8⋊D9, D18⋊C4, D18⋊C8
►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 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 83)(20 82)(21 81)(22 80)(23 79)(24 78)(25 77)(26 76)(27 75)(28 74)(29 73)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 84)(55 91)(56 108)(57 107)(58 106)(59 105)(60 104)(61 103)(62 102)(63 101)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(109 127)(110 144)(111 143)(112 142)(113 141)(114 140)(115 139)(116 138)(117 137)(118 136)(119 135)(120 134)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)
(1 84 57 111 44 28 108 135)(2 85 58 112 45 29 91 136)(3 86 59 113 46 30 92 137)(4 87 60 114 47 31 93 138)(5 88 61 115 48 32 94 139)(6 89 62 116 49 33 95 140)(7 90 63 117 50 34 96 141)(8 73 64 118 51 35 97 142)(9 74 65 119 52 36 98 143)(10 75 66 120 53 19 99 144)(11 76 67 121 54 20 100 127)(12 77 68 122 37 21 101 128)(13 78 69 123 38 22 102 129)(14 79 70 124 39 23 103 130)(15 80 71 125 40 24 104 131)(16 81 72 126 41 25 105 132)(17 82 55 109 42 26 106 133)(18 83 56 110 43 27 107 134)
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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,83)(20,82)(21,81)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(55,91)(56,108)(57,107)(58,106)(59,105)(60,104)(61,103)(62,102)(63,101)(64,100)(65,99)(66,98)(67,97)(68,96)(69,95)(70,94)(71,93)(72,92)(109,127)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128), (1,84,57,111,44,28,108,135)(2,85,58,112,45,29,91,136)(3,86,59,113,46,30,92,137)(4,87,60,114,47,31,93,138)(5,88,61,115,48,32,94,139)(6,89,62,116,49,33,95,140)(7,90,63,117,50,34,96,141)(8,73,64,118,51,35,97,142)(9,74,65,119,52,36,98,143)(10,75,66,120,53,19,99,144)(11,76,67,121,54,20,100,127)(12,77,68,122,37,21,101,128)(13,78,69,123,38,22,102,129)(14,79,70,124,39,23,103,130)(15,80,71,125,40,24,104,131)(16,81,72,126,41,25,105,132)(17,82,55,109,42,26,106,133)(18,83,56,110,43,27,107,134)>;
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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,83)(20,82)(21,81)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(55,91)(56,108)(57,107)(58,106)(59,105)(60,104)(61,103)(62,102)(63,101)(64,100)(65,99)(66,98)(67,97)(68,96)(69,95)(70,94)(71,93)(72,92)(109,127)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128), (1,84,57,111,44,28,108,135)(2,85,58,112,45,29,91,136)(3,86,59,113,46,30,92,137)(4,87,60,114,47,31,93,138)(5,88,61,115,48,32,94,139)(6,89,62,116,49,33,95,140)(7,90,63,117,50,34,96,141)(8,73,64,118,51,35,97,142)(9,74,65,119,52,36,98,143)(10,75,66,120,53,19,99,144)(11,76,67,121,54,20,100,127)(12,77,68,122,37,21,101,128)(13,78,69,123,38,22,102,129)(14,79,70,124,39,23,103,130)(15,80,71,125,40,24,104,131)(16,81,72,126,41,25,105,132)(17,82,55,109,42,26,106,133)(18,83,56,110,43,27,107,134) );
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)], [(1,43),(2,42),(3,41),(4,40),(5,39),(6,38),(7,37),(8,54),(9,53),(10,52),(11,51),(12,50),(13,49),(14,48),(15,47),(16,46),(17,45),(18,44),(19,83),(20,82),(21,81),(22,80),(23,79),(24,78),(25,77),(26,76),(27,75),(28,74),(29,73),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,84),(55,91),(56,108),(57,107),(58,106),(59,105),(60,104),(61,103),(62,102),(63,101),(64,100),(65,99),(66,98),(67,97),(68,96),(69,95),(70,94),(71,93),(72,92),(109,127),(110,144),(111,143),(112,142),(113,141),(114,140),(115,139),(116,138),(117,137),(118,136),(119,135),(120,134),(121,133),(122,132),(123,131),(124,130),(125,129),(126,128)], [(1,84,57,111,44,28,108,135),(2,85,58,112,45,29,91,136),(3,86,59,113,46,30,92,137),(4,87,60,114,47,31,93,138),(5,88,61,115,48,32,94,139),(6,89,62,116,49,33,95,140),(7,90,63,117,50,34,96,141),(8,73,64,118,51,35,97,142),(9,74,65,119,52,36,98,143),(10,75,66,120,53,19,99,144),(11,76,67,121,54,20,100,127),(12,77,68,122,37,21,101,128),(13,78,69,123,38,22,102,129),(14,79,70,124,39,23,103,130),(15,80,71,125,40,24,104,131),(16,81,72,126,41,25,105,132),(17,82,55,109,42,26,106,133),(18,83,56,110,43,27,107,134)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D6 | M4(2) | D9 | D12 | C3⋊D4 | C4×S3 | D18 | S3×C8 | C8⋊S3 | D36 | C9⋊D4 | C4×D9 | C8×D9 | C8⋊D9 |
| kernel | D18⋊C8 | C2×C9⋊C8 | C2×C72 | C2×C4×D9 | C2×Dic9 | C22×D9 | D18 | C2×C24 | C36 | C2×C12 | C18 | C2×C8 | C12 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 3 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 12 | 12 |
Matrix representation of D18⋊C8 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 3 | 28 |
| 0 | 45 | 31 |
| 1 | 0 | 0 |
| 0 | 28 | 42 |
| 0 | 70 | 45 |
| 10 | 0 | 0 |
| 0 | 65 | 16 |
| 0 | 57 | 8 |
G:=sub<GL(3,GF(73))| [1,0,0,0,3,45,0,28,31],[1,0,0,0,28,70,0,42,45],[10,0,0,0,65,57,0,16,8] >;
D18⋊C8 in GAP, Magma, Sage, TeX
D_{18}\rtimes C_8 % in TeX
G:=Group("D18:C8"); // GroupNames label
G:=SmallGroup(288,27);
// by ID
G=gap.SmallGroup(288,27);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^18=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^9*b>;
// generators/relations