direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C18, C36.42D4, C72⋊12C22, C36.44C23, C3.(C6×D8), (C6×D8).C3, C8⋊2(C2×C18), (C2×C8)⋊3C18, C4.6(D4×C9), (C2×C72)⋊11C2, D4⋊1(C2×C18), (C2×D4)⋊4C18, (C3×D8).6C6, C6.74(C6×D4), C6.18(C3×D8), (D4×C18)⋊13C2, C24.24(C2×C6), (C2×C24).14C6, (C6×D4).14C6, (C2×C18).52D4, C2.11(D4×C18), C18.74(C2×D4), C12.42(C3×D4), (D4×C9)⋊10C22, C4.1(C22×C18), C22.14(D4×C9), C12.44(C22×C6), (C2×C36).127C22, (C2×C6).61(C3×D4), (C2×C4).26(C2×C18), (C3×D4).11(C2×C6), (C2×C12).144(C2×C6), SmallGroup(288,182)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C18
G = < a,b,c | a18=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 210 in 114 conjugacy classes, 66 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, D4, D4, C23, C9, C12, C2×C6, C2×C6, C2×C8, D8, C2×D4, C18, C18, C18, C24, C2×C12, C3×D4, C3×D4, C22×C6, C2×D8, C36, C2×C18, C2×C18, C2×C24, C3×D8, C6×D4, C72, C2×C36, D4×C9, D4×C9, C22×C18, C6×D8, C2×C72, C9×D8, D4×C18, D8×C18
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, D8, C2×D4, C18, C3×D4, C22×C6, C2×D8, C2×C18, C3×D8, C6×D4, D4×C9, C22×C18, C6×D8, C9×D8, D4×C18, D8×C18
►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 76 93 123 22 132 55 41)(2 77 94 124 23 133 56 42)(3 78 95 125 24 134 57 43)(4 79 96 126 25 135 58 44)(5 80 97 109 26 136 59 45)(6 81 98 110 27 137 60 46)(7 82 99 111 28 138 61 47)(8 83 100 112 29 139 62 48)(9 84 101 113 30 140 63 49)(10 85 102 114 31 141 64 50)(11 86 103 115 32 142 65 51)(12 87 104 116 33 143 66 52)(13 88 105 117 34 144 67 53)(14 89 106 118 35 127 68 54)(15 90 107 119 36 128 69 37)(16 73 108 120 19 129 70 38)(17 74 91 121 20 130 71 39)(18 75 92 122 21 131 72 40)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 37)(16 38)(17 39)(18 40)(19 120)(20 121)(21 122)(22 123)(23 124)(24 125)(25 126)(26 109)(27 110)(28 111)(29 112)(30 113)(31 114)(32 115)(33 116)(34 117)(35 118)(36 119)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 73)(71 74)(72 75)(91 130)(92 131)(93 132)(94 133)(95 134)(96 135)(97 136)(98 137)(99 138)(100 139)(101 140)(102 141)(103 142)(104 143)(105 144)(106 127)(107 128)(108 129)
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), (1,76,93,123,22,132,55,41)(2,77,94,124,23,133,56,42)(3,78,95,125,24,134,57,43)(4,79,96,126,25,135,58,44)(5,80,97,109,26,136,59,45)(6,81,98,110,27,137,60,46)(7,82,99,111,28,138,61,47)(8,83,100,112,29,139,62,48)(9,84,101,113,30,140,63,49)(10,85,102,114,31,141,64,50)(11,86,103,115,32,142,65,51)(12,87,104,116,33,143,66,52)(13,88,105,117,34,144,67,53)(14,89,106,118,35,127,68,54)(15,90,107,119,36,128,69,37)(16,73,108,120,19,129,70,38)(17,74,91,121,20,130,71,39)(18,75,92,122,21,131,72,40), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,37)(16,38)(17,39)(18,40)(19,120)(20,121)(21,122)(22,123)(23,124)(24,125)(25,126)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,73)(71,74)(72,75)(91,130)(92,131)(93,132)(94,133)(95,134)(96,135)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,127)(107,128)(108,129)>;
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), (1,76,93,123,22,132,55,41)(2,77,94,124,23,133,56,42)(3,78,95,125,24,134,57,43)(4,79,96,126,25,135,58,44)(5,80,97,109,26,136,59,45)(6,81,98,110,27,137,60,46)(7,82,99,111,28,138,61,47)(8,83,100,112,29,139,62,48)(9,84,101,113,30,140,63,49)(10,85,102,114,31,141,64,50)(11,86,103,115,32,142,65,51)(12,87,104,116,33,143,66,52)(13,88,105,117,34,144,67,53)(14,89,106,118,35,127,68,54)(15,90,107,119,36,128,69,37)(16,73,108,120,19,129,70,38)(17,74,91,121,20,130,71,39)(18,75,92,122,21,131,72,40), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,37)(16,38)(17,39)(18,40)(19,120)(20,121)(21,122)(22,123)(23,124)(24,125)(25,126)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,73)(71,74)(72,75)(91,130)(92,131)(93,132)(94,133)(95,134)(96,135)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,127)(107,128)(108,129) );
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)], [(1,76,93,123,22,132,55,41),(2,77,94,124,23,133,56,42),(3,78,95,125,24,134,57,43),(4,79,96,126,25,135,58,44),(5,80,97,109,26,136,59,45),(6,81,98,110,27,137,60,46),(7,82,99,111,28,138,61,47),(8,83,100,112,29,139,62,48),(9,84,101,113,30,140,63,49),(10,85,102,114,31,141,64,50),(11,86,103,115,32,142,65,51),(12,87,104,116,33,143,66,52),(13,88,105,117,34,144,67,53),(14,89,106,118,35,127,68,54),(15,90,107,119,36,128,69,37),(16,73,108,120,19,129,70,38),(17,74,91,121,20,130,71,39),(18,75,92,122,21,131,72,40)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,37),(16,38),(17,39),(18,40),(19,120),(20,121),(21,122),(22,123),(23,124),(24,125),(25,126),(26,109),(27,110),(28,111),(29,112),(30,113),(31,114),(32,115),(33,116),(34,117),(35,118),(36,119),(55,76),(56,77),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,73),(71,74),(72,75),(91,130),(92,131),(93,132),(94,133),(95,134),(96,135),(97,136),(98,137),(99,138),(100,139),(101,140),(102,141),(103,142),(104,143),(105,144),(106,127),(107,128),(108,129)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6N | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 18A | ··· | 18R | 18S | ··· | 18AP | 24A | ··· | 24H | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | D4 | D4 | D8 | C3×D4 | C3×D4 | C3×D8 | D4×C9 | D4×C9 | C9×D8 |
| kernel | D8×C18 | C2×C72 | C9×D8 | D4×C18 | C6×D8 | C2×C24 | C3×D8 | C6×D4 | C2×D8 | C2×C8 | D8 | C2×D4 | C36 | C2×C18 | C18 | C12 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 6 | 6 | 24 | 12 | 1 | 1 | 4 | 2 | 2 | 8 | 6 | 6 | 24 |
Matrix representation of D8×C18 ►in GL4(𝔽73) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 57 | 0 |
| 0 | 0 | 0 | 57 |
| 42 | 4 | 0 | 0 |
| 15 | 31 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 57 | 57 |
| 42 | 4 | 0 | 0 |
| 52 | 31 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(73))| [36,0,0,0,0,36,0,0,0,0,57,0,0,0,0,57],[42,15,0,0,4,31,0,0,0,0,57,57,0,0,16,57],[42,52,0,0,4,31,0,0,0,0,57,16,0,0,16,16] >;
D8×C18 in GAP, Magma, Sage, TeX
D_8\times C_{18} % in TeX
G:=Group("D8xC18"); // GroupNames label
G:=SmallGroup(288,182);
// by ID
G=gap.SmallGroup(288,182);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,365,192,5884,2951,242]);
// Polycyclic
G:=Group<a,b,c|a^18=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations