metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.8D18, C36.50D4, Q8.13D18, C36.17C23, Dic18.11C22, D4.D9⋊6C2, C4○D4.4D9, C9⋊Q16⋊6C2, (C2×C18).8D4, C9⋊C8.4C22, (C3×D4).32D6, (C2×C12).68D6, (C2×C4).20D18, C18.59(C2×D4), C9⋊5(C8.C22), C4.Dic9⋊9C2, (C3×Q8).56D6, C4.24(C9⋊D4), C3.(Q8.14D6), (D4×C9).8C22, C4.17(C22×D9), (C2×Dic18)⋊11C2, (Q8×C9).8C22, (C2×C36).45C22, C12.56(C22×S3), C22.5(C9⋊D4), C12.112(C3⋊D4), (C9×C4○D4).3C2, C2.23(C2×C9⋊D4), (C3×C4○D4).11S3, (C2×C6).7(C3⋊D4), C6.107(C2×C3⋊D4), SmallGroup(288,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D18
G = < a,b,c,d | a4=b2=c18=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >
Subgroups: 316 in 90 conjugacy classes, 38 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C9, Dic3, C12, C12, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C18, C18, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C8.C22, Dic9, C36, C36, C2×C18, C2×C18, C4.Dic3, D4.S3, C3⋊Q16, C2×Dic6, C3×C4○D4, C9⋊C8, Dic18, Dic18, C2×Dic9, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, Q8.14D6, C4.Dic9, D4.D9, C9⋊Q16, C2×Dic18, C9×C4○D4, D4.D18
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, C8.C22, D18, C2×C3⋊D4, C9⋊D4, C22×D9, Q8.14D6, C2×C9⋊D4, D4.D18
►On 144 points
Generators in S144
(1 100 113 84)(2 101 114 85)(3 102 115 86)(4 103 116 87)(5 104 117 88)(6 105 118 89)(7 106 119 90)(8 107 120 73)(9 108 121 74)(10 91 122 75)(11 92 123 76)(12 93 124 77)(13 94 125 78)(14 95 126 79)(15 96 109 80)(16 97 110 81)(17 98 111 82)(18 99 112 83)(19 141 39 60)(20 142 40 61)(21 143 41 62)(22 144 42 63)(23 127 43 64)(24 128 44 65)(25 129 45 66)(26 130 46 67)(27 131 47 68)(28 132 48 69)(29 133 49 70)(30 134 50 71)(31 135 51 72)(32 136 52 55)(33 137 53 56)(34 138 54 57)(35 139 37 58)(36 140 38 59)
(1 84)(2 101)(3 86)(4 103)(5 88)(6 105)(7 90)(8 107)(9 74)(10 91)(11 76)(12 93)(13 78)(14 95)(15 80)(16 97)(17 82)(18 99)(19 39)(21 41)(23 43)(25 45)(27 47)(29 49)(31 51)(33 53)(35 37)(55 136)(57 138)(59 140)(61 142)(63 144)(65 128)(67 130)(69 132)(71 134)(73 120)(75 122)(77 124)(79 126)(81 110)(83 112)(85 114)(87 116)(89 118)(92 123)(94 125)(96 109)(98 111)(100 113)(102 115)(104 117)(106 119)(108 121)
(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 62 113 143)(2 61 114 142)(3 60 115 141)(4 59 116 140)(5 58 117 139)(6 57 118 138)(7 56 119 137)(8 55 120 136)(9 72 121 135)(10 71 122 134)(11 70 123 133)(12 69 124 132)(13 68 125 131)(14 67 126 130)(15 66 109 129)(16 65 110 128)(17 64 111 127)(18 63 112 144)(19 102 39 86)(20 101 40 85)(21 100 41 84)(22 99 42 83)(23 98 43 82)(24 97 44 81)(25 96 45 80)(26 95 46 79)(27 94 47 78)(28 93 48 77)(29 92 49 76)(30 91 50 75)(31 108 51 74)(32 107 52 73)(33 106 53 90)(34 105 54 89)(35 104 37 88)(36 103 38 87)
G:=sub<Sym(144)| (1,100,113,84)(2,101,114,85)(3,102,115,86)(4,103,116,87)(5,104,117,88)(6,105,118,89)(7,106,119,90)(8,107,120,73)(9,108,121,74)(10,91,122,75)(11,92,123,76)(12,93,124,77)(13,94,125,78)(14,95,126,79)(15,96,109,80)(16,97,110,81)(17,98,111,82)(18,99,112,83)(19,141,39,60)(20,142,40,61)(21,143,41,62)(22,144,42,63)(23,127,43,64)(24,128,44,65)(25,129,45,66)(26,130,46,67)(27,131,47,68)(28,132,48,69)(29,133,49,70)(30,134,50,71)(31,135,51,72)(32,136,52,55)(33,137,53,56)(34,138,54,57)(35,139,37,58)(36,140,38,59), (1,84)(2,101)(3,86)(4,103)(5,88)(6,105)(7,90)(8,107)(9,74)(10,91)(11,76)(12,93)(13,78)(14,95)(15,80)(16,97)(17,82)(18,99)(19,39)(21,41)(23,43)(25,45)(27,47)(29,49)(31,51)(33,53)(35,37)(55,136)(57,138)(59,140)(61,142)(63,144)(65,128)(67,130)(69,132)(71,134)(73,120)(75,122)(77,124)(79,126)(81,110)(83,112)(85,114)(87,116)(89,118)(92,123)(94,125)(96,109)(98,111)(100,113)(102,115)(104,117)(106,119)(108,121), (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), (1,62,113,143)(2,61,114,142)(3,60,115,141)(4,59,116,140)(5,58,117,139)(6,57,118,138)(7,56,119,137)(8,55,120,136)(9,72,121,135)(10,71,122,134)(11,70,123,133)(12,69,124,132)(13,68,125,131)(14,67,126,130)(15,66,109,129)(16,65,110,128)(17,64,111,127)(18,63,112,144)(19,102,39,86)(20,101,40,85)(21,100,41,84)(22,99,42,83)(23,98,43,82)(24,97,44,81)(25,96,45,80)(26,95,46,79)(27,94,47,78)(28,93,48,77)(29,92,49,76)(30,91,50,75)(31,108,51,74)(32,107,52,73)(33,106,53,90)(34,105,54,89)(35,104,37,88)(36,103,38,87)>;
G:=Group( (1,100,113,84)(2,101,114,85)(3,102,115,86)(4,103,116,87)(5,104,117,88)(6,105,118,89)(7,106,119,90)(8,107,120,73)(9,108,121,74)(10,91,122,75)(11,92,123,76)(12,93,124,77)(13,94,125,78)(14,95,126,79)(15,96,109,80)(16,97,110,81)(17,98,111,82)(18,99,112,83)(19,141,39,60)(20,142,40,61)(21,143,41,62)(22,144,42,63)(23,127,43,64)(24,128,44,65)(25,129,45,66)(26,130,46,67)(27,131,47,68)(28,132,48,69)(29,133,49,70)(30,134,50,71)(31,135,51,72)(32,136,52,55)(33,137,53,56)(34,138,54,57)(35,139,37,58)(36,140,38,59), (1,84)(2,101)(3,86)(4,103)(5,88)(6,105)(7,90)(8,107)(9,74)(10,91)(11,76)(12,93)(13,78)(14,95)(15,80)(16,97)(17,82)(18,99)(19,39)(21,41)(23,43)(25,45)(27,47)(29,49)(31,51)(33,53)(35,37)(55,136)(57,138)(59,140)(61,142)(63,144)(65,128)(67,130)(69,132)(71,134)(73,120)(75,122)(77,124)(79,126)(81,110)(83,112)(85,114)(87,116)(89,118)(92,123)(94,125)(96,109)(98,111)(100,113)(102,115)(104,117)(106,119)(108,121), (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), (1,62,113,143)(2,61,114,142)(3,60,115,141)(4,59,116,140)(5,58,117,139)(6,57,118,138)(7,56,119,137)(8,55,120,136)(9,72,121,135)(10,71,122,134)(11,70,123,133)(12,69,124,132)(13,68,125,131)(14,67,126,130)(15,66,109,129)(16,65,110,128)(17,64,111,127)(18,63,112,144)(19,102,39,86)(20,101,40,85)(21,100,41,84)(22,99,42,83)(23,98,43,82)(24,97,44,81)(25,96,45,80)(26,95,46,79)(27,94,47,78)(28,93,48,77)(29,92,49,76)(30,91,50,75)(31,108,51,74)(32,107,52,73)(33,106,53,90)(34,105,54,89)(35,104,37,88)(36,103,38,87) );
G=PermutationGroup([[(1,100,113,84),(2,101,114,85),(3,102,115,86),(4,103,116,87),(5,104,117,88),(6,105,118,89),(7,106,119,90),(8,107,120,73),(9,108,121,74),(10,91,122,75),(11,92,123,76),(12,93,124,77),(13,94,125,78),(14,95,126,79),(15,96,109,80),(16,97,110,81),(17,98,111,82),(18,99,112,83),(19,141,39,60),(20,142,40,61),(21,143,41,62),(22,144,42,63),(23,127,43,64),(24,128,44,65),(25,129,45,66),(26,130,46,67),(27,131,47,68),(28,132,48,69),(29,133,49,70),(30,134,50,71),(31,135,51,72),(32,136,52,55),(33,137,53,56),(34,138,54,57),(35,139,37,58),(36,140,38,59)], [(1,84),(2,101),(3,86),(4,103),(5,88),(6,105),(7,90),(8,107),(9,74),(10,91),(11,76),(12,93),(13,78),(14,95),(15,80),(16,97),(17,82),(18,99),(19,39),(21,41),(23,43),(25,45),(27,47),(29,49),(31,51),(33,53),(35,37),(55,136),(57,138),(59,140),(61,142),(63,144),(65,128),(67,130),(69,132),(71,134),(73,120),(75,122),(77,124),(79,126),(81,110),(83,112),(85,114),(87,116),(89,118),(92,123),(94,125),(96,109),(98,111),(100,113),(102,115),(104,117),(106,119),(108,121)], [(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)], [(1,62,113,143),(2,61,114,142),(3,60,115,141),(4,59,116,140),(5,58,117,139),(6,57,118,138),(7,56,119,137),(8,55,120,136),(9,72,121,135),(10,71,122,134),(11,70,123,133),(12,69,124,132),(13,68,125,131),(14,67,126,130),(15,66,109,129),(16,65,110,128),(17,64,111,127),(18,63,112,144),(19,102,39,86),(20,101,40,85),(21,100,41,84),(22,99,42,83),(23,98,43,82),(24,97,44,81),(25,96,45,80),(26,95,46,79),(27,94,47,78),(28,93,48,77),(29,92,49,76),(30,91,50,75),(31,108,51,74),(32,107,52,73),(33,106,53,90),(34,105,54,89),(35,104,37,88),(36,103,38,87)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 18A | 18B | 18C | 18D | ··· | 18L | 36A | ··· | 36F | 36G | ··· | 36O |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 36 | 36 | 2 | 4 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D9 | C3⋊D4 | C3⋊D4 | D18 | D18 | D18 | C9⋊D4 | C9⋊D4 | C8.C22 | Q8.14D6 | D4.D18 |
| kernel | D4.D18 | C4.Dic9 | D4.D9 | C9⋊Q16 | C2×Dic18 | C9×C4○D4 | C3×C4○D4 | C36 | C2×C18 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 1 | 2 | 6 |
Matrix representation of D4.D18 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 32 | 1 | 41 | 66 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 30 | 42 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 41 | 72 | 32 | 7 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 31 | 28 | 0 | 0 | 0 | 0 |
| 45 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 68 | 66 | 71 | 49 |
| 0 | 0 | 63 | 59 | 17 | 25 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 49 | 5 | 35 | 19 |
| 54 | 25 | 0 | 0 | 0 | 0 |
| 44 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 | 6 | 5 |
| 0 | 0 | 63 | 65 | 27 | 28 |
| 0 | 0 | 6 | 0 | 67 | 68 |
| 0 | 0 | 39 | 40 | 0 | 12 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,72,30,0,0,0,1,0,42,0,0,1,41,0,0,0,0,0,66,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,41,72,0,0,0,0,72,0,0,0,0,72,32,0,0,0,0,0,7,0,1],[31,45,0,0,0,0,28,3,0,0,0,0,0,0,68,63,7,49,0,0,66,59,7,5,0,0,71,17,0,35,0,0,49,25,0,19],[54,44,0,0,0,0,25,19,0,0,0,0,0,0,2,63,6,39,0,0,9,65,0,40,0,0,6,27,67,0,0,0,5,28,68,12] >;
D4.D18 in GAP, Magma, Sage, TeX
D_4.D_{18} % in TeX
G:=Group("D4.D18"); // GroupNames label
G:=SmallGroup(288,159);
// by ID
G=gap.SmallGroup(288,159);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,219,675,185,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^18=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations