metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.9D18, C36.57D4, Q8.14D18, C36.19C23, D36.12C22, Dic18.12C22, D4⋊D9⋊7C2, C4○D4⋊4D9, C9⋊5(C4○D8), D4.D9⋊7C2, C9⋊Q16⋊7C2, Q8⋊2D9⋊7C2, (C3×D4).34D6, C18.61(C2×D4), (C2×C12).70D6, (C2×C18).10D4, (C2×C4).59D18, C9⋊C8.10C22, (C3×Q8).58D6, D36⋊5C2⋊4C2, C4.32(C9⋊D4), C3.(Q8.13D6), (D4×C9).9C22, C4.19(C22×D9), (Q8×C9).9C22, C12.58(C22×S3), (C2×C36).47C22, C22.1(C9⋊D4), C12.128(C3⋊D4), (C2×C9⋊C8)⋊8C2, (C9×C4○D4)⋊2C2, C2.25(C2×C9⋊D4), (C3×C4○D4).13S3, C6.109(C2×C3⋊D4), (C2×C6).9(C3⋊D4), SmallGroup(288,161)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.9D18
G = < a,b,c,d | a4=b2=1, c18=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c17 >
Subgroups: 368 in 93 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C9, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, D9, C18, C18, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, Dic9, C36, C36, D18, C2×C18, C2×C18, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, C9⋊C8, Dic18, C4×D9, D36, C9⋊D4, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, Q8.13D6, C2×C9⋊C8, D4.D9, D4⋊D9, C9⋊Q16, Q8⋊2D9, D36⋊5C2, C9×C4○D4, D4.9D18
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, C4○D8, D18, C2×C3⋊D4, C9⋊D4, C22×D9, Q8.13D6, C2×C9⋊D4, D4.9D18
►On 144 points
Generators in S144
(1 65 19 47)(2 66 20 48)(3 67 21 49)(4 68 22 50)(5 69 23 51)(6 70 24 52)(7 71 25 53)(8 72 26 54)(9 37 27 55)(10 38 28 56)(11 39 29 57)(12 40 30 58)(13 41 31 59)(14 42 32 60)(15 43 33 61)(16 44 34 62)(17 45 35 63)(18 46 36 64)(73 114 91 132)(74 115 92 133)(75 116 93 134)(76 117 94 135)(77 118 95 136)(78 119 96 137)(79 120 97 138)(80 121 98 139)(81 122 99 140)(82 123 100 141)(83 124 101 142)(84 125 102 143)(85 126 103 144)(86 127 104 109)(87 128 105 110)(88 129 106 111)(89 130 107 112)(90 131 108 113)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 73)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 82)(26 83)(27 84)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 125)(38 126)(39 127)(40 128)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 143)(56 144)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 121)(70 122)(71 123)(72 124)
(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 18 19 36)(2 35 20 17)(3 16 21 34)(4 33 22 15)(5 14 23 32)(6 31 24 13)(7 12 25 30)(8 29 26 11)(9 10 27 28)(37 56 55 38)(39 54 57 72)(40 71 58 53)(41 52 59 70)(42 69 60 51)(43 50 61 68)(44 67 62 49)(45 48 63 66)(46 65 64 47)(73 137 91 119)(74 118 92 136)(75 135 93 117)(76 116 94 134)(77 133 95 115)(78 114 96 132)(79 131 97 113)(80 112 98 130)(81 129 99 111)(82 110 100 128)(83 127 101 109)(84 144 102 126)(85 125 103 143)(86 142 104 124)(87 123 105 141)(88 140 106 122)(89 121 107 139)(90 138 108 120)
G:=sub<Sym(144)| (1,65,19,47)(2,66,20,48)(3,67,21,49)(4,68,22,50)(5,69,23,51)(6,70,24,52)(7,71,25,53)(8,72,26,54)(9,37,27,55)(10,38,28,56)(11,39,29,57)(12,40,30,58)(13,41,31,59)(14,42,32,60)(15,43,33,61)(16,44,34,62)(17,45,35,63)(18,46,36,64)(73,114,91,132)(74,115,92,133)(75,116,93,134)(76,117,94,135)(77,118,95,136)(78,119,96,137)(79,120,97,138)(80,121,98,139)(81,122,99,140)(82,123,100,141)(83,124,101,142)(84,125,102,143)(85,126,103,144)(86,127,104,109)(87,128,105,110)(88,129,106,111)(89,130,107,112)(90,131,108,113), (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,73)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,81)(25,82)(26,83)(27,84)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122)(71,123)(72,124), (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), (1,18,19,36)(2,35,20,17)(3,16,21,34)(4,33,22,15)(5,14,23,32)(6,31,24,13)(7,12,25,30)(8,29,26,11)(9,10,27,28)(37,56,55,38)(39,54,57,72)(40,71,58,53)(41,52,59,70)(42,69,60,51)(43,50,61,68)(44,67,62,49)(45,48,63,66)(46,65,64,47)(73,137,91,119)(74,118,92,136)(75,135,93,117)(76,116,94,134)(77,133,95,115)(78,114,96,132)(79,131,97,113)(80,112,98,130)(81,129,99,111)(82,110,100,128)(83,127,101,109)(84,144,102,126)(85,125,103,143)(86,142,104,124)(87,123,105,141)(88,140,106,122)(89,121,107,139)(90,138,108,120)>;
G:=Group( (1,65,19,47)(2,66,20,48)(3,67,21,49)(4,68,22,50)(5,69,23,51)(6,70,24,52)(7,71,25,53)(8,72,26,54)(9,37,27,55)(10,38,28,56)(11,39,29,57)(12,40,30,58)(13,41,31,59)(14,42,32,60)(15,43,33,61)(16,44,34,62)(17,45,35,63)(18,46,36,64)(73,114,91,132)(74,115,92,133)(75,116,93,134)(76,117,94,135)(77,118,95,136)(78,119,96,137)(79,120,97,138)(80,121,98,139)(81,122,99,140)(82,123,100,141)(83,124,101,142)(84,125,102,143)(85,126,103,144)(86,127,104,109)(87,128,105,110)(88,129,106,111)(89,130,107,112)(90,131,108,113), (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,73)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,81)(25,82)(26,83)(27,84)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122)(71,123)(72,124), (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), (1,18,19,36)(2,35,20,17)(3,16,21,34)(4,33,22,15)(5,14,23,32)(6,31,24,13)(7,12,25,30)(8,29,26,11)(9,10,27,28)(37,56,55,38)(39,54,57,72)(40,71,58,53)(41,52,59,70)(42,69,60,51)(43,50,61,68)(44,67,62,49)(45,48,63,66)(46,65,64,47)(73,137,91,119)(74,118,92,136)(75,135,93,117)(76,116,94,134)(77,133,95,115)(78,114,96,132)(79,131,97,113)(80,112,98,130)(81,129,99,111)(82,110,100,128)(83,127,101,109)(84,144,102,126)(85,125,103,143)(86,142,104,124)(87,123,105,141)(88,140,106,122)(89,121,107,139)(90,138,108,120) );
G=PermutationGroup([[(1,65,19,47),(2,66,20,48),(3,67,21,49),(4,68,22,50),(5,69,23,51),(6,70,24,52),(7,71,25,53),(8,72,26,54),(9,37,27,55),(10,38,28,56),(11,39,29,57),(12,40,30,58),(13,41,31,59),(14,42,32,60),(15,43,33,61),(16,44,34,62),(17,45,35,63),(18,46,36,64),(73,114,91,132),(74,115,92,133),(75,116,93,134),(76,117,94,135),(77,118,95,136),(78,119,96,137),(79,120,97,138),(80,121,98,139),(81,122,99,140),(82,123,100,141),(83,124,101,142),(84,125,102,143),(85,126,103,144),(86,127,104,109),(87,128,105,110),(88,129,106,111),(89,130,107,112),(90,131,108,113)], [(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,100),(8,101),(9,102),(10,103),(11,104),(12,105),(13,106),(14,107),(15,108),(16,73),(17,74),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,81),(25,82),(26,83),(27,84),(28,85),(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,125),(38,126),(39,127),(40,128),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,141),(54,142),(55,143),(56,144),(57,109),(58,110),(59,111),(60,112),(61,113),(62,114),(63,115),(64,116),(65,117),(66,118),(67,119),(68,120),(69,121),(70,122),(71,123),(72,124)], [(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)], [(1,18,19,36),(2,35,20,17),(3,16,21,34),(4,33,22,15),(5,14,23,32),(6,31,24,13),(7,12,25,30),(8,29,26,11),(9,10,27,28),(37,56,55,38),(39,54,57,72),(40,71,58,53),(41,52,59,70),(42,69,60,51),(43,50,61,68),(44,67,62,49),(45,48,63,66),(46,65,64,47),(73,137,91,119),(74,118,92,136),(75,135,93,117),(76,116,94,134),(77,133,95,115),(78,114,96,132),(79,131,97,113),(80,112,98,130),(81,129,99,111),(82,110,100,128),(83,127,101,109),(84,144,102,126),(85,125,103,143),(86,142,104,124),(87,123,105,141),(88,140,106,122),(89,121,107,139),(90,138,108,120)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 18A | 18B | 18C | 18D | ··· | 18L | 36A | ··· | 36F | 36G | ··· | 36O |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 4 | 36 | 2 | 1 | 1 | 2 | 4 | 36 | 2 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D9 | C3⋊D4 | C3⋊D4 | C4○D8 | D18 | D18 | D18 | C9⋊D4 | C9⋊D4 | Q8.13D6 | D4.9D18 |
| kernel | D4.9D18 | C2×C9⋊C8 | D4.D9 | D4⋊D9 | C9⋊Q16 | Q8⋊2D9 | D36⋊5C2 | C9×C4○D4 | C3×C4○D4 | C36 | C2×C18 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C12 | C2×C6 | C9 | C2×C4 | D4 | Q8 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 4 | 3 | 3 | 3 | 6 | 6 | 2 | 6 |
Matrix representation of D4.9D18 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 72 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 32 | 32 |
| 0 | 0 | 57 | 41 |
| 70 | 42 | 0 | 0 |
| 31 | 28 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 45 | 42 | 0 | 0 |
| 70 | 28 | 0 | 0 |
| 0 | 0 | 27 | 54 |
| 0 | 0 | 0 | 46 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,72,0,0,2,72],[72,0,0,0,0,72,0,0,0,0,32,57,0,0,32,41],[70,31,0,0,42,28,0,0,0,0,46,0,0,0,0,46],[45,70,0,0,42,28,0,0,0,0,27,0,0,0,54,46] >;
D4.9D18 in GAP, Magma, Sage, TeX
D_4._9D_{18} % in TeX
G:=Group("D4.9D18"); // GroupNames label
G:=SmallGroup(288,161);
// by ID
G=gap.SmallGroup(288,161);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,675,185,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^18=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^17>;
// generators/relations