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 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8, C9, Dic3, C12 [×2], C12, D6, C2×C6, C2×C6, C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, D9, C18, C18 [×2], C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, Dic9, C36 [×2], C36, D18, C2×C18, C2×C18, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, C9⋊C8 [×2], 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 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C3⋊D4 [×2], C22×S3, C4○D8, D18 [×3], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, Q8.13D6, C2×C9⋊D4, D4.9D18
►On 144 points
Generators in S144
(1 47 19 65)(2 48 20 66)(3 49 21 67)(4 50 22 68)(5 51 23 69)(6 52 24 70)(7 53 25 71)(8 54 26 72)(9 55 27 37)(10 56 28 38)(11 57 29 39)(12 58 30 40)(13 59 31 41)(14 60 32 42)(15 61 33 43)(16 62 34 44)(17 63 35 45)(18 64 36 46)(73 143 91 125)(74 144 92 126)(75 109 93 127)(76 110 94 128)(77 111 95 129)(78 112 96 130)(79 113 97 131)(80 114 98 132)(81 115 99 133)(82 116 100 134)(83 117 101 135)(84 118 102 136)(85 119 103 137)(86 120 104 138)(87 121 105 139)(88 122 106 140)(89 123 107 141)(90 124 108 142)
(1 105)(2 106)(3 107)(4 108)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 111)(38 112)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 135)(62 136)(63 137)(64 138)(65 139)(66 140)(67 141)(68 142)(69 143)(70 144)(71 109)(72 110)
(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)(73 134 91 116)(74 115 92 133)(75 132 93 114)(76 113 94 131)(77 130 95 112)(78 111 96 129)(79 128 97 110)(80 109 98 127)(81 126 99 144)(82 143 100 125)(83 124 101 142)(84 141 102 123)(85 122 103 140)(86 139 104 121)(87 120 105 138)(88 137 106 119)(89 118 107 136)(90 135 108 117)
G:=sub<Sym(144)| (1,47,19,65)(2,48,20,66)(3,49,21,67)(4,50,22,68)(5,51,23,69)(6,52,24,70)(7,53,25,71)(8,54,26,72)(9,55,27,37)(10,56,28,38)(11,57,29,39)(12,58,30,40)(13,59,31,41)(14,60,32,42)(15,61,33,43)(16,62,34,44)(17,63,35,45)(18,64,36,46)(73,143,91,125)(74,144,92,126)(75,109,93,127)(76,110,94,128)(77,111,95,129)(78,112,96,130)(79,113,97,131)(80,114,98,132)(81,115,99,133)(82,116,100,134)(83,117,101,135)(84,118,102,136)(85,119,103,137)(86,120,104,138)(87,121,105,139)(88,122,106,140)(89,123,107,141)(90,124,108,142), (1,105)(2,106)(3,107)(4,108)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,109)(72,110), (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)(73,134,91,116)(74,115,92,133)(75,132,93,114)(76,113,94,131)(77,130,95,112)(78,111,96,129)(79,128,97,110)(80,109,98,127)(81,126,99,144)(82,143,100,125)(83,124,101,142)(84,141,102,123)(85,122,103,140)(86,139,104,121)(87,120,105,138)(88,137,106,119)(89,118,107,136)(90,135,108,117)>;
G:=Group( (1,47,19,65)(2,48,20,66)(3,49,21,67)(4,50,22,68)(5,51,23,69)(6,52,24,70)(7,53,25,71)(8,54,26,72)(9,55,27,37)(10,56,28,38)(11,57,29,39)(12,58,30,40)(13,59,31,41)(14,60,32,42)(15,61,33,43)(16,62,34,44)(17,63,35,45)(18,64,36,46)(73,143,91,125)(74,144,92,126)(75,109,93,127)(76,110,94,128)(77,111,95,129)(78,112,96,130)(79,113,97,131)(80,114,98,132)(81,115,99,133)(82,116,100,134)(83,117,101,135)(84,118,102,136)(85,119,103,137)(86,120,104,138)(87,121,105,139)(88,122,106,140)(89,123,107,141)(90,124,108,142), (1,105)(2,106)(3,107)(4,108)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,109)(72,110), (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)(73,134,91,116)(74,115,92,133)(75,132,93,114)(76,113,94,131)(77,130,95,112)(78,111,96,129)(79,128,97,110)(80,109,98,127)(81,126,99,144)(82,143,100,125)(83,124,101,142)(84,141,102,123)(85,122,103,140)(86,139,104,121)(87,120,105,138)(88,137,106,119)(89,118,107,136)(90,135,108,117) );
G=PermutationGroup([(1,47,19,65),(2,48,20,66),(3,49,21,67),(4,50,22,68),(5,51,23,69),(6,52,24,70),(7,53,25,71),(8,54,26,72),(9,55,27,37),(10,56,28,38),(11,57,29,39),(12,58,30,40),(13,59,31,41),(14,60,32,42),(15,61,33,43),(16,62,34,44),(17,63,35,45),(18,64,36,46),(73,143,91,125),(74,144,92,126),(75,109,93,127),(76,110,94,128),(77,111,95,129),(78,112,96,130),(79,113,97,131),(80,114,98,132),(81,115,99,133),(82,116,100,134),(83,117,101,135),(84,118,102,136),(85,119,103,137),(86,120,104,138),(87,121,105,139),(88,122,106,140),(89,123,107,141),(90,124,108,142)], [(1,105),(2,106),(3,107),(4,108),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,81),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,97),(30,98),(31,99),(32,100),(33,101),(34,102),(35,103),(36,104),(37,111),(38,112),(39,113),(40,114),(41,115),(42,116),(43,117),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,129),(56,130),(57,131),(58,132),(59,133),(60,134),(61,135),(62,136),(63,137),(64,138),(65,139),(66,140),(67,141),(68,142),(69,143),(70,144),(71,109),(72,110)], [(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),(73,134,91,116),(74,115,92,133),(75,132,93,114),(76,113,94,131),(77,130,95,112),(78,111,96,129),(79,128,97,110),(80,109,98,127),(81,126,99,144),(82,143,100,125),(83,124,101,142),(84,141,102,123),(85,122,103,140),(86,139,104,121),(87,120,105,138),(88,137,106,119),(89,118,107,136),(90,135,108,117)])
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