metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic9⋊3D4, C23.14D18, (C2×D4)⋊5D9, (C2×C18)⋊3D4, (D4×C18)⋊9C2, C9⋊5(C4⋊D4), C2.27(D4×D9), D18⋊C4⋊15C2, (C6×D4).21S3, (C2×C4).16D18, C18.38(C2×D4), C6.102(S3×D4), Dic9⋊C4⋊15C2, (C2×C12).217D6, C22⋊2(C9⋊D4), (C22×C6).52D6, C18.32(C4○D4), C3.(C23.14D6), (C2×C18).54C23, (C2×C36).61C22, (C22×Dic9)⋊6C2, C18.D4⋊12C2, C2.18(D4⋊2D9), C6.89(D4⋊2S3), C22.61(C22×D9), (C22×C18).21C22, (C2×Dic9).40C22, (C22×D9).11C22, (C2×C9⋊D4)⋊6C2, C6.99(C2×C3⋊D4), C2.15(C2×C9⋊D4), (C2×C6).6(C3⋊D4), (C2×C6).211(C22×S3), SmallGroup(288,149)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic9⋊D4
G = < a,b,c,d | a18=c4=d2=1, b2=a9, bab-1=a-1, ac=ca, ad=da, cbc-1=a9b, bd=db, dcd=c-1 >
Subgroups: 612 in 141 conjugacy classes, 46 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, D9, C18, C18, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4⋊D4, Dic9, Dic9, C36, D18, C2×C18, C2×C18, C2×C18, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C6×D4, C2×Dic9, C2×Dic9, C9⋊D4, C2×C36, D4×C9, C22×D9, C22×C18, C23.14D6, Dic9⋊C4, D18⋊C4, C18.D4, C22×Dic9, C2×C9⋊D4, D4×C18, Dic9⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, C3⋊D4, C22×S3, C4⋊D4, D18, S3×D4, D4⋊2S3, C2×C3⋊D4, C9⋊D4, C22×D9, C23.14D6, D4×D9, D4⋊2D9, C2×C9⋊D4, Dic9⋊D4
►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 70 10 61)(2 69 11 60)(3 68 12 59)(4 67 13 58)(5 66 14 57)(6 65 15 56)(7 64 16 55)(8 63 17 72)(9 62 18 71)(19 50 28 41)(20 49 29 40)(21 48 30 39)(22 47 31 38)(23 46 32 37)(24 45 33 54)(25 44 34 53)(26 43 35 52)(27 42 36 51)(73 134 82 143)(74 133 83 142)(75 132 84 141)(76 131 85 140)(77 130 86 139)(78 129 87 138)(79 128 88 137)(80 127 89 136)(81 144 90 135)(91 111 100 120)(92 110 101 119)(93 109 102 118)(94 126 103 117)(95 125 104 116)(96 124 105 115)(97 123 106 114)(98 122 107 113)(99 121 108 112)
(1 126 28 84)(2 109 29 85)(3 110 30 86)(4 111 31 87)(5 112 32 88)(6 113 33 89)(7 114 34 90)(8 115 35 73)(9 116 36 74)(10 117 19 75)(11 118 20 76)(12 119 21 77)(13 120 22 78)(14 121 23 79)(15 122 24 80)(16 123 25 81)(17 124 26 82)(18 125 27 83)(37 128 66 108)(38 129 67 91)(39 130 68 92)(40 131 69 93)(41 132 70 94)(42 133 71 95)(43 134 72 96)(44 135 55 97)(45 136 56 98)(46 137 57 99)(47 138 58 100)(48 139 59 101)(49 140 60 102)(50 141 61 103)(51 142 62 104)(52 143 63 105)(53 144 64 106)(54 127 65 107)
(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 91)(17 92)(18 93)(19 141)(20 142)(21 143)(22 144)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 109)(72 110)
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), (1,70,10,61)(2,69,11,60)(3,68,12,59)(4,67,13,58)(5,66,14,57)(6,65,15,56)(7,64,16,55)(8,63,17,72)(9,62,18,71)(19,50,28,41)(20,49,29,40)(21,48,30,39)(22,47,31,38)(23,46,32,37)(24,45,33,54)(25,44,34,53)(26,43,35,52)(27,42,36,51)(73,134,82,143)(74,133,83,142)(75,132,84,141)(76,131,85,140)(77,130,86,139)(78,129,87,138)(79,128,88,137)(80,127,89,136)(81,144,90,135)(91,111,100,120)(92,110,101,119)(93,109,102,118)(94,126,103,117)(95,125,104,116)(96,124,105,115)(97,123,106,114)(98,122,107,113)(99,121,108,112), (1,126,28,84)(2,109,29,85)(3,110,30,86)(4,111,31,87)(5,112,32,88)(6,113,33,89)(7,114,34,90)(8,115,35,73)(9,116,36,74)(10,117,19,75)(11,118,20,76)(12,119,21,77)(13,120,22,78)(14,121,23,79)(15,122,24,80)(16,123,25,81)(17,124,26,82)(18,125,27,83)(37,128,66,108)(38,129,67,91)(39,130,68,92)(40,131,69,93)(41,132,70,94)(42,133,71,95)(43,134,72,96)(44,135,55,97)(45,136,56,98)(46,137,57,99)(47,138,58,100)(48,139,59,101)(49,140,60,102)(50,141,61,103)(51,142,62,104)(52,143,63,105)(53,144,64,106)(54,127,65,107), (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)(16,91)(17,92)(18,93)(19,141)(20,142)(21,143)(22,144)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,109)(72,110)>;
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), (1,70,10,61)(2,69,11,60)(3,68,12,59)(4,67,13,58)(5,66,14,57)(6,65,15,56)(7,64,16,55)(8,63,17,72)(9,62,18,71)(19,50,28,41)(20,49,29,40)(21,48,30,39)(22,47,31,38)(23,46,32,37)(24,45,33,54)(25,44,34,53)(26,43,35,52)(27,42,36,51)(73,134,82,143)(74,133,83,142)(75,132,84,141)(76,131,85,140)(77,130,86,139)(78,129,87,138)(79,128,88,137)(80,127,89,136)(81,144,90,135)(91,111,100,120)(92,110,101,119)(93,109,102,118)(94,126,103,117)(95,125,104,116)(96,124,105,115)(97,123,106,114)(98,122,107,113)(99,121,108,112), (1,126,28,84)(2,109,29,85)(3,110,30,86)(4,111,31,87)(5,112,32,88)(6,113,33,89)(7,114,34,90)(8,115,35,73)(9,116,36,74)(10,117,19,75)(11,118,20,76)(12,119,21,77)(13,120,22,78)(14,121,23,79)(15,122,24,80)(16,123,25,81)(17,124,26,82)(18,125,27,83)(37,128,66,108)(38,129,67,91)(39,130,68,92)(40,131,69,93)(41,132,70,94)(42,133,71,95)(43,134,72,96)(44,135,55,97)(45,136,56,98)(46,137,57,99)(47,138,58,100)(48,139,59,101)(49,140,60,102)(50,141,61,103)(51,142,62,104)(52,143,63,105)(53,144,64,106)(54,127,65,107), (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)(16,91)(17,92)(18,93)(19,141)(20,142)(21,143)(22,144)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,109)(72,110) );
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)], [(1,70,10,61),(2,69,11,60),(3,68,12,59),(4,67,13,58),(5,66,14,57),(6,65,15,56),(7,64,16,55),(8,63,17,72),(9,62,18,71),(19,50,28,41),(20,49,29,40),(21,48,30,39),(22,47,31,38),(23,46,32,37),(24,45,33,54),(25,44,34,53),(26,43,35,52),(27,42,36,51),(73,134,82,143),(74,133,83,142),(75,132,84,141),(76,131,85,140),(77,130,86,139),(78,129,87,138),(79,128,88,137),(80,127,89,136),(81,144,90,135),(91,111,100,120),(92,110,101,119),(93,109,102,118),(94,126,103,117),(95,125,104,116),(96,124,105,115),(97,123,106,114),(98,122,107,113),(99,121,108,112)], [(1,126,28,84),(2,109,29,85),(3,110,30,86),(4,111,31,87),(5,112,32,88),(6,113,33,89),(7,114,34,90),(8,115,35,73),(9,116,36,74),(10,117,19,75),(11,118,20,76),(12,119,21,77),(13,120,22,78),(14,121,23,79),(15,122,24,80),(16,123,25,81),(17,124,26,82),(18,125,27,83),(37,128,66,108),(38,129,67,91),(39,130,68,92),(40,131,69,93),(41,132,70,94),(42,133,71,95),(43,134,72,96),(44,135,55,97),(45,136,56,98),(46,137,57,99),(47,138,58,100),(48,139,59,101),(49,140,60,102),(50,141,61,103),(51,142,62,104),(52,143,63,105),(53,144,64,106),(54,127,65,107)], [(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),(16,91),(17,92),(18,93),(19,141),(20,142),(21,143),(22,144),(23,127),(24,128),(25,129),(26,130),(27,131),(28,132),(29,133),(30,134),(31,135),(32,136),(33,137),(34,138),(35,139),(36,140),(37,80),(38,81),(39,82),(40,83),(41,84),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,73),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,111),(56,112),(57,113),(58,114),(59,115),(60,116),(61,117),(62,118),(63,119),(64,120),(65,121),(66,122),(67,123),(68,124),(69,125),(70,126),(71,109),(72,110)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 12A | 12B | 18A | ··· | 18I | 18J | ··· | 18U | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 36 | 2 | 4 | 18 | 18 | 18 | 18 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | D9 | C3⋊D4 | D18 | D18 | C9⋊D4 | S3×D4 | D4⋊2S3 | D4×D9 | D4⋊2D9 |
| kernel | Dic9⋊D4 | Dic9⋊C4 | D18⋊C4 | C18.D4 | C22×Dic9 | C2×C9⋊D4 | D4×C18 | C6×D4 | Dic9 | C2×C18 | C2×C12 | C22×C6 | C18 | C2×D4 | C2×C6 | C2×C4 | C23 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 3 | 4 | 3 | 6 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of Dic9⋊D4 ►in GL4(𝔽37) generated by
| 17 | 31 | 0 | 0 |
| 6 | 11 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 11 | 0 | 0 |
| 13 | 35 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 7 | 23 | 0 | 0 |
| 14 | 30 | 0 | 0 |
| 0 | 0 | 36 | 35 |
| 0 | 0 | 1 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 35 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(37))| [17,6,0,0,31,11,0,0,0,0,1,0,0,0,0,1],[2,13,0,0,11,35,0,0,0,0,36,0,0,0,0,36],[7,14,0,0,23,30,0,0,0,0,36,1,0,0,35,1],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,35,1] >;
Dic9⋊D4 in GAP, Magma, Sage, TeX
{\rm Dic}_9\rtimes D_4 % in TeX
G:=Group("Dic9:D4"); // GroupNames label
G:=SmallGroup(288,149);
// by ID
G=gap.SmallGroup(288,149);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,219,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^18=c^4=d^2=1,b^2=a^9,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations