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 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×2], C22 [×8], S3, C6 [×3], C6 [×3], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, C9, Dic3 [×4], C12, D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], D9, C18 [×3], C18 [×3], C2×Dic3 [×5], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], C4⋊D4, Dic9 [×2], Dic9 [×2], C36, D18 [×3], C2×C18, C2×C18 [×2], C2×C18 [×5], Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4 [×2], C6×D4, C2×Dic9 [×3], C2×Dic9 [×2], C9⋊D4 [×4], C2×C36, D4×C9 [×2], C22×D9, C22×C18 [×2], C23.14D6, Dic9⋊C4, D18⋊C4, C18.D4, C22×Dic9, C2×C9⋊D4 [×2], D4×C18, Dic9⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D9, C3⋊D4 [×2], C22×S3, C4⋊D4, D18 [×3], S3×D4, D4⋊2S3, C2×C3⋊D4, C9⋊D4 [×2], 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 33 10 24)(2 32 11 23)(3 31 12 22)(4 30 13 21)(5 29 14 20)(6 28 15 19)(7 27 16 36)(8 26 17 35)(9 25 18 34)(37 89 46 80)(38 88 47 79)(39 87 48 78)(40 86 49 77)(41 85 50 76)(42 84 51 75)(43 83 52 74)(44 82 53 73)(45 81 54 90)(55 98 64 107)(56 97 65 106)(57 96 66 105)(58 95 67 104)(59 94 68 103)(60 93 69 102)(61 92 70 101)(62 91 71 100)(63 108 72 99)(109 139 118 130)(110 138 119 129)(111 137 120 128)(112 136 121 127)(113 135 122 144)(114 134 123 143)(115 133 124 142)(116 132 125 141)(117 131 126 140)
(1 63 122 85)(2 64 123 86)(3 65 124 87)(4 66 125 88)(5 67 126 89)(6 68 109 90)(7 69 110 73)(8 70 111 74)(9 71 112 75)(10 72 113 76)(11 55 114 77)(12 56 115 78)(13 57 116 79)(14 58 117 80)(15 59 118 81)(16 60 119 82)(17 61 120 83)(18 62 121 84)(19 103 130 45)(20 104 131 46)(21 105 132 47)(22 106 133 48)(23 107 134 49)(24 108 135 50)(25 91 136 51)(26 92 137 52)(27 93 138 53)(28 94 139 54)(29 95 140 37)(30 96 141 38)(31 97 142 39)(32 98 143 40)(33 99 144 41)(34 100 127 42)(35 101 128 43)(36 102 129 44)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 105)(8 106)(9 107)(10 108)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(73 132)(74 133)(75 134)(76 135)(77 136)(78 137)(79 138)(80 139)(81 140)(82 141)(83 142)(84 143)(85 144)(86 127)(87 128)(88 129)(89 130)(90 131)
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), (1,33,10,24)(2,32,11,23)(3,31,12,22)(4,30,13,21)(5,29,14,20)(6,28,15,19)(7,27,16,36)(8,26,17,35)(9,25,18,34)(37,89,46,80)(38,88,47,79)(39,87,48,78)(40,86,49,77)(41,85,50,76)(42,84,51,75)(43,83,52,74)(44,82,53,73)(45,81,54,90)(55,98,64,107)(56,97,65,106)(57,96,66,105)(58,95,67,104)(59,94,68,103)(60,93,69,102)(61,92,70,101)(62,91,71,100)(63,108,72,99)(109,139,118,130)(110,138,119,129)(111,137,120,128)(112,136,121,127)(113,135,122,144)(114,134,123,143)(115,133,124,142)(116,132,125,141)(117,131,126,140), (1,63,122,85)(2,64,123,86)(3,65,124,87)(4,66,125,88)(5,67,126,89)(6,68,109,90)(7,69,110,73)(8,70,111,74)(9,71,112,75)(10,72,113,76)(11,55,114,77)(12,56,115,78)(13,57,116,79)(14,58,117,80)(15,59,118,81)(16,60,119,82)(17,61,120,83)(18,62,121,84)(19,103,130,45)(20,104,131,46)(21,105,132,47)(22,106,133,48)(23,107,134,49)(24,108,135,50)(25,91,136,51)(26,92,137,52)(27,93,138,53)(28,94,139,54)(29,95,140,37)(30,96,141,38)(31,97,142,39)(32,98,143,40)(33,99,144,41)(34,100,127,42)(35,101,128,43)(36,102,129,44), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,105)(8,106)(9,107)(10,108)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,127)(87,128)(88,129)(89,130)(90,131)>;
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), (1,33,10,24)(2,32,11,23)(3,31,12,22)(4,30,13,21)(5,29,14,20)(6,28,15,19)(7,27,16,36)(8,26,17,35)(9,25,18,34)(37,89,46,80)(38,88,47,79)(39,87,48,78)(40,86,49,77)(41,85,50,76)(42,84,51,75)(43,83,52,74)(44,82,53,73)(45,81,54,90)(55,98,64,107)(56,97,65,106)(57,96,66,105)(58,95,67,104)(59,94,68,103)(60,93,69,102)(61,92,70,101)(62,91,71,100)(63,108,72,99)(109,139,118,130)(110,138,119,129)(111,137,120,128)(112,136,121,127)(113,135,122,144)(114,134,123,143)(115,133,124,142)(116,132,125,141)(117,131,126,140), (1,63,122,85)(2,64,123,86)(3,65,124,87)(4,66,125,88)(5,67,126,89)(6,68,109,90)(7,69,110,73)(8,70,111,74)(9,71,112,75)(10,72,113,76)(11,55,114,77)(12,56,115,78)(13,57,116,79)(14,58,117,80)(15,59,118,81)(16,60,119,82)(17,61,120,83)(18,62,121,84)(19,103,130,45)(20,104,131,46)(21,105,132,47)(22,106,133,48)(23,107,134,49)(24,108,135,50)(25,91,136,51)(26,92,137,52)(27,93,138,53)(28,94,139,54)(29,95,140,37)(30,96,141,38)(31,97,142,39)(32,98,143,40)(33,99,144,41)(34,100,127,42)(35,101,128,43)(36,102,129,44), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,105)(8,106)(9,107)(10,108)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,127)(87,128)(88,129)(89,130)(90,131) );
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)], [(1,33,10,24),(2,32,11,23),(3,31,12,22),(4,30,13,21),(5,29,14,20),(6,28,15,19),(7,27,16,36),(8,26,17,35),(9,25,18,34),(37,89,46,80),(38,88,47,79),(39,87,48,78),(40,86,49,77),(41,85,50,76),(42,84,51,75),(43,83,52,74),(44,82,53,73),(45,81,54,90),(55,98,64,107),(56,97,65,106),(57,96,66,105),(58,95,67,104),(59,94,68,103),(60,93,69,102),(61,92,70,101),(62,91,71,100),(63,108,72,99),(109,139,118,130),(110,138,119,129),(111,137,120,128),(112,136,121,127),(113,135,122,144),(114,134,123,143),(115,133,124,142),(116,132,125,141),(117,131,126,140)], [(1,63,122,85),(2,64,123,86),(3,65,124,87),(4,66,125,88),(5,67,126,89),(6,68,109,90),(7,69,110,73),(8,70,111,74),(9,71,112,75),(10,72,113,76),(11,55,114,77),(12,56,115,78),(13,57,116,79),(14,58,117,80),(15,59,118,81),(16,60,119,82),(17,61,120,83),(18,62,121,84),(19,103,130,45),(20,104,131,46),(21,105,132,47),(22,106,133,48),(23,107,134,49),(24,108,135,50),(25,91,136,51),(26,92,137,52),(27,93,138,53),(28,94,139,54),(29,95,140,37),(30,96,141,38),(31,97,142,39),(32,98,143,40),(33,99,144,41),(34,100,127,42),(35,101,128,43),(36,102,129,44)], [(1,99),(2,100),(3,101),(4,102),(5,103),(6,104),(7,105),(8,106),(9,107),(10,108),(11,91),(12,92),(13,93),(14,94),(15,95),(16,96),(17,97),(18,98),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,118),(38,119),(39,120),(40,121),(41,122),(42,123),(43,124),(44,125),(45,126),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(73,132),(74,133),(75,134),(76,135),(77,136),(78,137),(79,138),(80,139),(81,140),(82,141),(83,142),(84,143),(85,144),(86,127),(87,128),(88,129),(89,130),(90,131)])
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