direct product, metabelian, supersoluble, monomial
Aliases: D9×Dic6, D18.8D6, C36.21D6, C12.34D18, Dic9.7D6, Dic3.1D18, (C3×D9)⋊Q8, C3⋊2(Q8×D9), C12.13S32, C4.5(S3×D9), (C4×D9).1S3, C9⋊1(C2×Dic6), (Dic3×D9).C2, (C9×Dic6)⋊3C2, (C12×D9).1C2, (C3×C12).89D6, C3.1(S3×Dic6), C12.D9⋊7C2, C6.1(C22×D9), C9⋊Dic6⋊3C2, C32.2(S3×Q8), (C3×C18).1C23, C18.1(C22×S3), (C3×Dic3).1D6, (C3×Dic6).5S3, (C6×D9).5C22, (C3×C36).24C22, C9⋊Dic3.1C22, (C3×Dic9).7C22, (C9×Dic3).1C22, (C3×C9)⋊1(C2×Q8), C2.5(C2×S3×D9), C6.20(C2×S32), (C3×C6).69(C22×S3), SmallGroup(432,280)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D9×Dic6
G = < a,b,c,d | a9=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 684 in 126 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×Q8, D9, C18, C18, C3×S3, C3×C6, Dic6, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C3×C9, Dic9, Dic9, C36, C36, D18, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×Dic6, S3×Q8, C3×D9, C3×C18, Dic18, C4×D9, C4×D9, Q8×C9, S3×Dic3, C32⋊2Q8, C3×Dic6, S3×C12, C32⋊4Q8, C3×Dic9, C9×Dic3, C9⋊Dic3, C3×C36, C6×D9, Q8×D9, S3×Dic6, C9⋊Dic6, Dic3×D9, C9×Dic6, C12×D9, C12.D9, D9×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, Dic6, C22×S3, D18, S32, C2×Dic6, S3×Q8, C22×D9, C2×S32, S3×D9, Q8×D9, S3×Dic6, C2×S3×D9, D9×Dic6
►On 144 points
Generators in S144
(1 40 67 5 44 71 9 48 63)(2 41 68 6 45 72 10 37 64)(3 42 69 7 46 61 11 38 65)(4 43 70 8 47 62 12 39 66)(13 50 30 21 58 26 17 54 34)(14 51 31 22 59 27 18 55 35)(15 52 32 23 60 28 19 56 36)(16 53 33 24 49 29 20 57 25)(73 132 96 77 124 88 81 128 92)(74 121 85 78 125 89 82 129 93)(75 122 86 79 126 90 83 130 94)(76 123 87 80 127 91 84 131 95)(97 113 140 105 109 136 101 117 144)(98 114 141 106 110 137 102 118 133)(99 115 142 107 111 138 103 119 134)(100 116 143 108 112 139 104 120 135)
(1 138)(2 139)(3 140)(4 141)(5 142)(6 143)(7 144)(8 133)(9 134)(10 135)(11 136)(12 137)(13 84)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 122)(26 123)(27 124)(28 125)(29 126)(30 127)(31 128)(32 129)(33 130)(34 131)(35 132)(36 121)(37 120)(38 109)(39 110)(40 111)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 85)(57 86)(58 87)(59 88)(60 89)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 97)(70 98)(71 99)(72 100)
(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 24 7 18)(2 23 8 17)(3 22 9 16)(4 21 10 15)(5 20 11 14)(6 19 12 13)(25 65 31 71)(26 64 32 70)(27 63 33 69)(28 62 34 68)(29 61 35 67)(30 72 36 66)(37 52 43 58)(38 51 44 57)(39 50 45 56)(40 49 46 55)(41 60 47 54)(42 59 48 53)(73 142 79 136)(74 141 80 135)(75 140 81 134)(76 139 82 133)(77 138 83 144)(78 137 84 143)(85 110 91 116)(86 109 92 115)(87 120 93 114)(88 119 94 113)(89 118 95 112)(90 117 96 111)(97 124 103 130)(98 123 104 129)(99 122 105 128)(100 121 106 127)(101 132 107 126)(102 131 108 125)
G:=sub<Sym(144)| (1,40,67,5,44,71,9,48,63)(2,41,68,6,45,72,10,37,64)(3,42,69,7,46,61,11,38,65)(4,43,70,8,47,62,12,39,66)(13,50,30,21,58,26,17,54,34)(14,51,31,22,59,27,18,55,35)(15,52,32,23,60,28,19,56,36)(16,53,33,24,49,29,20,57,25)(73,132,96,77,124,88,81,128,92)(74,121,85,78,125,89,82,129,93)(75,122,86,79,126,90,83,130,94)(76,123,87,80,127,91,84,131,95)(97,113,140,105,109,136,101,117,144)(98,114,141,106,110,137,102,118,133)(99,115,142,107,111,138,103,119,134)(100,116,143,108,112,139,104,120,135), (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,133)(9,134)(10,135)(11,136)(12,137)(13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,121)(37,120)(38,109)(39,110)(40,111)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,85)(57,86)(58,87)(59,88)(60,89)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,97)(70,98)(71,99)(72,100), (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), (1,24,7,18)(2,23,8,17)(3,22,9,16)(4,21,10,15)(5,20,11,14)(6,19,12,13)(25,65,31,71)(26,64,32,70)(27,63,33,69)(28,62,34,68)(29,61,35,67)(30,72,36,66)(37,52,43,58)(38,51,44,57)(39,50,45,56)(40,49,46,55)(41,60,47,54)(42,59,48,53)(73,142,79,136)(74,141,80,135)(75,140,81,134)(76,139,82,133)(77,138,83,144)(78,137,84,143)(85,110,91,116)(86,109,92,115)(87,120,93,114)(88,119,94,113)(89,118,95,112)(90,117,96,111)(97,124,103,130)(98,123,104,129)(99,122,105,128)(100,121,106,127)(101,132,107,126)(102,131,108,125)>;
G:=Group( (1,40,67,5,44,71,9,48,63)(2,41,68,6,45,72,10,37,64)(3,42,69,7,46,61,11,38,65)(4,43,70,8,47,62,12,39,66)(13,50,30,21,58,26,17,54,34)(14,51,31,22,59,27,18,55,35)(15,52,32,23,60,28,19,56,36)(16,53,33,24,49,29,20,57,25)(73,132,96,77,124,88,81,128,92)(74,121,85,78,125,89,82,129,93)(75,122,86,79,126,90,83,130,94)(76,123,87,80,127,91,84,131,95)(97,113,140,105,109,136,101,117,144)(98,114,141,106,110,137,102,118,133)(99,115,142,107,111,138,103,119,134)(100,116,143,108,112,139,104,120,135), (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,133)(9,134)(10,135)(11,136)(12,137)(13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,121)(37,120)(38,109)(39,110)(40,111)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,85)(57,86)(58,87)(59,88)(60,89)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,97)(70,98)(71,99)(72,100), (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), (1,24,7,18)(2,23,8,17)(3,22,9,16)(4,21,10,15)(5,20,11,14)(6,19,12,13)(25,65,31,71)(26,64,32,70)(27,63,33,69)(28,62,34,68)(29,61,35,67)(30,72,36,66)(37,52,43,58)(38,51,44,57)(39,50,45,56)(40,49,46,55)(41,60,47,54)(42,59,48,53)(73,142,79,136)(74,141,80,135)(75,140,81,134)(76,139,82,133)(77,138,83,144)(78,137,84,143)(85,110,91,116)(86,109,92,115)(87,120,93,114)(88,119,94,113)(89,118,95,112)(90,117,96,111)(97,124,103,130)(98,123,104,129)(99,122,105,128)(100,121,106,127)(101,132,107,126)(102,131,108,125) );
G=PermutationGroup([[(1,40,67,5,44,71,9,48,63),(2,41,68,6,45,72,10,37,64),(3,42,69,7,46,61,11,38,65),(4,43,70,8,47,62,12,39,66),(13,50,30,21,58,26,17,54,34),(14,51,31,22,59,27,18,55,35),(15,52,32,23,60,28,19,56,36),(16,53,33,24,49,29,20,57,25),(73,132,96,77,124,88,81,128,92),(74,121,85,78,125,89,82,129,93),(75,122,86,79,126,90,83,130,94),(76,123,87,80,127,91,84,131,95),(97,113,140,105,109,136,101,117,144),(98,114,141,106,110,137,102,118,133),(99,115,142,107,111,138,103,119,134),(100,116,143,108,112,139,104,120,135)], [(1,138),(2,139),(3,140),(4,141),(5,142),(6,143),(7,144),(8,133),(9,134),(10,135),(11,136),(12,137),(13,84),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,122),(26,123),(27,124),(28,125),(29,126),(30,127),(31,128),(32,129),(33,130),(34,131),(35,132),(36,121),(37,120),(38,109),(39,110),(40,111),(41,112),(42,113),(43,114),(44,115),(45,116),(46,117),(47,118),(48,119),(49,90),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,85),(57,86),(58,87),(59,88),(60,89),(61,101),(62,102),(63,103),(64,104),(65,105),(66,106),(67,107),(68,108),(69,97),(70,98),(71,99),(72,100)], [(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)], [(1,24,7,18),(2,23,8,17),(3,22,9,16),(4,21,10,15),(5,20,11,14),(6,19,12,13),(25,65,31,71),(26,64,32,70),(27,63,33,69),(28,62,34,68),(29,61,35,67),(30,72,36,66),(37,52,43,58),(38,51,44,57),(39,50,45,56),(40,49,46,55),(41,60,47,54),(42,59,48,53),(73,142,79,136),(74,141,80,135),(75,140,81,134),(76,139,82,133),(77,138,83,144),(78,137,84,143),(85,110,91,116),(86,109,92,115),(87,120,93,114),(88,119,94,113),(89,118,95,112),(90,117,96,111),(97,124,103,130),(98,123,104,129),(99,122,105,128),(100,121,106,127),(101,132,107,126),(102,131,108,125)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 18A | 18B | 18C | 18D | 18E | 18F | 36A | ··· | 36I | 36J | ··· | 36O |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 2 | 6 | 6 | 18 | 54 | 54 | 2 | 2 | 4 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | + | + | - | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | Q8 | D6 | D6 | D6 | D6 | D6 | D9 | Dic6 | D18 | D18 | S32 | S3×Q8 | C2×S32 | S3×D9 | Q8×D9 | S3×Dic6 | C2×S3×D9 | D9×Dic6 |
| kernel | D9×Dic6 | C9⋊Dic6 | Dic3×D9 | C9×Dic6 | C12×D9 | C12.D9 | C4×D9 | C3×Dic6 | C3×D9 | Dic9 | C36 | D18 | C3×Dic3 | C3×C12 | Dic6 | D9 | Dic3 | C12 | C12 | C32 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 3 | 4 | 6 | 3 | 1 | 1 | 1 | 3 | 3 | 2 | 3 | 6 |
Matrix representation of D9×Dic6 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 26 | 31 |
| 0 | 0 | 6 | 20 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 31 | 17 |
| 0 | 0 | 11 | 6 |
| 32 | 5 | 0 | 0 |
| 32 | 27 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 35 | 11 | 0 | 0 |
| 13 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,26,6,0,0,31,20],[36,0,0,0,0,36,0,0,0,0,31,11,0,0,17,6],[32,32,0,0,5,27,0,0,0,0,36,0,0,0,0,36],[35,13,0,0,11,2,0,0,0,0,1,0,0,0,0,1] >;
D9×Dic6 in GAP, Magma, Sage, TeX
D_9\times {\rm Dic}_6 % in TeX
G:=Group("D9xDic6"); // GroupNames label
G:=SmallGroup(432,280);
// by ID
G=gap.SmallGroup(432,280);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,254,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations