metabelian, supersoluble, monomial
Aliases: Dic6⋊2D9, D36.2S3, C36.13D6, C12.15D18, C12.5S32, (C3×C9)⋊5SD16, C4.17(S3×D9), (C3×C18).9D4, C9⋊2(D4.S3), (C9×Dic6)⋊1C2, (C3×D36).3C2, (C3×C12).77D6, C36.S3⋊3C2, C3⋊2(Q8⋊2D9), C6.10(C9⋊D4), C18.9(C3⋊D4), (C3×Dic6).1S3, C2.6(D6⋊D9), (C3×C36).12C22, C6.14(D6⋊S3), C3.2(Dic6⋊S3), C32.2(Q8⋊2S3), (C3×C6).45(C3⋊D4), SmallGroup(432,72)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊D9
G = < a,b,c,d | a12=c9=d2=1, b2=a6, bab-1=a-1, ac=ca, dad=a7, bc=cb, dbd=a3b, dcd=c-1 >
Subgroups: 376 in 68 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, SD16, D9, C18, C18, C3×S3, C3×C6, C3⋊C8, Dic6, D12, C3×D4, C3×Q8, C3×C9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, D4.S3, Q8⋊2S3, C3×D9, C3×C18, C9⋊C8, D36, Q8×C9, C32⋊4C8, C3×Dic6, C3×D12, C9×Dic3, C3×C36, C6×D9, Q8⋊2D9, Dic6⋊S3, C36.S3, C9×Dic6, C3×D36, Dic6⋊D9
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, C3⋊D4, D18, S32, D4.S3, Q8⋊2S3, C9⋊D4, D6⋊S3, S3×D9, Q8⋊2D9, Dic6⋊S3, D6⋊D9, Dic6⋊D9
►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 84 7 78)(2 83 8 77)(3 82 9 76)(4 81 10 75)(5 80 11 74)(6 79 12 73)(13 129 19 123)(14 128 20 122)(15 127 21 121)(16 126 22 132)(17 125 23 131)(18 124 24 130)(25 54 31 60)(26 53 32 59)(27 52 33 58)(28 51 34 57)(29 50 35 56)(30 49 36 55)(37 102 43 108)(38 101 44 107)(39 100 45 106)(40 99 46 105)(41 98 47 104)(42 97 48 103)(61 142 67 136)(62 141 68 135)(63 140 69 134)(64 139 70 133)(65 138 71 144)(66 137 72 143)(85 116 91 110)(86 115 92 109)(87 114 93 120)(88 113 94 119)(89 112 95 118)(90 111 96 117)
(1 59 21 5 51 13 9 55 17)(2 60 22 6 52 14 10 56 18)(3 49 23 7 53 15 11 57 19)(4 50 24 8 54 16 12 58 20)(25 132 79 33 128 75 29 124 83)(26 121 80 34 129 76 30 125 84)(27 122 81 35 130 77 31 126 73)(28 123 82 36 131 78 32 127 74)(37 71 120 41 63 112 45 67 116)(38 72 109 42 64 113 46 68 117)(39 61 110 43 65 114 47 69 118)(40 62 111 44 66 115 48 70 119)(85 108 138 93 104 134 89 100 142)(86 97 139 94 105 135 90 101 143)(87 98 140 95 106 136 91 102 144)(88 99 141 96 107 137 92 103 133)
(1 92)(2 87)(3 94)(4 89)(5 96)(6 91)(7 86)(8 93)(9 88)(10 95)(11 90)(12 85)(13 99)(14 106)(15 101)(16 108)(17 103)(18 98)(19 105)(20 100)(21 107)(22 102)(23 97)(24 104)(25 62)(26 69)(27 64)(28 71)(29 66)(30 61)(31 68)(32 63)(33 70)(34 65)(35 72)(36 67)(37 123)(38 130)(39 125)(40 132)(41 127)(42 122)(43 129)(44 124)(45 131)(46 126)(47 121)(48 128)(49 139)(50 134)(51 141)(52 136)(53 143)(54 138)(55 133)(56 140)(57 135)(58 142)(59 137)(60 144)(73 113)(74 120)(75 115)(76 110)(77 117)(78 112)(79 119)(80 114)(81 109)(82 116)(83 111)(84 118)
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), (1,84,7,78)(2,83,8,77)(3,82,9,76)(4,81,10,75)(5,80,11,74)(6,79,12,73)(13,129,19,123)(14,128,20,122)(15,127,21,121)(16,126,22,132)(17,125,23,131)(18,124,24,130)(25,54,31,60)(26,53,32,59)(27,52,33,58)(28,51,34,57)(29,50,35,56)(30,49,36,55)(37,102,43,108)(38,101,44,107)(39,100,45,106)(40,99,46,105)(41,98,47,104)(42,97,48,103)(61,142,67,136)(62,141,68,135)(63,140,69,134)(64,139,70,133)(65,138,71,144)(66,137,72,143)(85,116,91,110)(86,115,92,109)(87,114,93,120)(88,113,94,119)(89,112,95,118)(90,111,96,117), (1,59,21,5,51,13,9,55,17)(2,60,22,6,52,14,10,56,18)(3,49,23,7,53,15,11,57,19)(4,50,24,8,54,16,12,58,20)(25,132,79,33,128,75,29,124,83)(26,121,80,34,129,76,30,125,84)(27,122,81,35,130,77,31,126,73)(28,123,82,36,131,78,32,127,74)(37,71,120,41,63,112,45,67,116)(38,72,109,42,64,113,46,68,117)(39,61,110,43,65,114,47,69,118)(40,62,111,44,66,115,48,70,119)(85,108,138,93,104,134,89,100,142)(86,97,139,94,105,135,90,101,143)(87,98,140,95,106,136,91,102,144)(88,99,141,96,107,137,92,103,133), (1,92)(2,87)(3,94)(4,89)(5,96)(6,91)(7,86)(8,93)(9,88)(10,95)(11,90)(12,85)(13,99)(14,106)(15,101)(16,108)(17,103)(18,98)(19,105)(20,100)(21,107)(22,102)(23,97)(24,104)(25,62)(26,69)(27,64)(28,71)(29,66)(30,61)(31,68)(32,63)(33,70)(34,65)(35,72)(36,67)(37,123)(38,130)(39,125)(40,132)(41,127)(42,122)(43,129)(44,124)(45,131)(46,126)(47,121)(48,128)(49,139)(50,134)(51,141)(52,136)(53,143)(54,138)(55,133)(56,140)(57,135)(58,142)(59,137)(60,144)(73,113)(74,120)(75,115)(76,110)(77,117)(78,112)(79,119)(80,114)(81,109)(82,116)(83,111)(84,118)>;
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), (1,84,7,78)(2,83,8,77)(3,82,9,76)(4,81,10,75)(5,80,11,74)(6,79,12,73)(13,129,19,123)(14,128,20,122)(15,127,21,121)(16,126,22,132)(17,125,23,131)(18,124,24,130)(25,54,31,60)(26,53,32,59)(27,52,33,58)(28,51,34,57)(29,50,35,56)(30,49,36,55)(37,102,43,108)(38,101,44,107)(39,100,45,106)(40,99,46,105)(41,98,47,104)(42,97,48,103)(61,142,67,136)(62,141,68,135)(63,140,69,134)(64,139,70,133)(65,138,71,144)(66,137,72,143)(85,116,91,110)(86,115,92,109)(87,114,93,120)(88,113,94,119)(89,112,95,118)(90,111,96,117), (1,59,21,5,51,13,9,55,17)(2,60,22,6,52,14,10,56,18)(3,49,23,7,53,15,11,57,19)(4,50,24,8,54,16,12,58,20)(25,132,79,33,128,75,29,124,83)(26,121,80,34,129,76,30,125,84)(27,122,81,35,130,77,31,126,73)(28,123,82,36,131,78,32,127,74)(37,71,120,41,63,112,45,67,116)(38,72,109,42,64,113,46,68,117)(39,61,110,43,65,114,47,69,118)(40,62,111,44,66,115,48,70,119)(85,108,138,93,104,134,89,100,142)(86,97,139,94,105,135,90,101,143)(87,98,140,95,106,136,91,102,144)(88,99,141,96,107,137,92,103,133), (1,92)(2,87)(3,94)(4,89)(5,96)(6,91)(7,86)(8,93)(9,88)(10,95)(11,90)(12,85)(13,99)(14,106)(15,101)(16,108)(17,103)(18,98)(19,105)(20,100)(21,107)(22,102)(23,97)(24,104)(25,62)(26,69)(27,64)(28,71)(29,66)(30,61)(31,68)(32,63)(33,70)(34,65)(35,72)(36,67)(37,123)(38,130)(39,125)(40,132)(41,127)(42,122)(43,129)(44,124)(45,131)(46,126)(47,121)(48,128)(49,139)(50,134)(51,141)(52,136)(53,143)(54,138)(55,133)(56,140)(57,135)(58,142)(59,137)(60,144)(73,113)(74,120)(75,115)(76,110)(77,117)(78,112)(79,119)(80,114)(81,109)(82,116)(83,111)(84,118) );
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)], [(1,84,7,78),(2,83,8,77),(3,82,9,76),(4,81,10,75),(5,80,11,74),(6,79,12,73),(13,129,19,123),(14,128,20,122),(15,127,21,121),(16,126,22,132),(17,125,23,131),(18,124,24,130),(25,54,31,60),(26,53,32,59),(27,52,33,58),(28,51,34,57),(29,50,35,56),(30,49,36,55),(37,102,43,108),(38,101,44,107),(39,100,45,106),(40,99,46,105),(41,98,47,104),(42,97,48,103),(61,142,67,136),(62,141,68,135),(63,140,69,134),(64,139,70,133),(65,138,71,144),(66,137,72,143),(85,116,91,110),(86,115,92,109),(87,114,93,120),(88,113,94,119),(89,112,95,118),(90,111,96,117)], [(1,59,21,5,51,13,9,55,17),(2,60,22,6,52,14,10,56,18),(3,49,23,7,53,15,11,57,19),(4,50,24,8,54,16,12,58,20),(25,132,79,33,128,75,29,124,83),(26,121,80,34,129,76,30,125,84),(27,122,81,35,130,77,31,126,73),(28,123,82,36,131,78,32,127,74),(37,71,120,41,63,112,45,67,116),(38,72,109,42,64,113,46,68,117),(39,61,110,43,65,114,47,69,118),(40,62,111,44,66,115,48,70,119),(85,108,138,93,104,134,89,100,142),(86,97,139,94,105,135,90,101,143),(87,98,140,95,106,136,91,102,144),(88,99,141,96,107,137,92,103,133)], [(1,92),(2,87),(3,94),(4,89),(5,96),(6,91),(7,86),(8,93),(9,88),(10,95),(11,90),(12,85),(13,99),(14,106),(15,101),(16,108),(17,103),(18,98),(19,105),(20,100),(21,107),(22,102),(23,97),(24,104),(25,62),(26,69),(27,64),(28,71),(29,66),(30,61),(31,68),(32,63),(33,70),(34,65),(35,72),(36,67),(37,123),(38,130),(39,125),(40,132),(41,127),(42,122),(43,129),(44,124),(45,131),(46,126),(47,121),(48,128),(49,139),(50,134),(51,141),(52,136),(53,143),(54,138),(55,133),(56,140),(57,135),(58,142),(59,137),(60,144),(73,113),(74,120),(75,115),(76,110),(77,117),(78,112),(79,119),(80,114),(81,109),(82,116),(83,111),(84,118)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 18A | 18B | 18C | 18D | 18E | 18F | 36A | ··· | 36I | 36J | ··· | 36O |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 36 | 2 | 2 | 4 | 2 | 12 | 2 | 2 | 4 | 36 | 36 | 54 | 54 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | SD16 | D9 | C3⋊D4 | C3⋊D4 | D18 | C9⋊D4 | S32 | D4.S3 | Q8⋊2S3 | D6⋊S3 | S3×D9 | Q8⋊2D9 | Dic6⋊S3 | D6⋊D9 | Dic6⋊D9 |
| kernel | Dic6⋊D9 | C36.S3 | C9×Dic6 | C3×D36 | D36 | C3×Dic6 | C3×C18 | C36 | C3×C12 | C3×C9 | Dic6 | C18 | C3×C6 | C12 | C6 | C12 | C9 | C32 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 2 | 3 | 6 | 1 | 1 | 1 | 1 | 3 | 3 | 2 | 3 | 6 |
Matrix representation of Dic6⋊D9 ►in GL6(𝔽73)
| 1 | 71 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 67 | 59 | 0 | 0 | 0 | 0 |
| 60 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 13 |
| 0 | 0 | 0 | 0 | 60 | 43 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 28 |
| 0 | 0 | 0 | 0 | 45 | 70 |
| 11 | 52 | 0 | 0 | 0 | 0 |
| 37 | 62 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 41 |
| 0 | 0 | 0 | 0 | 51 | 63 |
G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,71,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[67,60,0,0,0,0,59,6,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,30,60,0,0,0,0,13,43],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,45,0,0,0,0,28,70],[11,37,0,0,0,0,52,62,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,10,51,0,0,0,0,41,63] >;
Dic6⋊D9 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes D_9 % in TeX
G:=Group("Dic6:D9"); // GroupNames label
G:=SmallGroup(432,72);
// by ID
G=gap.SmallGroup(432,72);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,571,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^9=d^2=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^7,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations