metabelian, supersoluble, monomial
Aliases: Dic3⋊Dic9, C6.16D36, C18.3Dic6, C6.3Dic18, C62.56D6, (C3×C18).3Q8, C18.19(C4×S3), C3⋊1(C4⋊Dic9), (C9×Dic3)⋊1C4, (C2×C18).11D6, (C2×C6).11D18, (C3×C6).33D12, (C3×C18).15D4, C2.4(S3×Dic9), C6.4(C2×Dic9), C22.7(S3×D9), C9⋊2(Dic3⋊C4), C18.5(C3⋊D4), (C6×C18).5C22, (C2×Dic9).3S3, (C6×Dic9).4C2, (C6×Dic3).8S3, (C2×Dic3).3D9, C6.25(S3×Dic3), (C3×C6).14Dic6, C6.5(C3⋊D12), C2.1(C3⋊D36), (Dic3×C18).5C2, C6.3(C32⋊2Q8), (C3×Dic3).2Dic3, C2.3(C9⋊Dic6), C32.2(C4⋊Dic3), C3.2(Dic3⋊Dic3), (C3×C9)⋊3(C4⋊C4), (C2×C6).17S32, (C3×C18).10(C2×C4), (C2×C9⋊Dic3).4C2, (C3×C6).32(C2×Dic3), SmallGroup(432,90)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊Dic9
G = < a,b,c,d | a6=c18=1, b2=a3, d2=c9, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 444 in 94 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C9, C9, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×C9, Dic9, C36, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C4⋊Dic3, C3×C18, C2×Dic9, C2×Dic9, C2×C36, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C3×Dic9, C9×Dic3, C9⋊Dic3, C6×C18, C4⋊Dic9, Dic3⋊Dic3, C6×Dic9, Dic3×C18, C2×C9⋊Dic3, Dic3⋊Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, D9, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, Dic9, D18, S32, Dic3⋊C4, C4⋊Dic3, Dic18, D36, C2×Dic9, S3×Dic3, C3⋊D12, C32⋊2Q8, S3×D9, C4⋊Dic9, Dic3⋊Dic3, C9⋊Dic6, C3⋊D36, S3×Dic9, Dic3⋊Dic9
►On 144 points
Generators in S144
(1 48 7 54 13 42)(2 49 8 37 14 43)(3 50 9 38 15 44)(4 51 10 39 16 45)(5 52 11 40 17 46)(6 53 12 41 18 47)(19 56 31 68 25 62)(20 57 32 69 26 63)(21 58 33 70 27 64)(22 59 34 71 28 65)(23 60 35 72 29 66)(24 61 36 55 30 67)(73 105 85 99 79 93)(74 106 86 100 80 94)(75 107 87 101 81 95)(76 108 88 102 82 96)(77 91 89 103 83 97)(78 92 90 104 84 98)(109 133 115 139 121 127)(110 134 116 140 122 128)(111 135 117 141 123 129)(112 136 118 142 124 130)(113 137 119 143 125 131)(114 138 120 144 126 132)
(1 80 54 106)(2 81 37 107)(3 82 38 108)(4 83 39 91)(5 84 40 92)(6 85 41 93)(7 86 42 94)(8 87 43 95)(9 88 44 96)(10 89 45 97)(11 90 46 98)(12 73 47 99)(13 74 48 100)(14 75 49 101)(15 76 50 102)(16 77 51 103)(17 78 52 104)(18 79 53 105)(19 133 68 121)(20 134 69 122)(21 135 70 123)(22 136 71 124)(23 137 72 125)(24 138 55 126)(25 139 56 109)(26 140 57 110)(27 141 58 111)(28 142 59 112)(29 143 60 113)(30 144 61 114)(31 127 62 115)(32 128 63 116)(33 129 64 117)(34 130 65 118)(35 131 66 119)(36 132 67 120)
(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 64 10 55)(2 63 11 72)(3 62 12 71)(4 61 13 70)(5 60 14 69)(6 59 15 68)(7 58 16 67)(8 57 17 66)(9 56 18 65)(19 41 28 50)(20 40 29 49)(21 39 30 48)(22 38 31 47)(23 37 32 46)(24 54 33 45)(25 53 34 44)(26 52 35 43)(27 51 36 42)(73 136 82 127)(74 135 83 144)(75 134 84 143)(76 133 85 142)(77 132 86 141)(78 131 87 140)(79 130 88 139)(80 129 89 138)(81 128 90 137)(91 114 100 123)(92 113 101 122)(93 112 102 121)(94 111 103 120)(95 110 104 119)(96 109 105 118)(97 126 106 117)(98 125 107 116)(99 124 108 115)
G:=sub<Sym(144)| (1,48,7,54,13,42)(2,49,8,37,14,43)(3,50,9,38,15,44)(4,51,10,39,16,45)(5,52,11,40,17,46)(6,53,12,41,18,47)(19,56,31,68,25,62)(20,57,32,69,26,63)(21,58,33,70,27,64)(22,59,34,71,28,65)(23,60,35,72,29,66)(24,61,36,55,30,67)(73,105,85,99,79,93)(74,106,86,100,80,94)(75,107,87,101,81,95)(76,108,88,102,82,96)(77,91,89,103,83,97)(78,92,90,104,84,98)(109,133,115,139,121,127)(110,134,116,140,122,128)(111,135,117,141,123,129)(112,136,118,142,124,130)(113,137,119,143,125,131)(114,138,120,144,126,132), (1,80,54,106)(2,81,37,107)(3,82,38,108)(4,83,39,91)(5,84,40,92)(6,85,41,93)(7,86,42,94)(8,87,43,95)(9,88,44,96)(10,89,45,97)(11,90,46,98)(12,73,47,99)(13,74,48,100)(14,75,49,101)(15,76,50,102)(16,77,51,103)(17,78,52,104)(18,79,53,105)(19,133,68,121)(20,134,69,122)(21,135,70,123)(22,136,71,124)(23,137,72,125)(24,138,55,126)(25,139,56,109)(26,140,57,110)(27,141,58,111)(28,142,59,112)(29,143,60,113)(30,144,61,114)(31,127,62,115)(32,128,63,116)(33,129,64,117)(34,130,65,118)(35,131,66,119)(36,132,67,120), (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), (1,64,10,55)(2,63,11,72)(3,62,12,71)(4,61,13,70)(5,60,14,69)(6,59,15,68)(7,58,16,67)(8,57,17,66)(9,56,18,65)(19,41,28,50)(20,40,29,49)(21,39,30,48)(22,38,31,47)(23,37,32,46)(24,54,33,45)(25,53,34,44)(26,52,35,43)(27,51,36,42)(73,136,82,127)(74,135,83,144)(75,134,84,143)(76,133,85,142)(77,132,86,141)(78,131,87,140)(79,130,88,139)(80,129,89,138)(81,128,90,137)(91,114,100,123)(92,113,101,122)(93,112,102,121)(94,111,103,120)(95,110,104,119)(96,109,105,118)(97,126,106,117)(98,125,107,116)(99,124,108,115)>;
G:=Group( (1,48,7,54,13,42)(2,49,8,37,14,43)(3,50,9,38,15,44)(4,51,10,39,16,45)(5,52,11,40,17,46)(6,53,12,41,18,47)(19,56,31,68,25,62)(20,57,32,69,26,63)(21,58,33,70,27,64)(22,59,34,71,28,65)(23,60,35,72,29,66)(24,61,36,55,30,67)(73,105,85,99,79,93)(74,106,86,100,80,94)(75,107,87,101,81,95)(76,108,88,102,82,96)(77,91,89,103,83,97)(78,92,90,104,84,98)(109,133,115,139,121,127)(110,134,116,140,122,128)(111,135,117,141,123,129)(112,136,118,142,124,130)(113,137,119,143,125,131)(114,138,120,144,126,132), (1,80,54,106)(2,81,37,107)(3,82,38,108)(4,83,39,91)(5,84,40,92)(6,85,41,93)(7,86,42,94)(8,87,43,95)(9,88,44,96)(10,89,45,97)(11,90,46,98)(12,73,47,99)(13,74,48,100)(14,75,49,101)(15,76,50,102)(16,77,51,103)(17,78,52,104)(18,79,53,105)(19,133,68,121)(20,134,69,122)(21,135,70,123)(22,136,71,124)(23,137,72,125)(24,138,55,126)(25,139,56,109)(26,140,57,110)(27,141,58,111)(28,142,59,112)(29,143,60,113)(30,144,61,114)(31,127,62,115)(32,128,63,116)(33,129,64,117)(34,130,65,118)(35,131,66,119)(36,132,67,120), (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), (1,64,10,55)(2,63,11,72)(3,62,12,71)(4,61,13,70)(5,60,14,69)(6,59,15,68)(7,58,16,67)(8,57,17,66)(9,56,18,65)(19,41,28,50)(20,40,29,49)(21,39,30,48)(22,38,31,47)(23,37,32,46)(24,54,33,45)(25,53,34,44)(26,52,35,43)(27,51,36,42)(73,136,82,127)(74,135,83,144)(75,134,84,143)(76,133,85,142)(77,132,86,141)(78,131,87,140)(79,130,88,139)(80,129,89,138)(81,128,90,137)(91,114,100,123)(92,113,101,122)(93,112,102,121)(94,111,103,120)(95,110,104,119)(96,109,105,118)(97,126,106,117)(98,125,107,116)(99,124,108,115) );
G=PermutationGroup([[(1,48,7,54,13,42),(2,49,8,37,14,43),(3,50,9,38,15,44),(4,51,10,39,16,45),(5,52,11,40,17,46),(6,53,12,41,18,47),(19,56,31,68,25,62),(20,57,32,69,26,63),(21,58,33,70,27,64),(22,59,34,71,28,65),(23,60,35,72,29,66),(24,61,36,55,30,67),(73,105,85,99,79,93),(74,106,86,100,80,94),(75,107,87,101,81,95),(76,108,88,102,82,96),(77,91,89,103,83,97),(78,92,90,104,84,98),(109,133,115,139,121,127),(110,134,116,140,122,128),(111,135,117,141,123,129),(112,136,118,142,124,130),(113,137,119,143,125,131),(114,138,120,144,126,132)], [(1,80,54,106),(2,81,37,107),(3,82,38,108),(4,83,39,91),(5,84,40,92),(6,85,41,93),(7,86,42,94),(8,87,43,95),(9,88,44,96),(10,89,45,97),(11,90,46,98),(12,73,47,99),(13,74,48,100),(14,75,49,101),(15,76,50,102),(16,77,51,103),(17,78,52,104),(18,79,53,105),(19,133,68,121),(20,134,69,122),(21,135,70,123),(22,136,71,124),(23,137,72,125),(24,138,55,126),(25,139,56,109),(26,140,57,110),(27,141,58,111),(28,142,59,112),(29,143,60,113),(30,144,61,114),(31,127,62,115),(32,128,63,116),(33,129,64,117),(34,130,65,118),(35,131,66,119),(36,132,67,120)], [(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)], [(1,64,10,55),(2,63,11,72),(3,62,12,71),(4,61,13,70),(5,60,14,69),(6,59,15,68),(7,58,16,67),(8,57,17,66),(9,56,18,65),(19,41,28,50),(20,40,29,49),(21,39,30,48),(22,38,31,47),(23,37,32,46),(24,54,33,45),(25,53,34,44),(26,52,35,43),(27,51,36,42),(73,136,82,127),(74,135,83,144),(75,134,84,143),(76,133,85,142),(77,132,86,141),(78,131,87,140),(79,130,88,139),(80,129,89,138),(81,128,90,137),(91,114,100,123),(92,113,101,122),(93,112,102,121),(94,111,103,120),(95,110,104,119),(96,109,105,118),(97,126,106,117),(98,125,107,116),(99,124,108,115)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | ··· | 18I | 18J | ··· | 18R | 36A | ··· | 36L |
| 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 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 18 | 18 | 54 | 54 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 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 | C4 | S3 | S3 | D4 | Q8 | D6 | Dic3 | D6 | D9 | Dic6 | C4×S3 | C3⋊D4 | Dic6 | D12 | Dic9 | D18 | Dic18 | D36 | S32 | S3×Dic3 | C3⋊D12 | C32⋊2Q8 | S3×D9 | C9⋊Dic6 | C3⋊D36 | S3×Dic9 |
| kernel | Dic3⋊Dic9 | C6×Dic9 | Dic3×C18 | C2×C9⋊Dic3 | C9×Dic3 | C2×Dic9 | C6×Dic3 | C3×C18 | C3×C18 | C2×C18 | C3×Dic3 | C62 | C2×Dic3 | C18 | C18 | C18 | C3×C6 | C3×C6 | Dic3 | C2×C6 | C6 | C6 | C2×C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 2 | 2 | 6 | 3 | 6 | 6 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
Matrix representation of Dic3⋊Dic9 ►in GL6(𝔽37)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 27 | 0 | 0 | 0 | 0 |
| 10 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 1 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 26 |
| 0 | 0 | 0 | 0 | 11 | 20 |
| 31 | 17 | 0 | 0 | 0 | 0 |
| 11 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 6 |
| 0 | 0 | 0 | 0 | 0 | 6 |
G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,36,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,10,0,0,0,0,27,32,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,11,0,0,0,0,26,20],[31,11,0,0,0,0,17,6,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,31,0,0,0,0,0,6,6] >;
Dic3⋊Dic9 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes {\rm Dic}_9 % in TeX
G:=Group("Dic3:Dic9"); // GroupNames label
G:=SmallGroup(432,90);
// by ID
G=gap.SmallGroup(432,90);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^18=1,b^2=a^3,d^2=c^9,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations