metabelian, supersoluble, monomial
Aliases: Dic9⋊Dic3, C18.16D12, C6.1Dic18, C18.1Dic6, C62.54D6, (C2×C18).9D6, C6.19(C4×D9), (C2×C6).9D18, (C3×C18).1Q8, C9⋊1(C4⋊Dic3), (C3×Dic9)⋊2C4, (C3×C18).13D4, C6.5(C9⋊D4), C2.4(Dic3×D9), C6.3(S3×Dic3), C22.5(S3×D9), C3⋊1(Dic9⋊C4), (C6×C18).3C22, (C2×Dic9).1S3, (C2×Dic3).1D9, (C6×Dic3).1S3, (C6×Dic9).2C2, (C3×C6).12Dic6, C18.4(C2×Dic3), C2.1(C9⋊D12), (Dic3×C18).1C2, C6.1(C32⋊2Q8), C6.18(C3⋊D12), C2.1(C9⋊Dic6), C3.1(Dic3⋊Dic3), C32.2(Dic3⋊C4), (C3×C9)⋊1(C4⋊C4), (C2×C6).15S32, (C3×C18).8(C2×C4), (C3×C6).38(C4×S3), (C2×C9⋊Dic3).2C2, (C3×C6).49(C3⋊D4), SmallGroup(432,88)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic9⋊Dic3
G = < a,b,c,d | a18=c6=1, b2=a9, d2=c3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a9b, dcd-1=c-1 >
Subgroups: 444 in 94 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×C9, Dic9, Dic9, C36, C2×C18, C2×C18, 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, Dic9⋊C4, Dic3⋊Dic3, C6×Dic9, Dic3×C18, C2×C9⋊Dic3, Dic9⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, D9, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, D18, S32, Dic3⋊C4, C4⋊Dic3, Dic18, C4×D9, C9⋊D4, S3×Dic3, C3⋊D12, C32⋊2Q8, S3×D9, Dic9⋊C4, Dic3⋊Dic3, C9⋊Dic6, Dic3×D9, C9⋊D12, Dic9⋊Dic3
►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)
(1 100 10 91)(2 99 11 108)(3 98 12 107)(4 97 13 106)(5 96 14 105)(6 95 15 104)(7 94 16 103)(8 93 17 102)(9 92 18 101)(19 88 28 79)(20 87 29 78)(21 86 30 77)(22 85 31 76)(23 84 32 75)(24 83 33 74)(25 82 34 73)(26 81 35 90)(27 80 36 89)(37 130 46 139)(38 129 47 138)(39 128 48 137)(40 127 49 136)(41 144 50 135)(42 143 51 134)(43 142 52 133)(44 141 53 132)(45 140 54 131)(55 114 64 123)(56 113 65 122)(57 112 66 121)(58 111 67 120)(59 110 68 119)(60 109 69 118)(61 126 70 117)(62 125 71 116)(63 124 72 115)
(1 29 7 35 13 23)(2 30 8 36 14 24)(3 31 9 19 15 25)(4 32 10 20 16 26)(5 33 11 21 17 27)(6 34 12 22 18 28)(37 68 49 62 43 56)(38 69 50 63 44 57)(39 70 51 64 45 58)(40 71 52 65 46 59)(41 72 53 66 47 60)(42 55 54 67 48 61)(73 107 85 101 79 95)(74 108 86 102 80 96)(75 91 87 103 81 97)(76 92 88 104 82 98)(77 93 89 105 83 99)(78 94 90 106 84 100)(109 144 115 132 121 138)(110 127 116 133 122 139)(111 128 117 134 123 140)(112 129 118 135 124 141)(113 130 119 136 125 142)(114 131 120 137 126 143)
(1 69 35 44)(2 70 36 45)(3 71 19 46)(4 72 20 47)(5 55 21 48)(6 56 22 49)(7 57 23 50)(8 58 24 51)(9 59 25 52)(10 60 26 53)(11 61 27 54)(12 62 28 37)(13 63 29 38)(14 64 30 39)(15 65 31 40)(16 66 32 41)(17 67 33 42)(18 68 34 43)(73 133 101 110)(74 134 102 111)(75 135 103 112)(76 136 104 113)(77 137 105 114)(78 138 106 115)(79 139 107 116)(80 140 108 117)(81 141 91 118)(82 142 92 119)(83 143 93 120)(84 144 94 121)(85 127 95 122)(86 128 96 123)(87 129 97 124)(88 130 98 125)(89 131 99 126)(90 132 100 109)
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), (1,100,10,91)(2,99,11,108)(3,98,12,107)(4,97,13,106)(5,96,14,105)(6,95,15,104)(7,94,16,103)(8,93,17,102)(9,92,18,101)(19,88,28,79)(20,87,29,78)(21,86,30,77)(22,85,31,76)(23,84,32,75)(24,83,33,74)(25,82,34,73)(26,81,35,90)(27,80,36,89)(37,130,46,139)(38,129,47,138)(39,128,48,137)(40,127,49,136)(41,144,50,135)(42,143,51,134)(43,142,52,133)(44,141,53,132)(45,140,54,131)(55,114,64,123)(56,113,65,122)(57,112,66,121)(58,111,67,120)(59,110,68,119)(60,109,69,118)(61,126,70,117)(62,125,71,116)(63,124,72,115), (1,29,7,35,13,23)(2,30,8,36,14,24)(3,31,9,19,15,25)(4,32,10,20,16,26)(5,33,11,21,17,27)(6,34,12,22,18,28)(37,68,49,62,43,56)(38,69,50,63,44,57)(39,70,51,64,45,58)(40,71,52,65,46,59)(41,72,53,66,47,60)(42,55,54,67,48,61)(73,107,85,101,79,95)(74,108,86,102,80,96)(75,91,87,103,81,97)(76,92,88,104,82,98)(77,93,89,105,83,99)(78,94,90,106,84,100)(109,144,115,132,121,138)(110,127,116,133,122,139)(111,128,117,134,123,140)(112,129,118,135,124,141)(113,130,119,136,125,142)(114,131,120,137,126,143), (1,69,35,44)(2,70,36,45)(3,71,19,46)(4,72,20,47)(5,55,21,48)(6,56,22,49)(7,57,23,50)(8,58,24,51)(9,59,25,52)(10,60,26,53)(11,61,27,54)(12,62,28,37)(13,63,29,38)(14,64,30,39)(15,65,31,40)(16,66,32,41)(17,67,33,42)(18,68,34,43)(73,133,101,110)(74,134,102,111)(75,135,103,112)(76,136,104,113)(77,137,105,114)(78,138,106,115)(79,139,107,116)(80,140,108,117)(81,141,91,118)(82,142,92,119)(83,143,93,120)(84,144,94,121)(85,127,95,122)(86,128,96,123)(87,129,97,124)(88,130,98,125)(89,131,99,126)(90,132,100,109)>;
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), (1,100,10,91)(2,99,11,108)(3,98,12,107)(4,97,13,106)(5,96,14,105)(6,95,15,104)(7,94,16,103)(8,93,17,102)(9,92,18,101)(19,88,28,79)(20,87,29,78)(21,86,30,77)(22,85,31,76)(23,84,32,75)(24,83,33,74)(25,82,34,73)(26,81,35,90)(27,80,36,89)(37,130,46,139)(38,129,47,138)(39,128,48,137)(40,127,49,136)(41,144,50,135)(42,143,51,134)(43,142,52,133)(44,141,53,132)(45,140,54,131)(55,114,64,123)(56,113,65,122)(57,112,66,121)(58,111,67,120)(59,110,68,119)(60,109,69,118)(61,126,70,117)(62,125,71,116)(63,124,72,115), (1,29,7,35,13,23)(2,30,8,36,14,24)(3,31,9,19,15,25)(4,32,10,20,16,26)(5,33,11,21,17,27)(6,34,12,22,18,28)(37,68,49,62,43,56)(38,69,50,63,44,57)(39,70,51,64,45,58)(40,71,52,65,46,59)(41,72,53,66,47,60)(42,55,54,67,48,61)(73,107,85,101,79,95)(74,108,86,102,80,96)(75,91,87,103,81,97)(76,92,88,104,82,98)(77,93,89,105,83,99)(78,94,90,106,84,100)(109,144,115,132,121,138)(110,127,116,133,122,139)(111,128,117,134,123,140)(112,129,118,135,124,141)(113,130,119,136,125,142)(114,131,120,137,126,143), (1,69,35,44)(2,70,36,45)(3,71,19,46)(4,72,20,47)(5,55,21,48)(6,56,22,49)(7,57,23,50)(8,58,24,51)(9,59,25,52)(10,60,26,53)(11,61,27,54)(12,62,28,37)(13,63,29,38)(14,64,30,39)(15,65,31,40)(16,66,32,41)(17,67,33,42)(18,68,34,43)(73,133,101,110)(74,134,102,111)(75,135,103,112)(76,136,104,113)(77,137,105,114)(78,138,106,115)(79,139,107,116)(80,140,108,117)(81,141,91,118)(82,142,92,119)(83,143,93,120)(84,144,94,121)(85,127,95,122)(86,128,96,123)(87,129,97,124)(88,130,98,125)(89,131,99,126)(90,132,100,109) );
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)], [(1,100,10,91),(2,99,11,108),(3,98,12,107),(4,97,13,106),(5,96,14,105),(6,95,15,104),(7,94,16,103),(8,93,17,102),(9,92,18,101),(19,88,28,79),(20,87,29,78),(21,86,30,77),(22,85,31,76),(23,84,32,75),(24,83,33,74),(25,82,34,73),(26,81,35,90),(27,80,36,89),(37,130,46,139),(38,129,47,138),(39,128,48,137),(40,127,49,136),(41,144,50,135),(42,143,51,134),(43,142,52,133),(44,141,53,132),(45,140,54,131),(55,114,64,123),(56,113,65,122),(57,112,66,121),(58,111,67,120),(59,110,68,119),(60,109,69,118),(61,126,70,117),(62,125,71,116),(63,124,72,115)], [(1,29,7,35,13,23),(2,30,8,36,14,24),(3,31,9,19,15,25),(4,32,10,20,16,26),(5,33,11,21,17,27),(6,34,12,22,18,28),(37,68,49,62,43,56),(38,69,50,63,44,57),(39,70,51,64,45,58),(40,71,52,65,46,59),(41,72,53,66,47,60),(42,55,54,67,48,61),(73,107,85,101,79,95),(74,108,86,102,80,96),(75,91,87,103,81,97),(76,92,88,104,82,98),(77,93,89,105,83,99),(78,94,90,106,84,100),(109,144,115,132,121,138),(110,127,116,133,122,139),(111,128,117,134,123,140),(112,129,118,135,124,141),(113,130,119,136,125,142),(114,131,120,137,126,143)], [(1,69,35,44),(2,70,36,45),(3,71,19,46),(4,72,20,47),(5,55,21,48),(6,56,22,49),(7,57,23,50),(8,58,24,51),(9,59,25,52),(10,60,26,53),(11,61,27,54),(12,62,28,37),(13,63,29,38),(14,64,30,39),(15,65,31,40),(16,66,32,41),(17,67,33,42),(18,68,34,43),(73,133,101,110),(74,134,102,111),(75,135,103,112),(76,136,104,113),(77,137,105,114),(78,138,106,115),(79,139,107,116),(80,140,108,117),(81,141,91,118),(82,142,92,119),(83,143,93,120),(84,144,94,121),(85,127,95,122),(86,128,96,123),(87,129,97,124),(88,130,98,125),(89,131,99,126),(90,132,100,109)]])
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 | Dic3 | D6 | D6 | D9 | Dic6 | D12 | Dic6 | C4×S3 | C3⋊D4 | D18 | Dic18 | C4×D9 | C9⋊D4 | S32 | S3×Dic3 | C3⋊D12 | C32⋊2Q8 | S3×D9 | C9⋊Dic6 | Dic3×D9 | C9⋊D12 |
| kernel | Dic9⋊Dic3 | C6×Dic9 | Dic3×C18 | C2×C9⋊Dic3 | C3×Dic9 | C2×Dic9 | C6×Dic3 | C3×C18 | C3×C18 | Dic9 | C2×C18 | C62 | C2×Dic3 | C18 | C18 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 | C2×C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 2 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
Matrix representation of Dic9⋊Dic3 ►in GL6(𝔽37)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 11 |
| 0 | 0 | 0 | 0 | 26 | 17 |
| 31 | 24 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 31 | 0 | 0 | 0 | 0 | 0 |
| 34 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 0 | 31 |
G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,26,0,0,0,0,11,17],[31,0,0,0,0,0,24,6,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[31,34,0,0,0,0,0,6,0,0,0,0,0,0,31,6,0,0,0,0,0,6,0,0,0,0,0,0,31,0,0,0,0,0,0,31] >;
Dic9⋊Dic3 in GAP, Magma, Sage, TeX
{\rm Dic}_9\rtimes {\rm Dic}_3 % in TeX
G:=Group("Dic9:Dic3"); // GroupNames label
G:=SmallGroup(432,88);
// by ID
G=gap.SmallGroup(432,88);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,571,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^18=c^6=1,b^2=a^9,d^2=c^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=c^-1>;
// generators/relations