metabelian, supersoluble, monomial
Aliases: D12⋊5D9, D18.6D6, D6.1D18, C36.25D6, C12.36D18, Dic9.11D6, C12.17S32, (C4×D9)⋊2S3, C4.7(S3×D9), (C9×D12)⋊3C2, (C12×D9)⋊2C2, (S3×C6).1D6, D6⋊D9⋊1C2, C9⋊3(C4○D12), (S3×Dic9)⋊1C2, (C3×D12).5S3, (C3×C12).94D6, C3⋊2(D4⋊2D9), C12.D9⋊8C2, C6.6(C22×D9), C18.6(C22×S3), (C3×C18).6C23, (C6×D9).7C22, (S3×C18).1C22, (C3×C36).28C22, C9⋊Dic3.3C22, C3.2(D12⋊5S3), (C3×Dic9).9C22, C32.2(D4⋊2S3), C6.25(C2×S32), C2.10(C2×S3×D9), (C3×C9)⋊3(C4○D4), (C3×C6).74(C22×S3), SmallGroup(432,285)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊5D9
G = < a,b,c,d | a12=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a6b, dcd=c-1 >
Subgroups: 732 in 132 conjugacy classes, 41 normal (29 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, D6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×C9, Dic9, Dic9, C36, C36, D18, C2×C18, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C4○D12, D4⋊2S3, C3×D9, S3×C9, C3×C18, Dic18, C4×D9, C2×Dic9, C9⋊D4, D4×C9, S3×Dic3, D6⋊S3, S3×C12, C3×D12, C32⋊4Q8, C3×Dic9, C9⋊Dic3, C3×C36, C6×D9, S3×C18, D4⋊2D9, D12⋊5S3, S3×Dic9, D6⋊D9, C12×D9, C9×D12, C12.D9, D12⋊5D9
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, S32, C4○D12, D4⋊2S3, C22×D9, C2×S32, S3×D9, D4⋊2D9, D12⋊5S3, C2×S3×D9, D12⋊5D9
►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)
(1 107)(2 106)(3 105)(4 104)(5 103)(6 102)(7 101)(8 100)(9 99)(10 98)(11 97)(12 108)(13 116)(14 115)(15 114)(16 113)(17 112)(18 111)(19 110)(20 109)(21 120)(22 119)(23 118)(24 117)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 48)(49 87)(50 86)(51 85)(52 96)(53 95)(54 94)(55 93)(56 92)(57 91)(58 90)(59 89)(60 88)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 84)(71 83)(72 82)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 144)(128 143)(129 142)(130 141)(131 140)(132 139)
(1 121 66 9 129 62 5 125 70)(2 122 67 10 130 63 6 126 71)(3 123 68 11 131 64 7 127 72)(4 124 69 12 132 65 8 128 61)(13 45 54 17 37 58 21 41 50)(14 46 55 18 38 59 22 42 51)(15 47 56 19 39 60 23 43 52)(16 48 57 20 40 49 24 44 53)(25 92 110 33 88 118 29 96 114)(26 93 111 34 89 119 30 85 115)(27 94 112 35 90 120 31 86 116)(28 95 113 36 91 109 32 87 117)(73 108 139 77 100 143 81 104 135)(74 97 140 78 101 144 82 105 136)(75 98 141 79 102 133 83 106 137)(76 99 142 80 103 134 84 107 138)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 49)(13 70)(14 71)(15 72)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 138)(26 139)(27 140)(28 141)(29 142)(30 143)(31 144)(32 133)(33 134)(34 135)(35 136)(36 137)(37 129)(38 130)(39 131)(40 132)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(73 111)(74 112)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)(81 119)(82 120)(83 109)(84 110)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 106)(92 107)(93 108)(94 97)(95 98)(96 99)
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), (1,107)(2,106)(3,105)(4,104)(5,103)(6,102)(7,101)(8,100)(9,99)(10,98)(11,97)(12,108)(13,116)(14,115)(15,114)(16,113)(17,112)(18,111)(19,110)(20,109)(21,120)(22,119)(23,118)(24,117)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(36,48)(49,87)(50,86)(51,85)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,84)(71,83)(72,82)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,144)(128,143)(129,142)(130,141)(131,140)(132,139), (1,121,66,9,129,62,5,125,70)(2,122,67,10,130,63,6,126,71)(3,123,68,11,131,64,7,127,72)(4,124,69,12,132,65,8,128,61)(13,45,54,17,37,58,21,41,50)(14,46,55,18,38,59,22,42,51)(15,47,56,19,39,60,23,43,52)(16,48,57,20,40,49,24,44,53)(25,92,110,33,88,118,29,96,114)(26,93,111,34,89,119,30,85,115)(27,94,112,35,90,120,31,86,116)(28,95,113,36,91,109,32,87,117)(73,108,139,77,100,143,81,104,135)(74,97,140,78,101,144,82,105,136)(75,98,141,79,102,133,83,106,137)(76,99,142,80,103,134,84,107,138), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,49)(13,70)(14,71)(15,72)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,133)(33,134)(34,135)(35,136)(36,137)(37,129)(38,130)(39,131)(40,132)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,109)(84,110)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,97)(95,98)(96,99)>;
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), (1,107)(2,106)(3,105)(4,104)(5,103)(6,102)(7,101)(8,100)(9,99)(10,98)(11,97)(12,108)(13,116)(14,115)(15,114)(16,113)(17,112)(18,111)(19,110)(20,109)(21,120)(22,119)(23,118)(24,117)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(36,48)(49,87)(50,86)(51,85)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,84)(71,83)(72,82)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,144)(128,143)(129,142)(130,141)(131,140)(132,139), (1,121,66,9,129,62,5,125,70)(2,122,67,10,130,63,6,126,71)(3,123,68,11,131,64,7,127,72)(4,124,69,12,132,65,8,128,61)(13,45,54,17,37,58,21,41,50)(14,46,55,18,38,59,22,42,51)(15,47,56,19,39,60,23,43,52)(16,48,57,20,40,49,24,44,53)(25,92,110,33,88,118,29,96,114)(26,93,111,34,89,119,30,85,115)(27,94,112,35,90,120,31,86,116)(28,95,113,36,91,109,32,87,117)(73,108,139,77,100,143,81,104,135)(74,97,140,78,101,144,82,105,136)(75,98,141,79,102,133,83,106,137)(76,99,142,80,103,134,84,107,138), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,49)(13,70)(14,71)(15,72)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,133)(33,134)(34,135)(35,136)(36,137)(37,129)(38,130)(39,131)(40,132)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,109)(84,110)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,97)(95,98)(96,99) );
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)], [(1,107),(2,106),(3,105),(4,104),(5,103),(6,102),(7,101),(8,100),(9,99),(10,98),(11,97),(12,108),(13,116),(14,115),(15,114),(16,113),(17,112),(18,111),(19,110),(20,109),(21,120),(22,119),(23,118),(24,117),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(36,48),(49,87),(50,86),(51,85),(52,96),(53,95),(54,94),(55,93),(56,92),(57,91),(58,90),(59,89),(60,88),(61,81),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,84),(71,83),(72,82),(121,138),(122,137),(123,136),(124,135),(125,134),(126,133),(127,144),(128,143),(129,142),(130,141),(131,140),(132,139)], [(1,121,66,9,129,62,5,125,70),(2,122,67,10,130,63,6,126,71),(3,123,68,11,131,64,7,127,72),(4,124,69,12,132,65,8,128,61),(13,45,54,17,37,58,21,41,50),(14,46,55,18,38,59,22,42,51),(15,47,56,19,39,60,23,43,52),(16,48,57,20,40,49,24,44,53),(25,92,110,33,88,118,29,96,114),(26,93,111,34,89,119,30,85,115),(27,94,112,35,90,120,31,86,116),(28,95,113,36,91,109,32,87,117),(73,108,139,77,100,143,81,104,135),(74,97,140,78,101,144,82,105,136),(75,98,141,79,102,133,83,106,137),(76,99,142,80,103,134,84,107,138)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,60),(12,49),(13,70),(14,71),(15,72),(16,61),(17,62),(18,63),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,138),(26,139),(27,140),(28,141),(29,142),(30,143),(31,144),(32,133),(33,134),(34,135),(35,136),(36,137),(37,129),(38,130),(39,131),(40,132),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(73,111),(74,112),(75,113),(76,114),(77,115),(78,116),(79,117),(80,118),(81,119),(82,120),(83,109),(84,110),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,106),(92,107),(93,108),(94,97),(95,98),(96,99)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L | 36A | ··· | 36I |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 6 | 6 | 18 | 2 | 2 | 4 | 2 | 9 | 9 | 54 | 54 | 2 | 2 | 4 | 12 | 12 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
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 | D6 | D6 | D6 | D6 | D6 | C4○D4 | D9 | D18 | D18 | C4○D12 | S32 | D4⋊2S3 | C2×S32 | S3×D9 | D4⋊2D9 | D12⋊5S3 | C2×S3×D9 | D12⋊5D9 |
| kernel | D12⋊5D9 | S3×Dic9 | D6⋊D9 | C12×D9 | C9×D12 | C12.D9 | C4×D9 | C3×D12 | Dic9 | C36 | D18 | C3×C12 | S3×C6 | C3×C9 | D12 | C12 | D6 | C9 | C12 | C32 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 4 | 1 | 1 | 1 | 3 | 3 | 2 | 3 | 6 |
Matrix representation of D12⋊5D9 ►in GL6(𝔽37)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 18 | 11 | 0 | 0 | 0 | 0 |
| 11 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 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 | 11 | 10 |
| 0 | 0 | 0 | 0 | 34 | 31 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 31 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 24 |
| 0 | 0 | 0 | 0 | 34 | 31 |
G:=sub<GL(6,GF(37))| [0,36,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,36,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,11,0,0,0,0,11,19,0,0,0,0,0,0,0,36,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,34,0,0,0,0,10,31],[0,31,0,0,0,0,6,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,6,34,0,0,0,0,24,31] >;
D12⋊5D9 in GAP, Magma, Sage, TeX
D_{12}\rtimes_5D_9 % in TeX
G:=Group("D12:5D9"); // GroupNames label
G:=SmallGroup(432,285);
// by ID
G=gap.SmallGroup(432,285);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations