direct product, metabelian, supersoluble, monomial, A-group
Aliases: D9×C2×C12, C62.129D6, C36⋊8(C2×C6), (C6×C36)⋊10C2, C18⋊4(C2×C12), (C2×C36)⋊13C6, C6.20(S3×C12), C12.96(S3×C6), (C6×C12).50S3, C9⋊4(C22×C12), Dic9⋊7(C2×C6), D18.9(C2×C6), (C2×C6).52D18, C22.9(C6×D9), (C3×C36)⋊11C22, (C2×Dic9)⋊11C6, (C6×Dic9)⋊11C2, (C3×C12).220D6, (C22×D9).4C6, C6.50(C22×D9), (C3×C18).39C23, (C6×C18).43C22, C18.16(C22×C6), (C6×D9).15C22, (C3×Dic9)⋊10C22, C2.1(C2×C6×D9), C6.20(S3×C2×C6), C3.1(S3×C2×C12), (C2×C6×D9).4C2, (C3×C18)⋊5(C2×C4), (C3×C9)⋊6(C22×C4), C32.5(S3×C2×C4), (C3×C6).73(C4×S3), (C2×C6).50(S3×C6), (C2×C12).35(C3×S3), (C2×C18).29(C2×C6), (C3×C6).153(C22×S3), SmallGroup(432,342)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C2×C12 |
Generators and relations for D9×C2×C12
G = < a,b,c,d | a2=b12=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 574 in 178 conjugacy classes, 86 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C22×C4, D9, C18, C18, C18, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C3×C9, Dic9, C36, C36, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, S3×C2×C4, C22×C12, C3×D9, C3×C18, C3×C18, C4×D9, C2×Dic9, C2×C36, C2×C36, C22×D9, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, C3×Dic9, C3×C36, C6×D9, C6×C18, C2×C4×D9, S3×C2×C12, C12×D9, C6×Dic9, C6×C36, C2×C6×D9, D9×C2×C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, D9, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, D18, S3×C6, S3×C2×C4, C22×C12, C3×D9, C4×D9, C22×D9, S3×C12, S3×C2×C6, C6×D9, C2×C4×D9, S3×C2×C12, C12×D9, C2×C6×D9, D9×C2×C12
►On 144 points
Generators in S144
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 37)(24 38)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 61)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 108)(86 97)(87 98)(88 99)(89 100)(90 101)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(121 143)(122 144)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(131 141)(132 142)
(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 53 130 5 57 122 9 49 126)(2 54 131 6 58 123 10 50 127)(3 55 132 7 59 124 11 51 128)(4 56 121 8 60 125 12 52 129)(13 98 109 21 106 117 17 102 113)(14 99 110 22 107 118 18 103 114)(15 100 111 23 108 119 19 104 115)(16 101 112 24 97 120 20 105 116)(25 72 134 29 64 138 33 68 142)(26 61 135 30 65 139 34 69 143)(27 62 136 31 66 140 35 70 144)(28 63 137 32 67 141 36 71 133)(37 85 77 45 93 73 41 89 81)(38 86 78 46 94 74 42 90 82)(39 87 79 47 95 75 43 91 83)(40 88 80 48 96 76 44 92 84)
(1 117)(2 118)(3 119)(4 120)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 122)(14 123)(15 124)(16 125)(17 126)(18 127)(19 128)(20 129)(21 130)(22 131)(23 132)(24 121)(25 81)(26 82)(27 83)(28 84)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 142)(38 143)(39 144)(40 133)(41 134)(42 135)(43 136)(44 137)(45 138)(46 139)(47 140)(48 141)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 97)(57 98)(58 99)(59 100)(60 101)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 85)(69 86)(70 87)(71 88)(72 89)
G:=sub<Sym(144)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,37)(24,38)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,61)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,108)(86,97)(87,98)(88,99)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(121,143)(122,144)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(131,141)(132,142), (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), (1,53,130,5,57,122,9,49,126)(2,54,131,6,58,123,10,50,127)(3,55,132,7,59,124,11,51,128)(4,56,121,8,60,125,12,52,129)(13,98,109,21,106,117,17,102,113)(14,99,110,22,107,118,18,103,114)(15,100,111,23,108,119,19,104,115)(16,101,112,24,97,120,20,105,116)(25,72,134,29,64,138,33,68,142)(26,61,135,30,65,139,34,69,143)(27,62,136,31,66,140,35,70,144)(28,63,137,32,67,141,36,71,133)(37,85,77,45,93,73,41,89,81)(38,86,78,46,94,74,42,90,82)(39,87,79,47,95,75,43,91,83)(40,88,80,48,96,76,44,92,84), (1,117)(2,118)(3,119)(4,120)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,122)(14,123)(15,124)(16,125)(17,126)(18,127)(19,128)(20,129)(21,130)(22,131)(23,132)(24,121)(25,81)(26,82)(27,83)(28,84)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,142)(38,143)(39,144)(40,133)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,141)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,97)(57,98)(58,99)(59,100)(60,101)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,85)(69,86)(70,87)(71,88)(72,89)>;
G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,37)(24,38)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,61)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,108)(86,97)(87,98)(88,99)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(121,143)(122,144)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(131,141)(132,142), (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), (1,53,130,5,57,122,9,49,126)(2,54,131,6,58,123,10,50,127)(3,55,132,7,59,124,11,51,128)(4,56,121,8,60,125,12,52,129)(13,98,109,21,106,117,17,102,113)(14,99,110,22,107,118,18,103,114)(15,100,111,23,108,119,19,104,115)(16,101,112,24,97,120,20,105,116)(25,72,134,29,64,138,33,68,142)(26,61,135,30,65,139,34,69,143)(27,62,136,31,66,140,35,70,144)(28,63,137,32,67,141,36,71,133)(37,85,77,45,93,73,41,89,81)(38,86,78,46,94,74,42,90,82)(39,87,79,47,95,75,43,91,83)(40,88,80,48,96,76,44,92,84), (1,117)(2,118)(3,119)(4,120)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,122)(14,123)(15,124)(16,125)(17,126)(18,127)(19,128)(20,129)(21,130)(22,131)(23,132)(24,121)(25,81)(26,82)(27,83)(28,84)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,142)(38,143)(39,144)(40,133)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,141)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,97)(57,98)(58,99)(59,100)(60,101)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,85)(69,86)(70,87)(71,88)(72,89) );
G=PermutationGroup([[(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,37),(24,38),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,61),(73,115),(74,116),(75,117),(76,118),(77,119),(78,120),(79,109),(80,110),(81,111),(82,112),(83,113),(84,114),(85,108),(86,97),(87,98),(88,99),(89,100),(90,101),(91,102),(92,103),(93,104),(94,105),(95,106),(96,107),(121,143),(122,144),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(131,141),(132,142)], [(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)], [(1,53,130,5,57,122,9,49,126),(2,54,131,6,58,123,10,50,127),(3,55,132,7,59,124,11,51,128),(4,56,121,8,60,125,12,52,129),(13,98,109,21,106,117,17,102,113),(14,99,110,22,107,118,18,103,114),(15,100,111,23,108,119,19,104,115),(16,101,112,24,97,120,20,105,116),(25,72,134,29,64,138,33,68,142),(26,61,135,30,65,139,34,69,143),(27,62,136,31,66,140,35,70,144),(28,63,137,32,67,141,36,71,133),(37,85,77,45,93,73,41,89,81),(38,86,78,46,94,74,42,90,82),(39,87,79,47,95,75,43,91,83),(40,88,80,48,96,76,44,92,84)], [(1,117),(2,118),(3,119),(4,120),(5,109),(6,110),(7,111),(8,112),(9,113),(10,114),(11,115),(12,116),(13,122),(14,123),(15,124),(16,125),(17,126),(18,127),(19,128),(20,129),(21,130),(22,131),(23,132),(24,121),(25,81),(26,82),(27,83),(28,84),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,142),(38,143),(39,144),(40,133),(41,134),(42,135),(43,136),(44,137),(45,138),(46,139),(47,140),(48,141),(49,102),(50,103),(51,104),(52,105),(53,106),(54,107),(55,108),(56,97),(57,98),(58,99),(59,100),(60,101),(61,90),(62,91),(63,92),(64,93),(65,94),(66,95),(67,96),(68,85),(69,86),(70,87),(71,88),(72,89)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 9A | ··· | 9I | 12A | ··· | 12H | 12I | ··· | 12T | 12U | ··· | 12AB | 18A | ··· | 18AA | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | S3 | D6 | D6 | D9 | C3×S3 | C4×S3 | D18 | S3×C6 | D18 | S3×C6 | C3×D9 | C4×D9 | S3×C12 | C6×D9 | C6×D9 | C12×D9 |
| kernel | D9×C2×C12 | C12×D9 | C6×Dic9 | C6×C36 | C2×C6×D9 | C2×C4×D9 | C6×D9 | C4×D9 | C2×Dic9 | C2×C36 | C22×D9 | D18 | C6×C12 | C3×C12 | C62 | C2×C12 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 2 | 16 | 1 | 2 | 1 | 3 | 2 | 4 | 6 | 4 | 3 | 2 | 6 | 12 | 8 | 12 | 6 | 24 |
Matrix representation of D9×C2×C12 ►in GL5(𝔽37)
| 36 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 27 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 0 | 26 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 33 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 8 | 12 | 0 | 0 |
| 0 | 4 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 |
| 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[27,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,26,0,0,0,0,0,26],[1,0,0,0,0,0,0,36,0,0,0,1,36,0,0,0,0,0,9,0,0,0,0,0,33],[1,0,0,0,0,0,8,4,0,0,0,12,29,0,0,0,0,0,0,9,0,0,0,33,0] >;
D9×C2×C12 in GAP, Magma, Sage, TeX
D_9\times C_2\times C_{12} % in TeX
G:=Group("D9xC2xC12"); // GroupNames label
G:=SmallGroup(432,342);
// by ID
G=gap.SmallGroup(432,342);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,142,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations