metabelian, supersoluble, monomial
Aliases: D12.2D9, C36.11D6, Dic18⋊2S3, C12.14D18, C12.3S32, (C3×C9)⋊3SD16, C4.16(S3×D9), (C3×C18).7D4, C3⋊2(D4.D9), (C9×D12).1C2, (C3×D12).3S3, (C3×C12).75D6, C6.9(C9⋊D4), C36.S3⋊2C2, C9⋊2(Q8⋊2S3), (C3×Dic18)⋊5C2, C18.8(C3⋊D4), C2.5(D6⋊D9), (C3×C36).10C22, C6.13(D6⋊S3), C32.2(D4.S3), C3.3(Dic6⋊S3), (C3×C6).43(C3⋊D4), SmallGroup(432,70)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.D9
G = < a,b,c,d | a12=b2=c9=1, d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 320 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, C18, C18, C3×S3, C3×C6, C3⋊C8, Dic6, D12, C3×D4, C3×Q8, C3×C9, Dic9, C36, C36, C2×C18, C3×Dic3, C3×C12, S3×C6, D4.S3, Q8⋊2S3, S3×C9, C3×C18, C9⋊C8, Dic18, D4×C9, C32⋊4C8, C3×Dic6, C3×D12, C3×Dic9, C3×C36, S3×C18, D4.D9, Dic6⋊S3, C36.S3, C3×Dic18, C9×D12, D12.D9
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, C3⋊D4, D18, S32, D4.S3, Q8⋊2S3, C9⋊D4, D6⋊S3, S3×D9, D4.D9, Dic6⋊S3, D6⋊D9, D12.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)
(1 127)(2 126)(3 125)(4 124)(5 123)(6 122)(7 121)(8 132)(9 131)(10 130)(11 129)(12 128)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 60)(23 59)(24 58)(25 110)(26 109)(27 120)(28 119)(29 118)(30 117)(31 116)(32 115)(33 114)(34 113)(35 112)(36 111)(37 75)(38 74)(39 73)(40 84)(41 83)(42 82)(43 81)(44 80)(45 79)(46 78)(47 77)(48 76)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 108)(70 107)(71 106)(72 105)(85 141)(86 140)(87 139)(88 138)(89 137)(90 136)(91 135)(92 134)(93 133)(94 144)(95 143)(96 142)
(1 43 99 9 39 107 5 47 103)(2 44 100 10 40 108 6 48 104)(3 45 101 11 41 97 7 37 105)(4 46 102 12 42 98 8 38 106)(13 110 137 21 118 133 17 114 141)(14 111 138 22 119 134 18 115 142)(15 112 139 23 120 135 19 116 143)(16 113 140 24 109 136 20 117 144)(25 89 49 29 93 53 33 85 57)(26 90 50 30 94 54 34 86 58)(27 91 51 31 95 55 35 87 59)(28 92 52 32 96 56 36 88 60)(61 126 80 65 130 84 69 122 76)(62 127 81 66 131 73 70 123 77)(63 128 82 67 132 74 71 124 78)(64 129 83 68 121 75 72 125 79)
(1 55 7 49)(2 50 8 56)(3 57 9 51)(4 52 10 58)(5 59 11 53)(6 54 12 60)(13 128 19 122)(14 123 20 129)(15 130 21 124)(16 125 22 131)(17 132 23 126)(18 127 24 121)(25 99 31 105)(26 106 32 100)(27 101 33 107)(28 108 34 102)(29 103 35 97)(30 98 36 104)(37 89 43 95)(38 96 44 90)(39 91 45 85)(40 86 46 92)(41 93 47 87)(42 88 48 94)(61 114 67 120)(62 109 68 115)(63 116 69 110)(64 111 70 117)(65 118 71 112)(66 113 72 119)(73 144 79 138)(74 139 80 133)(75 134 81 140)(76 141 82 135)(77 136 83 142)(78 143 84 137)
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), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,132)(9,131)(10,130)(11,129)(12,128)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,60)(23,59)(24,58)(25,110)(26,109)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,75)(38,74)(39,73)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,108)(70,107)(71,106)(72,105)(85,141)(86,140)(87,139)(88,138)(89,137)(90,136)(91,135)(92,134)(93,133)(94,144)(95,143)(96,142), (1,43,99,9,39,107,5,47,103)(2,44,100,10,40,108,6,48,104)(3,45,101,11,41,97,7,37,105)(4,46,102,12,42,98,8,38,106)(13,110,137,21,118,133,17,114,141)(14,111,138,22,119,134,18,115,142)(15,112,139,23,120,135,19,116,143)(16,113,140,24,109,136,20,117,144)(25,89,49,29,93,53,33,85,57)(26,90,50,30,94,54,34,86,58)(27,91,51,31,95,55,35,87,59)(28,92,52,32,96,56,36,88,60)(61,126,80,65,130,84,69,122,76)(62,127,81,66,131,73,70,123,77)(63,128,82,67,132,74,71,124,78)(64,129,83,68,121,75,72,125,79), (1,55,7,49)(2,50,8,56)(3,57,9,51)(4,52,10,58)(5,59,11,53)(6,54,12,60)(13,128,19,122)(14,123,20,129)(15,130,21,124)(16,125,22,131)(17,132,23,126)(18,127,24,121)(25,99,31,105)(26,106,32,100)(27,101,33,107)(28,108,34,102)(29,103,35,97)(30,98,36,104)(37,89,43,95)(38,96,44,90)(39,91,45,85)(40,86,46,92)(41,93,47,87)(42,88,48,94)(61,114,67,120)(62,109,68,115)(63,116,69,110)(64,111,70,117)(65,118,71,112)(66,113,72,119)(73,144,79,138)(74,139,80,133)(75,134,81,140)(76,141,82,135)(77,136,83,142)(78,143,84,137)>;
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), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,132)(9,131)(10,130)(11,129)(12,128)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,60)(23,59)(24,58)(25,110)(26,109)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,75)(38,74)(39,73)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,108)(70,107)(71,106)(72,105)(85,141)(86,140)(87,139)(88,138)(89,137)(90,136)(91,135)(92,134)(93,133)(94,144)(95,143)(96,142), (1,43,99,9,39,107,5,47,103)(2,44,100,10,40,108,6,48,104)(3,45,101,11,41,97,7,37,105)(4,46,102,12,42,98,8,38,106)(13,110,137,21,118,133,17,114,141)(14,111,138,22,119,134,18,115,142)(15,112,139,23,120,135,19,116,143)(16,113,140,24,109,136,20,117,144)(25,89,49,29,93,53,33,85,57)(26,90,50,30,94,54,34,86,58)(27,91,51,31,95,55,35,87,59)(28,92,52,32,96,56,36,88,60)(61,126,80,65,130,84,69,122,76)(62,127,81,66,131,73,70,123,77)(63,128,82,67,132,74,71,124,78)(64,129,83,68,121,75,72,125,79), (1,55,7,49)(2,50,8,56)(3,57,9,51)(4,52,10,58)(5,59,11,53)(6,54,12,60)(13,128,19,122)(14,123,20,129)(15,130,21,124)(16,125,22,131)(17,132,23,126)(18,127,24,121)(25,99,31,105)(26,106,32,100)(27,101,33,107)(28,108,34,102)(29,103,35,97)(30,98,36,104)(37,89,43,95)(38,96,44,90)(39,91,45,85)(40,86,46,92)(41,93,47,87)(42,88,48,94)(61,114,67,120)(62,109,68,115)(63,116,69,110)(64,111,70,117)(65,118,71,112)(66,113,72,119)(73,144,79,138)(74,139,80,133)(75,134,81,140)(76,141,82,135)(77,136,83,142)(78,143,84,137) );
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)], [(1,127),(2,126),(3,125),(4,124),(5,123),(6,122),(7,121),(8,132),(9,131),(10,130),(11,129),(12,128),(13,57),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,60),(23,59),(24,58),(25,110),(26,109),(27,120),(28,119),(29,118),(30,117),(31,116),(32,115),(33,114),(34,113),(35,112),(36,111),(37,75),(38,74),(39,73),(40,84),(41,83),(42,82),(43,81),(44,80),(45,79),(46,78),(47,77),(48,76),(61,104),(62,103),(63,102),(64,101),(65,100),(66,99),(67,98),(68,97),(69,108),(70,107),(71,106),(72,105),(85,141),(86,140),(87,139),(88,138),(89,137),(90,136),(91,135),(92,134),(93,133),(94,144),(95,143),(96,142)], [(1,43,99,9,39,107,5,47,103),(2,44,100,10,40,108,6,48,104),(3,45,101,11,41,97,7,37,105),(4,46,102,12,42,98,8,38,106),(13,110,137,21,118,133,17,114,141),(14,111,138,22,119,134,18,115,142),(15,112,139,23,120,135,19,116,143),(16,113,140,24,109,136,20,117,144),(25,89,49,29,93,53,33,85,57),(26,90,50,30,94,54,34,86,58),(27,91,51,31,95,55,35,87,59),(28,92,52,32,96,56,36,88,60),(61,126,80,65,130,84,69,122,76),(62,127,81,66,131,73,70,123,77),(63,128,82,67,132,74,71,124,78),(64,129,83,68,121,75,72,125,79)], [(1,55,7,49),(2,50,8,56),(3,57,9,51),(4,52,10,58),(5,59,11,53),(6,54,12,60),(13,128,19,122),(14,123,20,129),(15,130,21,124),(16,125,22,131),(17,132,23,126),(18,127,24,121),(25,99,31,105),(26,106,32,100),(27,101,33,107),(28,108,34,102),(29,103,35,97),(30,98,36,104),(37,89,43,95),(38,96,44,90),(39,91,45,85),(40,86,46,92),(41,93,47,87),(42,88,48,94),(61,114,67,120),(62,109,68,115),(63,116,69,110),(64,111,70,117),(65,118,71,112),(66,113,72,119),(73,144,79,138),(74,139,80,133),(75,134,81,140),(76,141,82,135),(77,136,83,142),(78,143,84,137)]])
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 | 18G | ··· | 18L | 36A | ··· | 36I |
| 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 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 12 | 2 | 2 | 4 | 2 | 36 | 2 | 2 | 4 | 12 | 12 | 54 | 54 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
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 | Q8⋊2S3 | D4.S3 | D6⋊S3 | S3×D9 | D4.D9 | Dic6⋊S3 | D6⋊D9 | D12.D9 |
| kernel | D12.D9 | C36.S3 | C3×Dic18 | C9×D12 | Dic18 | C3×D12 | C3×C18 | C36 | C3×C12 | C3×C9 | D12 | 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 D12.D9 ►in GL6(𝔽73)
| 69 | 3 | 0 | 0 | 0 | 0 |
| 43 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 20 | 1 | 0 | 0 | 0 | 0 |
| 39 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 3 | 0 | 0 |
| 0 | 0 | 31 | 45 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 13 |
| 0 | 0 | 0 | 0 | 60 | 30 |
| 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 | 70 | 28 |
| 0 | 0 | 0 | 0 | 45 | 42 |
| 50 | 60 | 0 | 0 | 0 | 0 |
| 52 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 13 | 0 | 0 |
| 0 | 0 | 60 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 25 |
| 0 | 0 | 0 | 0 | 54 | 29 |
G:=sub<GL(6,GF(73))| [69,43,0,0,0,0,3,4,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[20,39,0,0,0,0,1,53,0,0,0,0,0,0,28,31,0,0,0,0,3,45,0,0,0,0,0,0,43,60,0,0,0,0,13,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,70,45,0,0,0,0,28,42],[50,52,0,0,0,0,60,23,0,0,0,0,0,0,30,60,0,0,0,0,13,43,0,0,0,0,0,0,44,54,0,0,0,0,25,29] >;
D12.D9 in GAP, Magma, Sage, TeX
D_{12}.D_9 % in TeX
G:=Group("D12.D9"); // GroupNames label
G:=SmallGroup(432,70);
// by ID
G=gap.SmallGroup(432,70);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^9=1,d^2=a^6,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations