metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C45⋊2D4, D10⋊1D9, D18⋊2D5, C30.6D6, C18.6D10, C10.6D18, Dic45⋊4C2, C90.6C22, C5⋊2(C9⋊D4), C9⋊2(C5⋊D4), C2.6(D5×D9), (C10×D9)⋊2C2, (D5×C18)⋊1C2, C3.(C15⋊D4), (C6×D5).2S3, C6.13(S3×D5), C15.1(C3⋊D4), SmallGroup(360,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊D9
G = < a,b,c,d | a10=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a5b, dcd=c-1 >
Smallest permutation representation of D10⋊D9
►On 180 points
Generators in S180
(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)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170)(171 172 173 174 175 176 177 178 179 180)
(1 102)(2 101)(3 110)(4 109)(5 108)(6 107)(7 106)(8 105)(9 104)(10 103)(11 141)(12 150)(13 149)(14 148)(15 147)(16 146)(17 145)(18 144)(19 143)(20 142)(21 156)(22 155)(23 154)(24 153)(25 152)(26 151)(27 160)(28 159)(29 158)(30 157)(31 123)(32 122)(33 121)(34 130)(35 129)(36 128)(37 127)(38 126)(39 125)(40 124)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 100)(48 99)(49 98)(50 97)(51 112)(52 111)(53 120)(54 119)(55 118)(56 117)(57 116)(58 115)(59 114)(60 113)(61 140)(62 139)(63 138)(64 137)(65 136)(66 135)(67 134)(68 133)(69 132)(70 131)(71 164)(72 163)(73 162)(74 161)(75 170)(76 169)(77 168)(78 167)(79 166)(80 165)(81 178)(82 177)(83 176)(84 175)(85 174)(86 173)(87 172)(88 171)(89 180)(90 179)
(1 147 30 33 115 68 83 74 100)(2 148 21 34 116 69 84 75 91)(3 149 22 35 117 70 85 76 92)(4 150 23 36 118 61 86 77 93)(5 141 24 37 119 62 87 78 94)(6 142 25 38 120 63 88 79 95)(7 143 26 39 111 64 89 80 96)(8 144 27 40 112 65 90 71 97)(9 145 28 31 113 66 81 72 98)(10 146 29 32 114 67 82 73 99)(11 153 127 54 139 172 167 43 108)(12 154 128 55 140 173 168 44 109)(13 155 129 56 131 174 169 45 110)(14 156 130 57 132 175 170 46 101)(15 157 121 58 133 176 161 47 102)(16 158 122 59 134 177 162 48 103)(17 159 123 60 135 178 163 49 104)(18 160 124 51 136 179 164 50 105)(19 151 125 52 137 180 165 41 106)(20 152 126 53 138 171 166 42 107)
(1 100)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 162)(12 163)(13 164)(14 165)(15 166)(16 167)(17 168)(18 169)(19 170)(20 161)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 81)(29 82)(30 83)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 65)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 56)(52 57)(53 58)(54 59)(55 60)(71 144)(72 145)(73 146)(74 147)(75 148)(76 149)(77 150)(78 141)(79 142)(80 143)(121 138)(122 139)(123 140)(124 131)(125 132)(126 133)(127 134)(128 135)(129 136)(130 137)(151 175)(152 176)(153 177)(154 178)(155 179)(156 180)(157 171)(158 172)(159 173)(160 174)
G:=sub<Sym(180)| (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)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180), (1,102)(2,101)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,141)(12,150)(13,149)(14,148)(15,147)(16,146)(17,145)(18,144)(19,143)(20,142)(21,156)(22,155)(23,154)(24,153)(25,152)(26,151)(27,160)(28,159)(29,158)(30,157)(31,123)(32,122)(33,121)(34,130)(35,129)(36,128)(37,127)(38,126)(39,125)(40,124)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,100)(48,99)(49,98)(50,97)(51,112)(52,111)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,140)(62,139)(63,138)(64,137)(65,136)(66,135)(67,134)(68,133)(69,132)(70,131)(71,164)(72,163)(73,162)(74,161)(75,170)(76,169)(77,168)(78,167)(79,166)(80,165)(81,178)(82,177)(83,176)(84,175)(85,174)(86,173)(87,172)(88,171)(89,180)(90,179), (1,147,30,33,115,68,83,74,100)(2,148,21,34,116,69,84,75,91)(3,149,22,35,117,70,85,76,92)(4,150,23,36,118,61,86,77,93)(5,141,24,37,119,62,87,78,94)(6,142,25,38,120,63,88,79,95)(7,143,26,39,111,64,89,80,96)(8,144,27,40,112,65,90,71,97)(9,145,28,31,113,66,81,72,98)(10,146,29,32,114,67,82,73,99)(11,153,127,54,139,172,167,43,108)(12,154,128,55,140,173,168,44,109)(13,155,129,56,131,174,169,45,110)(14,156,130,57,132,175,170,46,101)(15,157,121,58,133,176,161,47,102)(16,158,122,59,134,177,162,48,103)(17,159,123,60,135,178,163,49,104)(18,160,124,51,136,179,164,50,105)(19,151,125,52,137,180,165,41,106)(20,152,126,53,138,171,166,42,107), (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,168)(18,169)(19,170)(20,161)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,81)(29,82)(30,83)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,56)(52,57)(53,58)(54,59)(55,60)(71,144)(72,145)(73,146)(74,147)(75,148)(76,149)(77,150)(78,141)(79,142)(80,143)(121,138)(122,139)(123,140)(124,131)(125,132)(126,133)(127,134)(128,135)(129,136)(130,137)(151,175)(152,176)(153,177)(154,178)(155,179)(156,180)(157,171)(158,172)(159,173)(160,174)>;
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)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180), (1,102)(2,101)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,141)(12,150)(13,149)(14,148)(15,147)(16,146)(17,145)(18,144)(19,143)(20,142)(21,156)(22,155)(23,154)(24,153)(25,152)(26,151)(27,160)(28,159)(29,158)(30,157)(31,123)(32,122)(33,121)(34,130)(35,129)(36,128)(37,127)(38,126)(39,125)(40,124)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,100)(48,99)(49,98)(50,97)(51,112)(52,111)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,140)(62,139)(63,138)(64,137)(65,136)(66,135)(67,134)(68,133)(69,132)(70,131)(71,164)(72,163)(73,162)(74,161)(75,170)(76,169)(77,168)(78,167)(79,166)(80,165)(81,178)(82,177)(83,176)(84,175)(85,174)(86,173)(87,172)(88,171)(89,180)(90,179), (1,147,30,33,115,68,83,74,100)(2,148,21,34,116,69,84,75,91)(3,149,22,35,117,70,85,76,92)(4,150,23,36,118,61,86,77,93)(5,141,24,37,119,62,87,78,94)(6,142,25,38,120,63,88,79,95)(7,143,26,39,111,64,89,80,96)(8,144,27,40,112,65,90,71,97)(9,145,28,31,113,66,81,72,98)(10,146,29,32,114,67,82,73,99)(11,153,127,54,139,172,167,43,108)(12,154,128,55,140,173,168,44,109)(13,155,129,56,131,174,169,45,110)(14,156,130,57,132,175,170,46,101)(15,157,121,58,133,176,161,47,102)(16,158,122,59,134,177,162,48,103)(17,159,123,60,135,178,163,49,104)(18,160,124,51,136,179,164,50,105)(19,151,125,52,137,180,165,41,106)(20,152,126,53,138,171,166,42,107), (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,168)(18,169)(19,170)(20,161)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,81)(29,82)(30,83)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,56)(52,57)(53,58)(54,59)(55,60)(71,144)(72,145)(73,146)(74,147)(75,148)(76,149)(77,150)(78,141)(79,142)(80,143)(121,138)(122,139)(123,140)(124,131)(125,132)(126,133)(127,134)(128,135)(129,136)(130,137)(151,175)(152,176)(153,177)(154,178)(155,179)(156,180)(157,171)(158,172)(159,173)(160,174) );
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),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170),(171,172,173,174,175,176,177,178,179,180)], [(1,102),(2,101),(3,110),(4,109),(5,108),(6,107),(7,106),(8,105),(9,104),(10,103),(11,141),(12,150),(13,149),(14,148),(15,147),(16,146),(17,145),(18,144),(19,143),(20,142),(21,156),(22,155),(23,154),(24,153),(25,152),(26,151),(27,160),(28,159),(29,158),(30,157),(31,123),(32,122),(33,121),(34,130),(35,129),(36,128),(37,127),(38,126),(39,125),(40,124),(41,96),(42,95),(43,94),(44,93),(45,92),(46,91),(47,100),(48,99),(49,98),(50,97),(51,112),(52,111),(53,120),(54,119),(55,118),(56,117),(57,116),(58,115),(59,114),(60,113),(61,140),(62,139),(63,138),(64,137),(65,136),(66,135),(67,134),(68,133),(69,132),(70,131),(71,164),(72,163),(73,162),(74,161),(75,170),(76,169),(77,168),(78,167),(79,166),(80,165),(81,178),(82,177),(83,176),(84,175),(85,174),(86,173),(87,172),(88,171),(89,180),(90,179)], [(1,147,30,33,115,68,83,74,100),(2,148,21,34,116,69,84,75,91),(3,149,22,35,117,70,85,76,92),(4,150,23,36,118,61,86,77,93),(5,141,24,37,119,62,87,78,94),(6,142,25,38,120,63,88,79,95),(7,143,26,39,111,64,89,80,96),(8,144,27,40,112,65,90,71,97),(9,145,28,31,113,66,81,72,98),(10,146,29,32,114,67,82,73,99),(11,153,127,54,139,172,167,43,108),(12,154,128,55,140,173,168,44,109),(13,155,129,56,131,174,169,45,110),(14,156,130,57,132,175,170,46,101),(15,157,121,58,133,176,161,47,102),(16,158,122,59,134,177,162,48,103),(17,159,123,60,135,178,163,49,104),(18,160,124,51,136,179,164,50,105),(19,151,125,52,137,180,165,41,106),(20,152,126,53,138,171,166,42,107)], [(1,100),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,162),(12,163),(13,164),(14,165),(15,166),(16,167),(17,168),(18,169),(19,170),(20,161),(21,84),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,81),(29,82),(30,83),(31,66),(32,67),(33,68),(34,69),(35,70),(36,61),(37,62),(38,63),(39,64),(40,65),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,56),(52,57),(53,58),(54,59),(55,60),(71,144),(72,145),(73,146),(74,147),(75,148),(76,149),(77,150),(78,141),(79,142),(80,143),(121,138),(122,139),(123,140),(124,131),(125,132),(126,133),(127,134),(128,135),(129,136),(130,137),(151,175),(152,176),(153,177),(154,178),(155,179),(156,180),(157,171),(158,172),(159,173),(160,174)])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6A | 6B | 6C | 9A | 9B | 9C | 10A | 10B | 10C | 10D | 10E | 10F | 15A | 15B | 18A | 18B | 18C | 18D | ··· | 18I | 30A | 30B | 45A | ··· | 45F | 90A | ··· | 90F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 10 | 10 | 15 | 15 | 18 | 18 | 18 | 18 | ··· | 18 | 30 | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 10 | 18 | 2 | 90 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 4 | 4 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D9 | D10 | C3⋊D4 | D18 | C5⋊D4 | C9⋊D4 | S3×D5 | C15⋊D4 | D5×D9 | D10⋊D9 |
| kernel | D10⋊D9 | Dic45 | D5×C18 | C10×D9 | C6×D5 | C45 | D18 | C30 | D10 | C18 | C15 | C10 | C9 | C5 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 3 | 4 | 6 | 2 | 2 | 6 | 6 |
Matrix representation of D10⋊D9 ►in GL4(𝔽181) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 122 | 122 |
| 180 | 0 | 0 | 0 |
| 0 | 180 | 0 | 0 |
| 0 | 0 | 37 | 118 |
| 0 | 0 | 151 | 144 |
| 50 | 177 | 0 | 0 |
| 4 | 54 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 131 | 0 | 0 |
| 127 | 177 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 73 | 180 |
G:=sub<GL(4,GF(181))| [1,0,0,0,0,1,0,0,0,0,46,122,0,0,0,122],[180,0,0,0,0,180,0,0,0,0,37,151,0,0,118,144],[50,4,0,0,177,54,0,0,0,0,1,0,0,0,0,1],[4,127,0,0,131,177,0,0,0,0,1,73,0,0,0,180] >;
D10⋊D9 in GAP, Magma, Sage, TeX
D_{10}\rtimes D_9 % in TeX
G:=Group("D10:D9"); // GroupNames label
G:=SmallGroup(360,12);
// by ID
G=gap.SmallGroup(360,12);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-3,73,1641,741,2884,4331]);
// Polycyclic
G:=Group<a,b,c,d|a^10=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^5*b,d*c*d=c^-1>;
// generators/relations