direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×Dic9, C30.5D6, D10.2D9, C10.5D18, C18.5D10, Dic45⋊3C2, C90.5C22, C9⋊3(C4×D5), C45⋊6(C2×C4), (C9×D5)⋊1C4, C2.3(D5×D9), C5⋊2(C2×Dic9), C3.(D5×Dic3), (C6×D5).1S3, C6.12(S3×D5), (C5×Dic9)⋊2C2, (D5×C18).1C2, (C3×D5).1Dic3, C15.2(C2×Dic3), SmallGroup(360,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C45 — D5×Dic9 |
Generators and relations for D5×Dic9
G = < a,b,c,d | a5=b2=c18=1, d2=c9, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of D5×Dic9
►On 180 points
Generators in S180
(1 107 57 29 161)(2 108 58 30 162)(3 91 59 31 145)(4 92 60 32 146)(5 93 61 33 147)(6 94 62 34 148)(7 95 63 35 149)(8 96 64 36 150)(9 97 65 19 151)(10 98 66 20 152)(11 99 67 21 153)(12 100 68 22 154)(13 101 69 23 155)(14 102 70 24 156)(15 103 71 25 157)(16 104 72 26 158)(17 105 55 27 159)(18 106 56 28 160)(37 112 75 143 177)(38 113 76 144 178)(39 114 77 127 179)(40 115 78 128 180)(41 116 79 129 163)(42 117 80 130 164)(43 118 81 131 165)(44 119 82 132 166)(45 120 83 133 167)(46 121 84 134 168)(47 122 85 135 169)(48 123 86 136 170)(49 124 87 137 171)(50 125 88 138 172)(51 126 89 139 173)(52 109 90 140 174)(53 110 73 141 175)(54 111 74 142 176)
(1 161)(2 162)(3 145)(4 146)(5 147)(6 148)(7 149)(8 150)(9 151)(10 152)(11 153)(12 154)(13 155)(14 156)(15 157)(16 158)(17 159)(18 160)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 112)(38 113)(39 114)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 109)(53 110)(54 111)(73 175)(74 176)(75 177)(76 178)(77 179)(78 180)(79 163)(80 164)(81 165)(82 166)(83 167)(84 168)(85 169)(86 170)(87 171)(88 172)(89 173)(90 174)
(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)
(1 116 10 125)(2 115 11 124)(3 114 12 123)(4 113 13 122)(5 112 14 121)(6 111 15 120)(7 110 16 119)(8 109 17 118)(9 126 18 117)(19 173 28 164)(20 172 29 163)(21 171 30 180)(22 170 31 179)(23 169 32 178)(24 168 33 177)(25 167 34 176)(26 166 35 175)(27 165 36 174)(37 156 46 147)(38 155 47 146)(39 154 48 145)(40 153 49 162)(41 152 50 161)(42 151 51 160)(43 150 52 159)(44 149 53 158)(45 148 54 157)(55 131 64 140)(56 130 65 139)(57 129 66 138)(58 128 67 137)(59 127 68 136)(60 144 69 135)(61 143 70 134)(62 142 71 133)(63 141 72 132)(73 104 82 95)(74 103 83 94)(75 102 84 93)(76 101 85 92)(77 100 86 91)(78 99 87 108)(79 98 88 107)(80 97 89 106)(81 96 90 105)
G:=sub<Sym(180)| (1,107,57,29,161)(2,108,58,30,162)(3,91,59,31,145)(4,92,60,32,146)(5,93,61,33,147)(6,94,62,34,148)(7,95,63,35,149)(8,96,64,36,150)(9,97,65,19,151)(10,98,66,20,152)(11,99,67,21,153)(12,100,68,22,154)(13,101,69,23,155)(14,102,70,24,156)(15,103,71,25,157)(16,104,72,26,158)(17,105,55,27,159)(18,106,56,28,160)(37,112,75,143,177)(38,113,76,144,178)(39,114,77,127,179)(40,115,78,128,180)(41,116,79,129,163)(42,117,80,130,164)(43,118,81,131,165)(44,119,82,132,166)(45,120,83,133,167)(46,121,84,134,168)(47,122,85,135,169)(48,123,86,136,170)(49,124,87,137,171)(50,125,88,138,172)(51,126,89,139,173)(52,109,90,140,174)(53,110,73,141,175)(54,111,74,142,176), (1,161)(2,162)(3,145)(4,146)(5,147)(6,148)(7,149)(8,150)(9,151)(10,152)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,109)(53,110)(54,111)(73,175)(74,176)(75,177)(76,178)(77,179)(78,180)(79,163)(80,164)(81,165)(82,166)(83,167)(84,168)(85,169)(86,170)(87,171)(88,172)(89,173)(90,174), (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), (1,116,10,125)(2,115,11,124)(3,114,12,123)(4,113,13,122)(5,112,14,121)(6,111,15,120)(7,110,16,119)(8,109,17,118)(9,126,18,117)(19,173,28,164)(20,172,29,163)(21,171,30,180)(22,170,31,179)(23,169,32,178)(24,168,33,177)(25,167,34,176)(26,166,35,175)(27,165,36,174)(37,156,46,147)(38,155,47,146)(39,154,48,145)(40,153,49,162)(41,152,50,161)(42,151,51,160)(43,150,52,159)(44,149,53,158)(45,148,54,157)(55,131,64,140)(56,130,65,139)(57,129,66,138)(58,128,67,137)(59,127,68,136)(60,144,69,135)(61,143,70,134)(62,142,71,133)(63,141,72,132)(73,104,82,95)(74,103,83,94)(75,102,84,93)(76,101,85,92)(77,100,86,91)(78,99,87,108)(79,98,88,107)(80,97,89,106)(81,96,90,105)>;
G:=Group( (1,107,57,29,161)(2,108,58,30,162)(3,91,59,31,145)(4,92,60,32,146)(5,93,61,33,147)(6,94,62,34,148)(7,95,63,35,149)(8,96,64,36,150)(9,97,65,19,151)(10,98,66,20,152)(11,99,67,21,153)(12,100,68,22,154)(13,101,69,23,155)(14,102,70,24,156)(15,103,71,25,157)(16,104,72,26,158)(17,105,55,27,159)(18,106,56,28,160)(37,112,75,143,177)(38,113,76,144,178)(39,114,77,127,179)(40,115,78,128,180)(41,116,79,129,163)(42,117,80,130,164)(43,118,81,131,165)(44,119,82,132,166)(45,120,83,133,167)(46,121,84,134,168)(47,122,85,135,169)(48,123,86,136,170)(49,124,87,137,171)(50,125,88,138,172)(51,126,89,139,173)(52,109,90,140,174)(53,110,73,141,175)(54,111,74,142,176), (1,161)(2,162)(3,145)(4,146)(5,147)(6,148)(7,149)(8,150)(9,151)(10,152)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,109)(53,110)(54,111)(73,175)(74,176)(75,177)(76,178)(77,179)(78,180)(79,163)(80,164)(81,165)(82,166)(83,167)(84,168)(85,169)(86,170)(87,171)(88,172)(89,173)(90,174), (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), (1,116,10,125)(2,115,11,124)(3,114,12,123)(4,113,13,122)(5,112,14,121)(6,111,15,120)(7,110,16,119)(8,109,17,118)(9,126,18,117)(19,173,28,164)(20,172,29,163)(21,171,30,180)(22,170,31,179)(23,169,32,178)(24,168,33,177)(25,167,34,176)(26,166,35,175)(27,165,36,174)(37,156,46,147)(38,155,47,146)(39,154,48,145)(40,153,49,162)(41,152,50,161)(42,151,51,160)(43,150,52,159)(44,149,53,158)(45,148,54,157)(55,131,64,140)(56,130,65,139)(57,129,66,138)(58,128,67,137)(59,127,68,136)(60,144,69,135)(61,143,70,134)(62,142,71,133)(63,141,72,132)(73,104,82,95)(74,103,83,94)(75,102,84,93)(76,101,85,92)(77,100,86,91)(78,99,87,108)(79,98,88,107)(80,97,89,106)(81,96,90,105) );
G=PermutationGroup([[(1,107,57,29,161),(2,108,58,30,162),(3,91,59,31,145),(4,92,60,32,146),(5,93,61,33,147),(6,94,62,34,148),(7,95,63,35,149),(8,96,64,36,150),(9,97,65,19,151),(10,98,66,20,152),(11,99,67,21,153),(12,100,68,22,154),(13,101,69,23,155),(14,102,70,24,156),(15,103,71,25,157),(16,104,72,26,158),(17,105,55,27,159),(18,106,56,28,160),(37,112,75,143,177),(38,113,76,144,178),(39,114,77,127,179),(40,115,78,128,180),(41,116,79,129,163),(42,117,80,130,164),(43,118,81,131,165),(44,119,82,132,166),(45,120,83,133,167),(46,121,84,134,168),(47,122,85,135,169),(48,123,86,136,170),(49,124,87,137,171),(50,125,88,138,172),(51,126,89,139,173),(52,109,90,140,174),(53,110,73,141,175),(54,111,74,142,176)], [(1,161),(2,162),(3,145),(4,146),(5,147),(6,148),(7,149),(8,150),(9,151),(10,152),(11,153),(12,154),(13,155),(14,156),(15,157),(16,158),(17,159),(18,160),(19,97),(20,98),(21,99),(22,100),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,107),(30,108),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,112),(38,113),(39,114),(40,115),(41,116),(42,117),(43,118),(44,119),(45,120),(46,121),(47,122),(48,123),(49,124),(50,125),(51,126),(52,109),(53,110),(54,111),(73,175),(74,176),(75,177),(76,178),(77,179),(78,180),(79,163),(80,164),(81,165),(82,166),(83,167),(84,168),(85,169),(86,170),(87,171),(88,172),(89,173),(90,174)], [(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)], [(1,116,10,125),(2,115,11,124),(3,114,12,123),(4,113,13,122),(5,112,14,121),(6,111,15,120),(7,110,16,119),(8,109,17,118),(9,126,18,117),(19,173,28,164),(20,172,29,163),(21,171,30,180),(22,170,31,179),(23,169,32,178),(24,168,33,177),(25,167,34,176),(26,166,35,175),(27,165,36,174),(37,156,46,147),(38,155,47,146),(39,154,48,145),(40,153,49,162),(41,152,50,161),(42,151,51,160),(43,150,52,159),(44,149,53,158),(45,148,54,157),(55,131,64,140),(56,130,65,139),(57,129,66,138),(58,128,67,137),(59,127,68,136),(60,144,69,135),(61,143,70,134),(62,142,71,133),(63,141,72,132),(73,104,82,95),(74,103,83,94),(75,102,84,93),(76,101,85,92),(77,100,86,91),(78,99,87,108),(79,98,88,107),(80,97,89,106),(81,96,90,105)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 9A | 9B | 9C | 10A | 10B | 15A | 15B | 18A | 18B | 18C | 18D | ··· | 18I | 20A | 20B | 20C | 20D | 30A | 30B | 45A | ··· | 45F | 90A | ··· | 90F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 9 | 9 | 9 | 10 | 10 | 15 | 15 | 18 | 18 | 18 | 18 | ··· | 18 | 20 | 20 | 20 | 20 | 30 | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 5 | 5 | 2 | 9 | 9 | 45 | 45 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 10 | ··· | 10 | 18 | 18 | 18 | 18 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D5 | Dic3 | D6 | D9 | D10 | Dic9 | D18 | C4×D5 | S3×D5 | D5×Dic3 | D5×D9 | D5×Dic9 |
| kernel | D5×Dic9 | C5×Dic9 | Dic45 | D5×C18 | C9×D5 | C6×D5 | Dic9 | C3×D5 | C30 | D10 | C18 | D5 | C10 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 1 | 3 | 2 | 6 | 3 | 4 | 2 | 2 | 6 | 6 |
Matrix representation of D5×Dic9 ►in GL4(𝔽181) generated by
| 166 | 3 | 0 | 0 |
| 55 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 180 | 0 | 0 | 0 |
| 55 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 180 | 0 | 0 | 0 |
| 0 | 180 | 0 | 0 |
| 0 | 0 | 131 | 54 |
| 0 | 0 | 127 | 4 |
| 19 | 0 | 0 | 0 |
| 0 | 19 | 0 | 0 |
| 0 | 0 | 159 | 11 |
| 0 | 0 | 170 | 22 |
G:=sub<GL(4,GF(181))| [166,55,0,0,3,1,0,0,0,0,1,0,0,0,0,1],[180,55,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[180,0,0,0,0,180,0,0,0,0,131,127,0,0,54,4],[19,0,0,0,0,19,0,0,0,0,159,170,0,0,11,22] >;
D5×Dic9 in GAP, Magma, Sage, TeX
D_5\times {\rm Dic}_9 % in TeX
G:=Group("D5xDic9"); // GroupNames label
G:=SmallGroup(360,11);
// by ID
G=gap.SmallGroup(360,11);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-3,31,1641,741,2884,4331]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^18=1,d^2=c^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export