direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D9×C20, C180⋊5C2, C36⋊2C10, D18.C10, C60.13S3, C30.57D6, Dic9⋊2C10, C10.14D18, C90.14C22, C9⋊1(C2×C20), C45⋊8(C2×C4), C3.(S3×C20), C15.5(C4×S3), C6.7(S3×C10), C12.5(C5×S3), C2.1(C10×D9), C18.2(C2×C10), (C5×Dic9)⋊5C2, (C10×D9).2C2, SmallGroup(360,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C20 |
Generators and relations for D9×C20
G = < a,b,c | a20=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D9×C20
►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 50 25 96 102 76 174 124 144)(2 51 26 97 103 77 175 125 145)(3 52 27 98 104 78 176 126 146)(4 53 28 99 105 79 177 127 147)(5 54 29 100 106 80 178 128 148)(6 55 30 81 107 61 179 129 149)(7 56 31 82 108 62 180 130 150)(8 57 32 83 109 63 161 131 151)(9 58 33 84 110 64 162 132 152)(10 59 34 85 111 65 163 133 153)(11 60 35 86 112 66 164 134 154)(12 41 36 87 113 67 165 135 155)(13 42 37 88 114 68 166 136 156)(14 43 38 89 115 69 167 137 157)(15 44 39 90 116 70 168 138 158)(16 45 40 91 117 71 169 139 159)(17 46 21 92 118 72 170 140 160)(18 47 22 93 119 73 171 121 141)(19 48 23 94 120 74 172 122 142)(20 49 24 95 101 75 173 123 143)
(1 154)(2 155)(3 156)(4 157)(5 158)(6 159)(7 160)(8 141)(9 142)(10 143)(11 144)(12 145)(13 146)(14 147)(15 148)(16 149)(17 150)(18 151)(19 152)(20 153)(21 180)(22 161)(23 162)(24 163)(25 164)(26 165)(27 166)(28 167)(29 168)(30 169)(31 170)(32 171)(33 172)(34 173)(35 174)(36 175)(37 176)(38 177)(39 178)(40 179)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(57 121)(58 122)(59 123)(60 124)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
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), (1,50,25,96,102,76,174,124,144)(2,51,26,97,103,77,175,125,145)(3,52,27,98,104,78,176,126,146)(4,53,28,99,105,79,177,127,147)(5,54,29,100,106,80,178,128,148)(6,55,30,81,107,61,179,129,149)(7,56,31,82,108,62,180,130,150)(8,57,32,83,109,63,161,131,151)(9,58,33,84,110,64,162,132,152)(10,59,34,85,111,65,163,133,153)(11,60,35,86,112,66,164,134,154)(12,41,36,87,113,67,165,135,155)(13,42,37,88,114,68,166,136,156)(14,43,38,89,115,69,167,137,157)(15,44,39,90,116,70,168,138,158)(16,45,40,91,117,71,169,139,159)(17,46,21,92,118,72,170,140,160)(18,47,22,93,119,73,171,121,141)(19,48,23,94,120,74,172,122,142)(20,49,24,95,101,75,173,123,143), (1,154)(2,155)(3,156)(4,157)(5,158)(6,159)(7,160)(8,141)(9,142)(10,143)(11,144)(12,145)(13,146)(14,147)(15,148)(16,149)(17,150)(18,151)(19,152)(20,153)(21,180)(22,161)(23,162)(24,163)(25,164)(26,165)(27,166)(28,167)(29,168)(30,169)(31,170)(32,171)(33,172)(34,173)(35,174)(36,175)(37,176)(38,177)(39,178)(40,179)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(57,121)(58,122)(59,123)(60,124)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)>;
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), (1,50,25,96,102,76,174,124,144)(2,51,26,97,103,77,175,125,145)(3,52,27,98,104,78,176,126,146)(4,53,28,99,105,79,177,127,147)(5,54,29,100,106,80,178,128,148)(6,55,30,81,107,61,179,129,149)(7,56,31,82,108,62,180,130,150)(8,57,32,83,109,63,161,131,151)(9,58,33,84,110,64,162,132,152)(10,59,34,85,111,65,163,133,153)(11,60,35,86,112,66,164,134,154)(12,41,36,87,113,67,165,135,155)(13,42,37,88,114,68,166,136,156)(14,43,38,89,115,69,167,137,157)(15,44,39,90,116,70,168,138,158)(16,45,40,91,117,71,169,139,159)(17,46,21,92,118,72,170,140,160)(18,47,22,93,119,73,171,121,141)(19,48,23,94,120,74,172,122,142)(20,49,24,95,101,75,173,123,143), (1,154)(2,155)(3,156)(4,157)(5,158)(6,159)(7,160)(8,141)(9,142)(10,143)(11,144)(12,145)(13,146)(14,147)(15,148)(16,149)(17,150)(18,151)(19,152)(20,153)(21,180)(22,161)(23,162)(24,163)(25,164)(26,165)(27,166)(28,167)(29,168)(30,169)(31,170)(32,171)(33,172)(34,173)(35,174)(36,175)(37,176)(38,177)(39,178)(40,179)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(57,121)(58,122)(59,123)(60,124)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120) );
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)], [(1,50,25,96,102,76,174,124,144),(2,51,26,97,103,77,175,125,145),(3,52,27,98,104,78,176,126,146),(4,53,28,99,105,79,177,127,147),(5,54,29,100,106,80,178,128,148),(6,55,30,81,107,61,179,129,149),(7,56,31,82,108,62,180,130,150),(8,57,32,83,109,63,161,131,151),(9,58,33,84,110,64,162,132,152),(10,59,34,85,111,65,163,133,153),(11,60,35,86,112,66,164,134,154),(12,41,36,87,113,67,165,135,155),(13,42,37,88,114,68,166,136,156),(14,43,38,89,115,69,167,137,157),(15,44,39,90,116,70,168,138,158),(16,45,40,91,117,71,169,139,159),(17,46,21,92,118,72,170,140,160),(18,47,22,93,119,73,171,121,141),(19,48,23,94,120,74,172,122,142),(20,49,24,95,101,75,173,123,143)], [(1,154),(2,155),(3,156),(4,157),(5,158),(6,159),(7,160),(8,141),(9,142),(10,143),(11,144),(12,145),(13,146),(14,147),(15,148),(16,149),(17,150),(18,151),(19,152),(20,153),(21,180),(22,161),(23,162),(24,163),(25,164),(26,165),(27,166),(28,167),(29,168),(30,169),(31,170),(32,171),(33,172),(34,173),(35,174),(36,175),(37,176),(38,177),(39,178),(40,179),(41,125),(42,126),(43,127),(44,128),(45,129),(46,130),(47,131),(48,132),(49,133),(50,134),(51,135),(52,136),(53,137),(54,138),(55,139),(56,140),(57,121),(58,122),(59,123),(60,124),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,81),(72,82),(73,83),(74,84),(75,85),(76,86),(77,87),(78,88),(79,89),(80,90),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6 | 9A | 9B | 9C | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 15A | 15B | 15C | 15D | 18A | 18B | 18C | 20A | ··· | 20H | 20I | ··· | 20P | 30A | 30B | 30C | 30D | 36A | ··· | 36F | 45A | ··· | 45L | 60A | ··· | 60H | 90A | ··· | 90L | 180A | ··· | 180X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 18 | 18 | 18 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 60 | ··· | 60 | 90 | ··· | 90 | 180 | ··· | 180 |
| size | 1 | 1 | 9 | 9 | 2 | 1 | 1 | 9 | 9 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D6 | D9 | C4×S3 | C5×S3 | D18 | S3×C10 | C4×D9 | C5×D9 | S3×C20 | C10×D9 | D9×C20 |
| kernel | D9×C20 | C5×Dic9 | C180 | C10×D9 | C5×D9 | C4×D9 | Dic9 | C36 | D18 | D9 | C60 | C30 | C20 | C15 | C12 | C10 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 1 | 3 | 2 | 4 | 3 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of D9×C20 ►in GL3(𝔽181) generated by
| 139 | 0 | 0 |
| 0 | 162 | 0 |
| 0 | 0 | 162 |
| 1 | 0 | 0 |
| 0 | 177 | 131 |
| 0 | 50 | 127 |
| 180 | 0 | 0 |
| 0 | 54 | 177 |
| 0 | 50 | 127 |
G:=sub<GL(3,GF(181))| [139,0,0,0,162,0,0,0,162],[1,0,0,0,177,50,0,131,127],[180,0,0,0,54,50,0,177,127] >;
D9×C20 in GAP, Magma, Sage, TeX
D_9\times C_{20} % in TeX
G:=Group("D9xC20"); // GroupNames label
G:=SmallGroup(360,21);
// by ID
G=gap.SmallGroup(360,21);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-3,-3,127,6004,208,8645]);
// Polycyclic
G:=Group<a,b,c|a^20=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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