direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D9×C25, C9⋊C50, C225⋊3C2, C45.C10, C75.2S3, C5.(C5×D9), C3.(S3×C25), (C5×D9).C5, C15.2(C5×S3), SmallGroup(450,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C25 |
Generators and relations for D9×C25
G = < a,b,c | a25=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D9×C25
►On 225 points
Generators in S225
(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)(176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225)
(1 109 35 79 210 191 61 129 154)(2 110 36 80 211 192 62 130 155)(3 111 37 81 212 193 63 131 156)(4 112 38 82 213 194 64 132 157)(5 113 39 83 214 195 65 133 158)(6 114 40 84 215 196 66 134 159)(7 115 41 85 216 197 67 135 160)(8 116 42 86 217 198 68 136 161)(9 117 43 87 218 199 69 137 162)(10 118 44 88 219 200 70 138 163)(11 119 45 89 220 176 71 139 164)(12 120 46 90 221 177 72 140 165)(13 121 47 91 222 178 73 141 166)(14 122 48 92 223 179 74 142 167)(15 123 49 93 224 180 75 143 168)(16 124 50 94 225 181 51 144 169)(17 125 26 95 201 182 52 145 170)(18 101 27 96 202 183 53 146 171)(19 102 28 97 203 184 54 147 172)(20 103 29 98 204 185 55 148 173)(21 104 30 99 205 186 56 149 174)(22 105 31 100 206 187 57 150 175)(23 106 32 76 207 188 58 126 151)(24 107 33 77 208 189 59 127 152)(25 108 34 78 209 190 60 128 153)
(1 154)(2 155)(3 156)(4 157)(5 158)(6 159)(7 160)(8 161)(9 162)(10 163)(11 164)(12 165)(13 166)(14 167)(15 168)(16 169)(17 170)(18 171)(19 172)(20 173)(21 174)(22 175)(23 151)(24 152)(25 153)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 73)(48 74)(49 75)(50 51)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 176)(90 177)(91 178)(92 179)(93 180)(94 181)(95 182)(96 183)(97 184)(98 185)(99 186)(100 187)(101 146)(102 147)(103 148)(104 149)(105 150)(106 126)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)(121 141)(122 142)(123 143)(124 144)(125 145)
G:=sub<Sym(225)| (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)(176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225), (1,109,35,79,210,191,61,129,154)(2,110,36,80,211,192,62,130,155)(3,111,37,81,212,193,63,131,156)(4,112,38,82,213,194,64,132,157)(5,113,39,83,214,195,65,133,158)(6,114,40,84,215,196,66,134,159)(7,115,41,85,216,197,67,135,160)(8,116,42,86,217,198,68,136,161)(9,117,43,87,218,199,69,137,162)(10,118,44,88,219,200,70,138,163)(11,119,45,89,220,176,71,139,164)(12,120,46,90,221,177,72,140,165)(13,121,47,91,222,178,73,141,166)(14,122,48,92,223,179,74,142,167)(15,123,49,93,224,180,75,143,168)(16,124,50,94,225,181,51,144,169)(17,125,26,95,201,182,52,145,170)(18,101,27,96,202,183,53,146,171)(19,102,28,97,203,184,54,147,172)(20,103,29,98,204,185,55,148,173)(21,104,30,99,205,186,56,149,174)(22,105,31,100,206,187,57,150,175)(23,106,32,76,207,188,58,126,151)(24,107,33,77,208,189,59,127,152)(25,108,34,78,209,190,60,128,153), (1,154)(2,155)(3,156)(4,157)(5,158)(6,159)(7,160)(8,161)(9,162)(10,163)(11,164)(12,165)(13,166)(14,167)(15,168)(16,169)(17,170)(18,171)(19,172)(20,173)(21,174)(22,175)(23,151)(24,152)(25,153)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74)(49,75)(50,51)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,176)(90,177)(91,178)(92,179)(93,180)(94,181)(95,182)(96,183)(97,184)(98,185)(99,186)(100,187)(101,146)(102,147)(103,148)(104,149)(105,150)(106,126)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(121,141)(122,142)(123,143)(124,144)(125,145)>;
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)(176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225), (1,109,35,79,210,191,61,129,154)(2,110,36,80,211,192,62,130,155)(3,111,37,81,212,193,63,131,156)(4,112,38,82,213,194,64,132,157)(5,113,39,83,214,195,65,133,158)(6,114,40,84,215,196,66,134,159)(7,115,41,85,216,197,67,135,160)(8,116,42,86,217,198,68,136,161)(9,117,43,87,218,199,69,137,162)(10,118,44,88,219,200,70,138,163)(11,119,45,89,220,176,71,139,164)(12,120,46,90,221,177,72,140,165)(13,121,47,91,222,178,73,141,166)(14,122,48,92,223,179,74,142,167)(15,123,49,93,224,180,75,143,168)(16,124,50,94,225,181,51,144,169)(17,125,26,95,201,182,52,145,170)(18,101,27,96,202,183,53,146,171)(19,102,28,97,203,184,54,147,172)(20,103,29,98,204,185,55,148,173)(21,104,30,99,205,186,56,149,174)(22,105,31,100,206,187,57,150,175)(23,106,32,76,207,188,58,126,151)(24,107,33,77,208,189,59,127,152)(25,108,34,78,209,190,60,128,153), (1,154)(2,155)(3,156)(4,157)(5,158)(6,159)(7,160)(8,161)(9,162)(10,163)(11,164)(12,165)(13,166)(14,167)(15,168)(16,169)(17,170)(18,171)(19,172)(20,173)(21,174)(22,175)(23,151)(24,152)(25,153)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74)(49,75)(50,51)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,176)(90,177)(91,178)(92,179)(93,180)(94,181)(95,182)(96,183)(97,184)(98,185)(99,186)(100,187)(101,146)(102,147)(103,148)(104,149)(105,150)(106,126)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(121,141)(122,142)(123,143)(124,144)(125,145) );
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),(176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)], [(1,109,35,79,210,191,61,129,154),(2,110,36,80,211,192,62,130,155),(3,111,37,81,212,193,63,131,156),(4,112,38,82,213,194,64,132,157),(5,113,39,83,214,195,65,133,158),(6,114,40,84,215,196,66,134,159),(7,115,41,85,216,197,67,135,160),(8,116,42,86,217,198,68,136,161),(9,117,43,87,218,199,69,137,162),(10,118,44,88,219,200,70,138,163),(11,119,45,89,220,176,71,139,164),(12,120,46,90,221,177,72,140,165),(13,121,47,91,222,178,73,141,166),(14,122,48,92,223,179,74,142,167),(15,123,49,93,224,180,75,143,168),(16,124,50,94,225,181,51,144,169),(17,125,26,95,201,182,52,145,170),(18,101,27,96,202,183,53,146,171),(19,102,28,97,203,184,54,147,172),(20,103,29,98,204,185,55,148,173),(21,104,30,99,205,186,56,149,174),(22,105,31,100,206,187,57,150,175),(23,106,32,76,207,188,58,126,151),(24,107,33,77,208,189,59,127,152),(25,108,34,78,209,190,60,128,153)], [(1,154),(2,155),(3,156),(4,157),(5,158),(6,159),(7,160),(8,161),(9,162),(10,163),(11,164),(12,165),(13,166),(14,167),(15,168),(16,169),(17,170),(18,171),(19,172),(20,173),(21,174),(22,175),(23,151),(24,152),(25,153),(26,52),(27,53),(28,54),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66),(41,67),(42,68),(43,69),(44,70),(45,71),(46,72),(47,73),(48,74),(49,75),(50,51),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,176),(90,177),(91,178),(92,179),(93,180),(94,181),(95,182),(96,183),(97,184),(98,185),(99,186),(100,187),(101,146),(102,147),(103,148),(104,149),(105,150),(106,126),(107,127),(108,128),(109,129),(110,130),(111,131),(112,132),(113,133),(114,134),(115,135),(116,136),(117,137),(118,138),(119,139),(120,140),(121,141),(122,142),(123,143),(124,144),(125,145)]])
150 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 9A | 9B | 9C | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 25A | ··· | 25T | 45A | ··· | 45L | 50A | ··· | 50T | 75A | ··· | 75T | 225A | ··· | 225BH |
| order | 1 | 2 | 3 | 5 | 5 | 5 | 5 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 25 | ··· | 25 | 45 | ··· | 45 | 50 | ··· | 50 | 75 | ··· | 75 | 225 | ··· | 225 |
| size | 1 | 9 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C5 | C10 | C25 | C50 | S3 | D9 | C5×S3 | C5×D9 | S3×C25 | D9×C25 |
| kernel | D9×C25 | C225 | C5×D9 | C45 | D9 | C9 | C75 | C25 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 4 | 4 | 20 | 20 | 1 | 3 | 4 | 12 | 20 | 60 |
Matrix representation of D9×C25 ►in GL2(𝔽1801) generated by
| 128 | 0 |
| 0 | 128 |
| 1067 | 298 |
| 1503 | 1365 |
| 1503 | 1365 |
| 1067 | 298 |
G:=sub<GL(2,GF(1801))| [128,0,0,128],[1067,1503,298,1365],[1503,1067,1365,298] >;
D9×C25 in GAP, Magma, Sage, TeX
D_9\times C_{25} % in TeX
G:=Group("D9xC25"); // GroupNames label
G:=SmallGroup(450,1);
// by ID
G=gap.SmallGroup(450,1);
# by ID
G:=PCGroup([5,-2,-5,-5,-3,-3,56,5003,138,7504]);
// Polycyclic
G:=Group<a,b,c|a^25=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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