direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D9×C52, C45⋊2C10, C9⋊(C5×C10), (C5×C45)⋊3C2, C3.(S3×C52), C15.4(C5×S3), (C5×C15).4S3, SmallGroup(450,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C52 |
Generators and relations for D9×C52
G = < a,b,c,d | a5=b5=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of D9×C52
►On 225 points
Generators in S225
(1 218 164 110 56)(2 219 165 111 57)(3 220 166 112 58)(4 221 167 113 59)(5 222 168 114 60)(6 223 169 115 61)(7 224 170 116 62)(8 225 171 117 63)(9 217 163 109 55)(10 181 172 118 64)(11 182 173 119 65)(12 183 174 120 66)(13 184 175 121 67)(14 185 176 122 68)(15 186 177 123 69)(16 187 178 124 70)(17 188 179 125 71)(18 189 180 126 72)(19 190 136 127 73)(20 191 137 128 74)(21 192 138 129 75)(22 193 139 130 76)(23 194 140 131 77)(24 195 141 132 78)(25 196 142 133 79)(26 197 143 134 80)(27 198 144 135 81)(28 199 145 91 82)(29 200 146 92 83)(30 201 147 93 84)(31 202 148 94 85)(32 203 149 95 86)(33 204 150 96 87)(34 205 151 97 88)(35 206 152 98 89)(36 207 153 99 90)(37 208 154 100 46)(38 209 155 101 47)(39 210 156 102 48)(40 211 157 103 49)(41 212 158 104 50)(42 213 159 105 51)(43 214 160 106 52)(44 215 161 107 53)(45 216 162 108 54)
(1 38 29 20 11)(2 39 30 21 12)(3 40 31 22 13)(4 41 32 23 14)(5 42 33 24 15)(6 43 34 25 16)(7 44 35 26 17)(8 45 36 27 18)(9 37 28 19 10)(46 82 73 64 55)(47 83 74 65 56)(48 84 75 66 57)(49 85 76 67 58)(50 86 77 68 59)(51 87 78 69 60)(52 88 79 70 61)(53 89 80 71 62)(54 90 81 72 63)(91 127 118 109 100)(92 128 119 110 101)(93 129 120 111 102)(94 130 121 112 103)(95 131 122 113 104)(96 132 123 114 105)(97 133 124 115 106)(98 134 125 116 107)(99 135 126 117 108)(136 172 163 154 145)(137 173 164 155 146)(138 174 165 156 147)(139 175 166 157 148)(140 176 167 158 149)(141 177 168 159 150)(142 178 169 160 151)(143 179 170 161 152)(144 180 171 162 153)(181 217 208 199 190)(182 218 209 200 191)(183 219 210 201 192)(184 220 211 202 193)(185 221 212 203 194)(186 222 213 204 195)(187 223 214 205 196)(188 224 215 206 197)(189 225 216 207 198)
(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 9)(2 8)(3 7)(4 6)(10 11)(12 18)(13 17)(14 16)(19 20)(21 27)(22 26)(23 25)(28 29)(30 36)(31 35)(32 34)(37 38)(39 45)(40 44)(41 43)(46 47)(48 54)(49 53)(50 52)(55 56)(57 63)(58 62)(59 61)(64 65)(66 72)(67 71)(68 70)(73 74)(75 81)(76 80)(77 79)(82 83)(84 90)(85 89)(86 88)(91 92)(93 99)(94 98)(95 97)(100 101)(102 108)(103 107)(104 106)(109 110)(111 117)(112 116)(113 115)(118 119)(120 126)(121 125)(122 124)(127 128)(129 135)(130 134)(131 133)(136 137)(138 144)(139 143)(140 142)(145 146)(147 153)(148 152)(149 151)(154 155)(156 162)(157 161)(158 160)(163 164)(165 171)(166 170)(167 169)(172 173)(174 180)(175 179)(176 178)(181 182)(183 189)(184 188)(185 187)(190 191)(192 198)(193 197)(194 196)(199 200)(201 207)(202 206)(203 205)(208 209)(210 216)(211 215)(212 214)(217 218)(219 225)(220 224)(221 223)
G:=sub<Sym(225)| (1,218,164,110,56)(2,219,165,111,57)(3,220,166,112,58)(4,221,167,113,59)(5,222,168,114,60)(6,223,169,115,61)(7,224,170,116,62)(8,225,171,117,63)(9,217,163,109,55)(10,181,172,118,64)(11,182,173,119,65)(12,183,174,120,66)(13,184,175,121,67)(14,185,176,122,68)(15,186,177,123,69)(16,187,178,124,70)(17,188,179,125,71)(18,189,180,126,72)(19,190,136,127,73)(20,191,137,128,74)(21,192,138,129,75)(22,193,139,130,76)(23,194,140,131,77)(24,195,141,132,78)(25,196,142,133,79)(26,197,143,134,80)(27,198,144,135,81)(28,199,145,91,82)(29,200,146,92,83)(30,201,147,93,84)(31,202,148,94,85)(32,203,149,95,86)(33,204,150,96,87)(34,205,151,97,88)(35,206,152,98,89)(36,207,153,99,90)(37,208,154,100,46)(38,209,155,101,47)(39,210,156,102,48)(40,211,157,103,49)(41,212,158,104,50)(42,213,159,105,51)(43,214,160,106,52)(44,215,161,107,53)(45,216,162,108,54), (1,38,29,20,11)(2,39,30,21,12)(3,40,31,22,13)(4,41,32,23,14)(5,42,33,24,15)(6,43,34,25,16)(7,44,35,26,17)(8,45,36,27,18)(9,37,28,19,10)(46,82,73,64,55)(47,83,74,65,56)(48,84,75,66,57)(49,85,76,67,58)(50,86,77,68,59)(51,87,78,69,60)(52,88,79,70,61)(53,89,80,71,62)(54,90,81,72,63)(91,127,118,109,100)(92,128,119,110,101)(93,129,120,111,102)(94,130,121,112,103)(95,131,122,113,104)(96,132,123,114,105)(97,133,124,115,106)(98,134,125,116,107)(99,135,126,117,108)(136,172,163,154,145)(137,173,164,155,146)(138,174,165,156,147)(139,175,166,157,148)(140,176,167,158,149)(141,177,168,159,150)(142,178,169,160,151)(143,179,170,161,152)(144,180,171,162,153)(181,217,208,199,190)(182,218,209,200,191)(183,219,210,201,192)(184,220,211,202,193)(185,221,212,203,194)(186,222,213,204,195)(187,223,214,205,196)(188,224,215,206,197)(189,225,216,207,198), (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), (1,9)(2,8)(3,7)(4,6)(10,11)(12,18)(13,17)(14,16)(19,20)(21,27)(22,26)(23,25)(28,29)(30,36)(31,35)(32,34)(37,38)(39,45)(40,44)(41,43)(46,47)(48,54)(49,53)(50,52)(55,56)(57,63)(58,62)(59,61)(64,65)(66,72)(67,71)(68,70)(73,74)(75,81)(76,80)(77,79)(82,83)(84,90)(85,89)(86,88)(91,92)(93,99)(94,98)(95,97)(100,101)(102,108)(103,107)(104,106)(109,110)(111,117)(112,116)(113,115)(118,119)(120,126)(121,125)(122,124)(127,128)(129,135)(130,134)(131,133)(136,137)(138,144)(139,143)(140,142)(145,146)(147,153)(148,152)(149,151)(154,155)(156,162)(157,161)(158,160)(163,164)(165,171)(166,170)(167,169)(172,173)(174,180)(175,179)(176,178)(181,182)(183,189)(184,188)(185,187)(190,191)(192,198)(193,197)(194,196)(199,200)(201,207)(202,206)(203,205)(208,209)(210,216)(211,215)(212,214)(217,218)(219,225)(220,224)(221,223)>;
G:=Group( (1,218,164,110,56)(2,219,165,111,57)(3,220,166,112,58)(4,221,167,113,59)(5,222,168,114,60)(6,223,169,115,61)(7,224,170,116,62)(8,225,171,117,63)(9,217,163,109,55)(10,181,172,118,64)(11,182,173,119,65)(12,183,174,120,66)(13,184,175,121,67)(14,185,176,122,68)(15,186,177,123,69)(16,187,178,124,70)(17,188,179,125,71)(18,189,180,126,72)(19,190,136,127,73)(20,191,137,128,74)(21,192,138,129,75)(22,193,139,130,76)(23,194,140,131,77)(24,195,141,132,78)(25,196,142,133,79)(26,197,143,134,80)(27,198,144,135,81)(28,199,145,91,82)(29,200,146,92,83)(30,201,147,93,84)(31,202,148,94,85)(32,203,149,95,86)(33,204,150,96,87)(34,205,151,97,88)(35,206,152,98,89)(36,207,153,99,90)(37,208,154,100,46)(38,209,155,101,47)(39,210,156,102,48)(40,211,157,103,49)(41,212,158,104,50)(42,213,159,105,51)(43,214,160,106,52)(44,215,161,107,53)(45,216,162,108,54), (1,38,29,20,11)(2,39,30,21,12)(3,40,31,22,13)(4,41,32,23,14)(5,42,33,24,15)(6,43,34,25,16)(7,44,35,26,17)(8,45,36,27,18)(9,37,28,19,10)(46,82,73,64,55)(47,83,74,65,56)(48,84,75,66,57)(49,85,76,67,58)(50,86,77,68,59)(51,87,78,69,60)(52,88,79,70,61)(53,89,80,71,62)(54,90,81,72,63)(91,127,118,109,100)(92,128,119,110,101)(93,129,120,111,102)(94,130,121,112,103)(95,131,122,113,104)(96,132,123,114,105)(97,133,124,115,106)(98,134,125,116,107)(99,135,126,117,108)(136,172,163,154,145)(137,173,164,155,146)(138,174,165,156,147)(139,175,166,157,148)(140,176,167,158,149)(141,177,168,159,150)(142,178,169,160,151)(143,179,170,161,152)(144,180,171,162,153)(181,217,208,199,190)(182,218,209,200,191)(183,219,210,201,192)(184,220,211,202,193)(185,221,212,203,194)(186,222,213,204,195)(187,223,214,205,196)(188,224,215,206,197)(189,225,216,207,198), (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), (1,9)(2,8)(3,7)(4,6)(10,11)(12,18)(13,17)(14,16)(19,20)(21,27)(22,26)(23,25)(28,29)(30,36)(31,35)(32,34)(37,38)(39,45)(40,44)(41,43)(46,47)(48,54)(49,53)(50,52)(55,56)(57,63)(58,62)(59,61)(64,65)(66,72)(67,71)(68,70)(73,74)(75,81)(76,80)(77,79)(82,83)(84,90)(85,89)(86,88)(91,92)(93,99)(94,98)(95,97)(100,101)(102,108)(103,107)(104,106)(109,110)(111,117)(112,116)(113,115)(118,119)(120,126)(121,125)(122,124)(127,128)(129,135)(130,134)(131,133)(136,137)(138,144)(139,143)(140,142)(145,146)(147,153)(148,152)(149,151)(154,155)(156,162)(157,161)(158,160)(163,164)(165,171)(166,170)(167,169)(172,173)(174,180)(175,179)(176,178)(181,182)(183,189)(184,188)(185,187)(190,191)(192,198)(193,197)(194,196)(199,200)(201,207)(202,206)(203,205)(208,209)(210,216)(211,215)(212,214)(217,218)(219,225)(220,224)(221,223) );
G=PermutationGroup([[(1,218,164,110,56),(2,219,165,111,57),(3,220,166,112,58),(4,221,167,113,59),(5,222,168,114,60),(6,223,169,115,61),(7,224,170,116,62),(8,225,171,117,63),(9,217,163,109,55),(10,181,172,118,64),(11,182,173,119,65),(12,183,174,120,66),(13,184,175,121,67),(14,185,176,122,68),(15,186,177,123,69),(16,187,178,124,70),(17,188,179,125,71),(18,189,180,126,72),(19,190,136,127,73),(20,191,137,128,74),(21,192,138,129,75),(22,193,139,130,76),(23,194,140,131,77),(24,195,141,132,78),(25,196,142,133,79),(26,197,143,134,80),(27,198,144,135,81),(28,199,145,91,82),(29,200,146,92,83),(30,201,147,93,84),(31,202,148,94,85),(32,203,149,95,86),(33,204,150,96,87),(34,205,151,97,88),(35,206,152,98,89),(36,207,153,99,90),(37,208,154,100,46),(38,209,155,101,47),(39,210,156,102,48),(40,211,157,103,49),(41,212,158,104,50),(42,213,159,105,51),(43,214,160,106,52),(44,215,161,107,53),(45,216,162,108,54)], [(1,38,29,20,11),(2,39,30,21,12),(3,40,31,22,13),(4,41,32,23,14),(5,42,33,24,15),(6,43,34,25,16),(7,44,35,26,17),(8,45,36,27,18),(9,37,28,19,10),(46,82,73,64,55),(47,83,74,65,56),(48,84,75,66,57),(49,85,76,67,58),(50,86,77,68,59),(51,87,78,69,60),(52,88,79,70,61),(53,89,80,71,62),(54,90,81,72,63),(91,127,118,109,100),(92,128,119,110,101),(93,129,120,111,102),(94,130,121,112,103),(95,131,122,113,104),(96,132,123,114,105),(97,133,124,115,106),(98,134,125,116,107),(99,135,126,117,108),(136,172,163,154,145),(137,173,164,155,146),(138,174,165,156,147),(139,175,166,157,148),(140,176,167,158,149),(141,177,168,159,150),(142,178,169,160,151),(143,179,170,161,152),(144,180,171,162,153),(181,217,208,199,190),(182,218,209,200,191),(183,219,210,201,192),(184,220,211,202,193),(185,221,212,203,194),(186,222,213,204,195),(187,223,214,205,196),(188,224,215,206,197),(189,225,216,207,198)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,11),(12,18),(13,17),(14,16),(19,20),(21,27),(22,26),(23,25),(28,29),(30,36),(31,35),(32,34),(37,38),(39,45),(40,44),(41,43),(46,47),(48,54),(49,53),(50,52),(55,56),(57,63),(58,62),(59,61),(64,65),(66,72),(67,71),(68,70),(73,74),(75,81),(76,80),(77,79),(82,83),(84,90),(85,89),(86,88),(91,92),(93,99),(94,98),(95,97),(100,101),(102,108),(103,107),(104,106),(109,110),(111,117),(112,116),(113,115),(118,119),(120,126),(121,125),(122,124),(127,128),(129,135),(130,134),(131,133),(136,137),(138,144),(139,143),(140,142),(145,146),(147,153),(148,152),(149,151),(154,155),(156,162),(157,161),(158,160),(163,164),(165,171),(166,170),(167,169),(172,173),(174,180),(175,179),(176,178),(181,182),(183,189),(184,188),(185,187),(190,191),(192,198),(193,197),(194,196),(199,200),(201,207),(202,206),(203,205),(208,209),(210,216),(211,215),(212,214),(217,218),(219,225),(220,224),(221,223)]])
150 conjugacy classes
| class | 1 | 2 | 3 | 5A | ··· | 5X | 9A | 9B | 9C | 10A | ··· | 10X | 15A | ··· | 15X | 45A | ··· | 45BT |
| order | 1 | 2 | 3 | 5 | ··· | 5 | 9 | 9 | 9 | 10 | ··· | 10 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 9 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C5 | C10 | S3 | D9 | C5×S3 | C5×D9 |
| kernel | D9×C52 | C5×C45 | C5×D9 | C45 | C5×C15 | C52 | C15 | C5 |
| # reps | 1 | 1 | 24 | 24 | 1 | 3 | 24 | 72 |
Matrix representation of D9×C52 ►in GL3(𝔽181) generated by
| 125 | 0 | 0 |
| 0 | 42 | 0 |
| 0 | 0 | 42 |
| 125 | 0 | 0 |
| 0 | 135 | 0 |
| 0 | 0 | 135 |
| 1 | 0 | 0 |
| 0 | 50 | 177 |
| 0 | 4 | 54 |
| 180 | 0 | 0 |
| 0 | 50 | 177 |
| 0 | 127 | 131 |
G:=sub<GL(3,GF(181))| [125,0,0,0,42,0,0,0,42],[125,0,0,0,135,0,0,0,135],[1,0,0,0,50,4,0,177,54],[180,0,0,0,50,127,0,177,131] >;
D9×C52 in GAP, Magma, Sage, TeX
D_9\times C_5^2
% in TeX
G:=Group("D9xC5^2"); // GroupNames label
G:=SmallGroup(450,16);
// by ID
G=gap.SmallGroup(450,16);
# by ID
G:=PCGroup([5,-2,-5,-5,-3,-3,5003,138,7504]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export