direct product, metabelian, supersoluble, monomial, A-group
Aliases: C9×C5⋊D5, C45⋊3D5, C52⋊4C18, C5⋊(C9×D5), (C5×C45)⋊5C2, (C5×C15).4C6, C15.4(C3×D5), C3.(C3×C5⋊D5), (C3×C5⋊D5).2C3, SmallGroup(450,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C9×C5⋊D5 |
Generators and relations for C9×C5⋊D5
G = < a,b,c,d | a9=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Smallest permutation representation of C9×C5⋊D5
►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 22 131 218 34)(2 23 132 219 35)(3 24 133 220 36)(4 25 134 221 28)(5 26 135 222 29)(6 27 127 223 30)(7 19 128 224 31)(8 20 129 225 32)(9 21 130 217 33)(10 150 206 39 119)(11 151 207 40 120)(12 152 199 41 121)(13 153 200 42 122)(14 145 201 43 123)(15 146 202 44 124)(16 147 203 45 125)(17 148 204 37 126)(18 149 205 38 118)(46 56 210 159 116)(47 57 211 160 117)(48 58 212 161 109)(49 59 213 162 110)(50 60 214 154 111)(51 61 215 155 112)(52 62 216 156 113)(53 63 208 157 114)(54 55 209 158 115)(64 178 95 142 88)(65 179 96 143 89)(66 180 97 144 90)(67 172 98 136 82)(68 173 99 137 83)(69 174 91 138 84)(70 175 92 139 85)(71 176 93 140 86)(72 177 94 141 87)(73 185 171 101 193)(74 186 163 102 194)(75 187 164 103 195)(76 188 165 104 196)(77 189 166 105 197)(78 181 167 106 198)(79 182 168 107 190)(80 183 169 108 191)(81 184 170 100 192)
(1 153 50 181 92)(2 145 51 182 93)(3 146 52 183 94)(4 147 53 184 95)(5 148 54 185 96)(6 149 46 186 97)(7 150 47 187 98)(8 151 48 188 99)(9 152 49 189 91)(10 117 75 172 31)(11 109 76 173 32)(12 110 77 174 33)(13 111 78 175 34)(14 112 79 176 35)(15 113 80 177 36)(16 114 81 178 28)(17 115 73 179 29)(18 116 74 180 30)(19 206 57 164 136)(20 207 58 165 137)(21 199 59 166 138)(22 200 60 167 139)(23 201 61 168 140)(24 202 62 169 141)(25 203 63 170 142)(26 204 55 171 143)(27 205 56 163 144)(37 209 101 89 135)(38 210 102 90 127)(39 211 103 82 128)(40 212 104 83 129)(41 213 105 84 130)(42 214 106 85 131)(43 215 107 86 132)(44 216 108 87 133)(45 208 100 88 134)(64 221 125 157 192)(65 222 126 158 193)(66 223 118 159 194)(67 224 119 160 195)(68 225 120 161 196)(69 217 121 162 197)(70 218 122 154 198)(71 219 123 155 190)(72 220 124 156 191)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 91)(10 164)(11 165)(12 166)(13 167)(14 168)(15 169)(16 170)(17 171)(18 163)(19 172)(20 173)(21 174)(22 175)(23 176)(24 177)(25 178)(26 179)(27 180)(28 142)(29 143)(30 144)(31 136)(32 137)(33 138)(34 139)(35 140)(36 141)(37 193)(38 194)(39 195)(40 196)(41 197)(42 198)(43 190)(44 191)(45 192)(55 115)(56 116)(57 117)(58 109)(59 110)(60 111)(61 112)(62 113)(63 114)(64 134)(65 135)(66 127)(67 128)(68 129)(69 130)(70 131)(71 132)(72 133)(73 204)(74 205)(75 206)(76 207)(77 199)(78 200)(79 201)(80 202)(81 203)(82 224)(83 225)(84 217)(85 218)(86 219)(87 220)(88 221)(89 222)(90 223)(100 125)(101 126)(102 118)(103 119)(104 120)(105 121)(106 122)(107 123)(108 124)(145 182)(146 183)(147 184)(148 185)(149 186)(150 187)(151 188)(152 189)(153 181)(154 214)(155 215)(156 216)(157 208)(158 209)(159 210)(160 211)(161 212)(162 213)
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), (1,22,131,218,34)(2,23,132,219,35)(3,24,133,220,36)(4,25,134,221,28)(5,26,135,222,29)(6,27,127,223,30)(7,19,128,224,31)(8,20,129,225,32)(9,21,130,217,33)(10,150,206,39,119)(11,151,207,40,120)(12,152,199,41,121)(13,153,200,42,122)(14,145,201,43,123)(15,146,202,44,124)(16,147,203,45,125)(17,148,204,37,126)(18,149,205,38,118)(46,56,210,159,116)(47,57,211,160,117)(48,58,212,161,109)(49,59,213,162,110)(50,60,214,154,111)(51,61,215,155,112)(52,62,216,156,113)(53,63,208,157,114)(54,55,209,158,115)(64,178,95,142,88)(65,179,96,143,89)(66,180,97,144,90)(67,172,98,136,82)(68,173,99,137,83)(69,174,91,138,84)(70,175,92,139,85)(71,176,93,140,86)(72,177,94,141,87)(73,185,171,101,193)(74,186,163,102,194)(75,187,164,103,195)(76,188,165,104,196)(77,189,166,105,197)(78,181,167,106,198)(79,182,168,107,190)(80,183,169,108,191)(81,184,170,100,192), (1,153,50,181,92)(2,145,51,182,93)(3,146,52,183,94)(4,147,53,184,95)(5,148,54,185,96)(6,149,46,186,97)(7,150,47,187,98)(8,151,48,188,99)(9,152,49,189,91)(10,117,75,172,31)(11,109,76,173,32)(12,110,77,174,33)(13,111,78,175,34)(14,112,79,176,35)(15,113,80,177,36)(16,114,81,178,28)(17,115,73,179,29)(18,116,74,180,30)(19,206,57,164,136)(20,207,58,165,137)(21,199,59,166,138)(22,200,60,167,139)(23,201,61,168,140)(24,202,62,169,141)(25,203,63,170,142)(26,204,55,171,143)(27,205,56,163,144)(37,209,101,89,135)(38,210,102,90,127)(39,211,103,82,128)(40,212,104,83,129)(41,213,105,84,130)(42,214,106,85,131)(43,215,107,86,132)(44,216,108,87,133)(45,208,100,88,134)(64,221,125,157,192)(65,222,126,158,193)(66,223,118,159,194)(67,224,119,160,195)(68,225,120,161,196)(69,217,121,162,197)(70,218,122,154,198)(71,219,123,155,190)(72,220,124,156,191), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,91)(10,164)(11,165)(12,166)(13,167)(14,168)(15,169)(16,170)(17,171)(18,163)(19,172)(20,173)(21,174)(22,175)(23,176)(24,177)(25,178)(26,179)(27,180)(28,142)(29,143)(30,144)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,193)(38,194)(39,195)(40,196)(41,197)(42,198)(43,190)(44,191)(45,192)(55,115)(56,116)(57,117)(58,109)(59,110)(60,111)(61,112)(62,113)(63,114)(64,134)(65,135)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,133)(73,204)(74,205)(75,206)(76,207)(77,199)(78,200)(79,201)(80,202)(81,203)(82,224)(83,225)(84,217)(85,218)(86,219)(87,220)(88,221)(89,222)(90,223)(100,125)(101,126)(102,118)(103,119)(104,120)(105,121)(106,122)(107,123)(108,124)(145,182)(146,183)(147,184)(148,185)(149,186)(150,187)(151,188)(152,189)(153,181)(154,214)(155,215)(156,216)(157,208)(158,209)(159,210)(160,211)(161,212)(162,213)>;
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), (1,22,131,218,34)(2,23,132,219,35)(3,24,133,220,36)(4,25,134,221,28)(5,26,135,222,29)(6,27,127,223,30)(7,19,128,224,31)(8,20,129,225,32)(9,21,130,217,33)(10,150,206,39,119)(11,151,207,40,120)(12,152,199,41,121)(13,153,200,42,122)(14,145,201,43,123)(15,146,202,44,124)(16,147,203,45,125)(17,148,204,37,126)(18,149,205,38,118)(46,56,210,159,116)(47,57,211,160,117)(48,58,212,161,109)(49,59,213,162,110)(50,60,214,154,111)(51,61,215,155,112)(52,62,216,156,113)(53,63,208,157,114)(54,55,209,158,115)(64,178,95,142,88)(65,179,96,143,89)(66,180,97,144,90)(67,172,98,136,82)(68,173,99,137,83)(69,174,91,138,84)(70,175,92,139,85)(71,176,93,140,86)(72,177,94,141,87)(73,185,171,101,193)(74,186,163,102,194)(75,187,164,103,195)(76,188,165,104,196)(77,189,166,105,197)(78,181,167,106,198)(79,182,168,107,190)(80,183,169,108,191)(81,184,170,100,192), (1,153,50,181,92)(2,145,51,182,93)(3,146,52,183,94)(4,147,53,184,95)(5,148,54,185,96)(6,149,46,186,97)(7,150,47,187,98)(8,151,48,188,99)(9,152,49,189,91)(10,117,75,172,31)(11,109,76,173,32)(12,110,77,174,33)(13,111,78,175,34)(14,112,79,176,35)(15,113,80,177,36)(16,114,81,178,28)(17,115,73,179,29)(18,116,74,180,30)(19,206,57,164,136)(20,207,58,165,137)(21,199,59,166,138)(22,200,60,167,139)(23,201,61,168,140)(24,202,62,169,141)(25,203,63,170,142)(26,204,55,171,143)(27,205,56,163,144)(37,209,101,89,135)(38,210,102,90,127)(39,211,103,82,128)(40,212,104,83,129)(41,213,105,84,130)(42,214,106,85,131)(43,215,107,86,132)(44,216,108,87,133)(45,208,100,88,134)(64,221,125,157,192)(65,222,126,158,193)(66,223,118,159,194)(67,224,119,160,195)(68,225,120,161,196)(69,217,121,162,197)(70,218,122,154,198)(71,219,123,155,190)(72,220,124,156,191), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,91)(10,164)(11,165)(12,166)(13,167)(14,168)(15,169)(16,170)(17,171)(18,163)(19,172)(20,173)(21,174)(22,175)(23,176)(24,177)(25,178)(26,179)(27,180)(28,142)(29,143)(30,144)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,193)(38,194)(39,195)(40,196)(41,197)(42,198)(43,190)(44,191)(45,192)(55,115)(56,116)(57,117)(58,109)(59,110)(60,111)(61,112)(62,113)(63,114)(64,134)(65,135)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,133)(73,204)(74,205)(75,206)(76,207)(77,199)(78,200)(79,201)(80,202)(81,203)(82,224)(83,225)(84,217)(85,218)(86,219)(87,220)(88,221)(89,222)(90,223)(100,125)(101,126)(102,118)(103,119)(104,120)(105,121)(106,122)(107,123)(108,124)(145,182)(146,183)(147,184)(148,185)(149,186)(150,187)(151,188)(152,189)(153,181)(154,214)(155,215)(156,216)(157,208)(158,209)(159,210)(160,211)(161,212)(162,213) );
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)], [(1,22,131,218,34),(2,23,132,219,35),(3,24,133,220,36),(4,25,134,221,28),(5,26,135,222,29),(6,27,127,223,30),(7,19,128,224,31),(8,20,129,225,32),(9,21,130,217,33),(10,150,206,39,119),(11,151,207,40,120),(12,152,199,41,121),(13,153,200,42,122),(14,145,201,43,123),(15,146,202,44,124),(16,147,203,45,125),(17,148,204,37,126),(18,149,205,38,118),(46,56,210,159,116),(47,57,211,160,117),(48,58,212,161,109),(49,59,213,162,110),(50,60,214,154,111),(51,61,215,155,112),(52,62,216,156,113),(53,63,208,157,114),(54,55,209,158,115),(64,178,95,142,88),(65,179,96,143,89),(66,180,97,144,90),(67,172,98,136,82),(68,173,99,137,83),(69,174,91,138,84),(70,175,92,139,85),(71,176,93,140,86),(72,177,94,141,87),(73,185,171,101,193),(74,186,163,102,194),(75,187,164,103,195),(76,188,165,104,196),(77,189,166,105,197),(78,181,167,106,198),(79,182,168,107,190),(80,183,169,108,191),(81,184,170,100,192)], [(1,153,50,181,92),(2,145,51,182,93),(3,146,52,183,94),(4,147,53,184,95),(5,148,54,185,96),(6,149,46,186,97),(7,150,47,187,98),(8,151,48,188,99),(9,152,49,189,91),(10,117,75,172,31),(11,109,76,173,32),(12,110,77,174,33),(13,111,78,175,34),(14,112,79,176,35),(15,113,80,177,36),(16,114,81,178,28),(17,115,73,179,29),(18,116,74,180,30),(19,206,57,164,136),(20,207,58,165,137),(21,199,59,166,138),(22,200,60,167,139),(23,201,61,168,140),(24,202,62,169,141),(25,203,63,170,142),(26,204,55,171,143),(27,205,56,163,144),(37,209,101,89,135),(38,210,102,90,127),(39,211,103,82,128),(40,212,104,83,129),(41,213,105,84,130),(42,214,106,85,131),(43,215,107,86,132),(44,216,108,87,133),(45,208,100,88,134),(64,221,125,157,192),(65,222,126,158,193),(66,223,118,159,194),(67,224,119,160,195),(68,225,120,161,196),(69,217,121,162,197),(70,218,122,154,198),(71,219,123,155,190),(72,220,124,156,191)], [(1,92),(2,93),(3,94),(4,95),(5,96),(6,97),(7,98),(8,99),(9,91),(10,164),(11,165),(12,166),(13,167),(14,168),(15,169),(16,170),(17,171),(18,163),(19,172),(20,173),(21,174),(22,175),(23,176),(24,177),(25,178),(26,179),(27,180),(28,142),(29,143),(30,144),(31,136),(32,137),(33,138),(34,139),(35,140),(36,141),(37,193),(38,194),(39,195),(40,196),(41,197),(42,198),(43,190),(44,191),(45,192),(55,115),(56,116),(57,117),(58,109),(59,110),(60,111),(61,112),(62,113),(63,114),(64,134),(65,135),(66,127),(67,128),(68,129),(69,130),(70,131),(71,132),(72,133),(73,204),(74,205),(75,206),(76,207),(77,199),(78,200),(79,201),(80,202),(81,203),(82,224),(83,225),(84,217),(85,218),(86,219),(87,220),(88,221),(89,222),(90,223),(100,125),(101,126),(102,118),(103,119),(104,120),(105,121),(106,122),(107,123),(108,124),(145,182),(146,183),(147,184),(148,185),(149,186),(150,187),(151,188),(152,189),(153,181),(154,214),(155,215),(156,216),(157,208),(158,209),(159,210),(160,211),(161,212),(162,213)]])
126 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | ··· | 5L | 6A | 6B | 9A | ··· | 9F | 15A | ··· | 15X | 18A | ··· | 18F | 45A | ··· | 45BT |
| order | 1 | 2 | 3 | 3 | 5 | ··· | 5 | 6 | 6 | 9 | ··· | 9 | 15 | ··· | 15 | 18 | ··· | 18 | 45 | ··· | 45 |
| size | 1 | 25 | 1 | 1 | 2 | ··· | 2 | 25 | 25 | 1 | ··· | 1 | 2 | ··· | 2 | 25 | ··· | 25 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | D5 | C3×D5 | C9×D5 |
| kernel | C9×C5⋊D5 | C5×C45 | C3×C5⋊D5 | C5×C15 | C5⋊D5 | C52 | C45 | C15 | C5 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 24 | 72 |
Matrix representation of C9×C5⋊D5 ►in GL4(𝔽181) generated by
| 73 | 0 | 0 | 0 |
| 0 | 73 | 0 | 0 |
| 0 | 0 | 73 | 0 |
| 0 | 0 | 0 | 73 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 180 | 1 |
| 0 | 0 | 166 | 14 |
| 180 | 1 | 0 | 0 |
| 166 | 14 | 0 | 0 |
| 0 | 0 | 14 | 180 |
| 0 | 0 | 15 | 180 |
| 180 | 0 | 0 | 0 |
| 166 | 1 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 14 | 0 |
G:=sub<GL(4,GF(181))| [73,0,0,0,0,73,0,0,0,0,73,0,0,0,0,73],[1,0,0,0,0,1,0,0,0,0,180,166,0,0,1,14],[180,166,0,0,1,14,0,0,0,0,14,15,0,0,180,180],[180,166,0,0,0,1,0,0,0,0,0,14,0,0,13,0] >;
C9×C5⋊D5 in GAP, Magma, Sage, TeX
C_9\times C_5\rtimes D_5
% in TeX
G:=Group("C9xC5:D5"); // GroupNames label
G:=SmallGroup(450,15);
// by ID
G=gap.SmallGroup(450,15);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,-5,36,1443,9004]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^5=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export