direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C32×D25, C75⋊2C6, C25⋊(C3×C6), (C3×C75)⋊3C2, C5.(C32×D5), C15.3(C3×D5), (C3×C15).3D5, SmallGroup(450,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C32×D25 |
Generators and relations for C32×D25
G = < a,b,c,d | a3=b3=c25=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of C32×D25
►On 225 points
Generators in S225
(1 206 119)(2 207 120)(3 208 121)(4 209 122)(5 210 123)(6 211 124)(7 212 125)(8 213 101)(9 214 102)(10 215 103)(11 216 104)(12 217 105)(13 218 106)(14 219 107)(15 220 108)(16 221 109)(17 222 110)(18 223 111)(19 224 112)(20 225 113)(21 201 114)(22 202 115)(23 203 116)(24 204 117)(25 205 118)(26 171 150)(27 172 126)(28 173 127)(29 174 128)(30 175 129)(31 151 130)(32 152 131)(33 153 132)(34 154 133)(35 155 134)(36 156 135)(37 157 136)(38 158 137)(39 159 138)(40 160 139)(41 161 140)(42 162 141)(43 163 142)(44 164 143)(45 165 144)(46 166 145)(47 167 146)(48 168 147)(49 169 148)(50 170 149)(51 188 88)(52 189 89)(53 190 90)(54 191 91)(55 192 92)(56 193 93)(57 194 94)(58 195 95)(59 196 96)(60 197 97)(61 198 98)(62 199 99)(63 200 100)(64 176 76)(65 177 77)(66 178 78)(67 179 79)(68 180 80)(69 181 81)(70 182 82)(71 183 83)(72 184 84)(73 185 85)(74 186 86)(75 187 87)
(1 73 44)(2 74 45)(3 75 46)(4 51 47)(5 52 48)(6 53 49)(7 54 50)(8 55 26)(9 56 27)(10 57 28)(11 58 29)(12 59 30)(13 60 31)(14 61 32)(15 62 33)(16 63 34)(17 64 35)(18 65 36)(19 66 37)(20 67 38)(21 68 39)(22 69 40)(23 70 41)(24 71 42)(25 72 43)(76 134 110)(77 135 111)(78 136 112)(79 137 113)(80 138 114)(81 139 115)(82 140 116)(83 141 117)(84 142 118)(85 143 119)(86 144 120)(87 145 121)(88 146 122)(89 147 123)(90 148 124)(91 149 125)(92 150 101)(93 126 102)(94 127 103)(95 128 104)(96 129 105)(97 130 106)(98 131 107)(99 132 108)(100 133 109)(151 218 197)(152 219 198)(153 220 199)(154 221 200)(155 222 176)(156 223 177)(157 224 178)(158 225 179)(159 201 180)(160 202 181)(161 203 182)(162 204 183)(163 205 184)(164 206 185)(165 207 186)(166 208 187)(167 209 188)(168 210 189)(169 211 190)(170 212 191)(171 213 192)(172 214 193)(173 215 194)(174 216 195)(175 217 196)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 36)(27 35)(28 34)(29 33)(30 32)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(70 75)(71 74)(72 73)(76 93)(77 92)(78 91)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)(94 100)(95 99)(96 98)(101 111)(102 110)(103 109)(104 108)(105 107)(112 125)(113 124)(114 123)(115 122)(116 121)(117 120)(118 119)(126 134)(127 133)(128 132)(129 131)(135 150)(136 149)(137 148)(138 147)(139 146)(140 145)(141 144)(142 143)(152 175)(153 174)(154 173)(155 172)(156 171)(157 170)(158 169)(159 168)(160 167)(161 166)(162 165)(163 164)(176 193)(177 192)(178 191)(179 190)(180 189)(181 188)(182 187)(183 186)(184 185)(194 200)(195 199)(196 198)(201 210)(202 209)(203 208)(204 207)(205 206)(211 225)(212 224)(213 223)(214 222)(215 221)(216 220)(217 219)
G:=sub<Sym(225)| (1,206,119)(2,207,120)(3,208,121)(4,209,122)(5,210,123)(6,211,124)(7,212,125)(8,213,101)(9,214,102)(10,215,103)(11,216,104)(12,217,105)(13,218,106)(14,219,107)(15,220,108)(16,221,109)(17,222,110)(18,223,111)(19,224,112)(20,225,113)(21,201,114)(22,202,115)(23,203,116)(24,204,117)(25,205,118)(26,171,150)(27,172,126)(28,173,127)(29,174,128)(30,175,129)(31,151,130)(32,152,131)(33,153,132)(34,154,133)(35,155,134)(36,156,135)(37,157,136)(38,158,137)(39,159,138)(40,160,139)(41,161,140)(42,162,141)(43,163,142)(44,164,143)(45,165,144)(46,166,145)(47,167,146)(48,168,147)(49,169,148)(50,170,149)(51,188,88)(52,189,89)(53,190,90)(54,191,91)(55,192,92)(56,193,93)(57,194,94)(58,195,95)(59,196,96)(60,197,97)(61,198,98)(62,199,99)(63,200,100)(64,176,76)(65,177,77)(66,178,78)(67,179,79)(68,180,80)(69,181,81)(70,182,82)(71,183,83)(72,184,84)(73,185,85)(74,186,86)(75,187,87), (1,73,44)(2,74,45)(3,75,46)(4,51,47)(5,52,48)(6,53,49)(7,54,50)(8,55,26)(9,56,27)(10,57,28)(11,58,29)(12,59,30)(13,60,31)(14,61,32)(15,62,33)(16,63,34)(17,64,35)(18,65,36)(19,66,37)(20,67,38)(21,68,39)(22,69,40)(23,70,41)(24,71,42)(25,72,43)(76,134,110)(77,135,111)(78,136,112)(79,137,113)(80,138,114)(81,139,115)(82,140,116)(83,141,117)(84,142,118)(85,143,119)(86,144,120)(87,145,121)(88,146,122)(89,147,123)(90,148,124)(91,149,125)(92,150,101)(93,126,102)(94,127,103)(95,128,104)(96,129,105)(97,130,106)(98,131,107)(99,132,108)(100,133,109)(151,218,197)(152,219,198)(153,220,199)(154,221,200)(155,222,176)(156,223,177)(157,224,178)(158,225,179)(159,201,180)(160,202,181)(161,203,182)(162,204,183)(163,205,184)(164,206,185)(165,207,186)(166,208,187)(167,209,188)(168,210,189)(169,211,190)(170,212,191)(171,213,192)(172,214,193)(173,215,194)(174,216,195)(175,217,196), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,36)(27,35)(28,34)(29,33)(30,32)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(70,75)(71,74)(72,73)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(94,100)(95,99)(96,98)(101,111)(102,110)(103,109)(104,108)(105,107)(112,125)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(126,134)(127,133)(128,132)(129,131)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,144)(142,143)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)(176,193)(177,192)(178,191)(179,190)(180,189)(181,188)(182,187)(183,186)(184,185)(194,200)(195,199)(196,198)(201,210)(202,209)(203,208)(204,207)(205,206)(211,225)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219)>;
G:=Group( (1,206,119)(2,207,120)(3,208,121)(4,209,122)(5,210,123)(6,211,124)(7,212,125)(8,213,101)(9,214,102)(10,215,103)(11,216,104)(12,217,105)(13,218,106)(14,219,107)(15,220,108)(16,221,109)(17,222,110)(18,223,111)(19,224,112)(20,225,113)(21,201,114)(22,202,115)(23,203,116)(24,204,117)(25,205,118)(26,171,150)(27,172,126)(28,173,127)(29,174,128)(30,175,129)(31,151,130)(32,152,131)(33,153,132)(34,154,133)(35,155,134)(36,156,135)(37,157,136)(38,158,137)(39,159,138)(40,160,139)(41,161,140)(42,162,141)(43,163,142)(44,164,143)(45,165,144)(46,166,145)(47,167,146)(48,168,147)(49,169,148)(50,170,149)(51,188,88)(52,189,89)(53,190,90)(54,191,91)(55,192,92)(56,193,93)(57,194,94)(58,195,95)(59,196,96)(60,197,97)(61,198,98)(62,199,99)(63,200,100)(64,176,76)(65,177,77)(66,178,78)(67,179,79)(68,180,80)(69,181,81)(70,182,82)(71,183,83)(72,184,84)(73,185,85)(74,186,86)(75,187,87), (1,73,44)(2,74,45)(3,75,46)(4,51,47)(5,52,48)(6,53,49)(7,54,50)(8,55,26)(9,56,27)(10,57,28)(11,58,29)(12,59,30)(13,60,31)(14,61,32)(15,62,33)(16,63,34)(17,64,35)(18,65,36)(19,66,37)(20,67,38)(21,68,39)(22,69,40)(23,70,41)(24,71,42)(25,72,43)(76,134,110)(77,135,111)(78,136,112)(79,137,113)(80,138,114)(81,139,115)(82,140,116)(83,141,117)(84,142,118)(85,143,119)(86,144,120)(87,145,121)(88,146,122)(89,147,123)(90,148,124)(91,149,125)(92,150,101)(93,126,102)(94,127,103)(95,128,104)(96,129,105)(97,130,106)(98,131,107)(99,132,108)(100,133,109)(151,218,197)(152,219,198)(153,220,199)(154,221,200)(155,222,176)(156,223,177)(157,224,178)(158,225,179)(159,201,180)(160,202,181)(161,203,182)(162,204,183)(163,205,184)(164,206,185)(165,207,186)(166,208,187)(167,209,188)(168,210,189)(169,211,190)(170,212,191)(171,213,192)(172,214,193)(173,215,194)(174,216,195)(175,217,196), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,36)(27,35)(28,34)(29,33)(30,32)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(70,75)(71,74)(72,73)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(94,100)(95,99)(96,98)(101,111)(102,110)(103,109)(104,108)(105,107)(112,125)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(126,134)(127,133)(128,132)(129,131)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,144)(142,143)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)(176,193)(177,192)(178,191)(179,190)(180,189)(181,188)(182,187)(183,186)(184,185)(194,200)(195,199)(196,198)(201,210)(202,209)(203,208)(204,207)(205,206)(211,225)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219) );
G=PermutationGroup([[(1,206,119),(2,207,120),(3,208,121),(4,209,122),(5,210,123),(6,211,124),(7,212,125),(8,213,101),(9,214,102),(10,215,103),(11,216,104),(12,217,105),(13,218,106),(14,219,107),(15,220,108),(16,221,109),(17,222,110),(18,223,111),(19,224,112),(20,225,113),(21,201,114),(22,202,115),(23,203,116),(24,204,117),(25,205,118),(26,171,150),(27,172,126),(28,173,127),(29,174,128),(30,175,129),(31,151,130),(32,152,131),(33,153,132),(34,154,133),(35,155,134),(36,156,135),(37,157,136),(38,158,137),(39,159,138),(40,160,139),(41,161,140),(42,162,141),(43,163,142),(44,164,143),(45,165,144),(46,166,145),(47,167,146),(48,168,147),(49,169,148),(50,170,149),(51,188,88),(52,189,89),(53,190,90),(54,191,91),(55,192,92),(56,193,93),(57,194,94),(58,195,95),(59,196,96),(60,197,97),(61,198,98),(62,199,99),(63,200,100),(64,176,76),(65,177,77),(66,178,78),(67,179,79),(68,180,80),(69,181,81),(70,182,82),(71,183,83),(72,184,84),(73,185,85),(74,186,86),(75,187,87)], [(1,73,44),(2,74,45),(3,75,46),(4,51,47),(5,52,48),(6,53,49),(7,54,50),(8,55,26),(9,56,27),(10,57,28),(11,58,29),(12,59,30),(13,60,31),(14,61,32),(15,62,33),(16,63,34),(17,64,35),(18,65,36),(19,66,37),(20,67,38),(21,68,39),(22,69,40),(23,70,41),(24,71,42),(25,72,43),(76,134,110),(77,135,111),(78,136,112),(79,137,113),(80,138,114),(81,139,115),(82,140,116),(83,141,117),(84,142,118),(85,143,119),(86,144,120),(87,145,121),(88,146,122),(89,147,123),(90,148,124),(91,149,125),(92,150,101),(93,126,102),(94,127,103),(95,128,104),(96,129,105),(97,130,106),(98,131,107),(99,132,108),(100,133,109),(151,218,197),(152,219,198),(153,220,199),(154,221,200),(155,222,176),(156,223,177),(157,224,178),(158,225,179),(159,201,180),(160,202,181),(161,203,182),(162,204,183),(163,205,184),(164,206,185),(165,207,186),(166,208,187),(167,209,188),(168,210,189),(169,211,190),(170,212,191),(171,213,192),(172,214,193),(173,215,194),(174,216,195),(175,217,196)], [(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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(26,36),(27,35),(28,34),(29,33),(30,32),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(70,75),(71,74),(72,73),(76,93),(77,92),(78,91),(79,90),(80,89),(81,88),(82,87),(83,86),(84,85),(94,100),(95,99),(96,98),(101,111),(102,110),(103,109),(104,108),(105,107),(112,125),(113,124),(114,123),(115,122),(116,121),(117,120),(118,119),(126,134),(127,133),(128,132),(129,131),(135,150),(136,149),(137,148),(138,147),(139,146),(140,145),(141,144),(142,143),(152,175),(153,174),(154,173),(155,172),(156,171),(157,170),(158,169),(159,168),(160,167),(161,166),(162,165),(163,164),(176,193),(177,192),(178,191),(179,190),(180,189),(181,188),(182,187),(183,186),(184,185),(194,200),(195,199),(196,198),(201,210),(202,209),(203,208),(204,207),(205,206),(211,225),(212,224),(213,223),(214,222),(215,221),(216,220),(217,219)]])
126 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 5A | 5B | 6A | ··· | 6H | 15A | ··· | 15P | 25A | ··· | 25J | 75A | ··· | 75CB |
| order | 1 | 2 | 3 | ··· | 3 | 5 | 5 | 6 | ··· | 6 | 15 | ··· | 15 | 25 | ··· | 25 | 75 | ··· | 75 |
| size | 1 | 25 | 1 | ··· | 1 | 2 | 2 | 25 | ··· | 25 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C3 | C6 | D5 | C3×D5 | D25 | C3×D25 |
| kernel | C32×D25 | C3×C75 | C3×D25 | C75 | C3×C15 | C15 | C32 | C3 |
| # reps | 1 | 1 | 8 | 8 | 2 | 16 | 10 | 80 |
Matrix representation of C32×D25 ►in GL3(𝔽151) generated by
| 32 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 1 | 0 | 0 |
| 0 | 48 | 78 |
| 0 | 73 | 40 |
| 150 | 0 | 0 |
| 0 | 48 | 78 |
| 0 | 15 | 103 |
G:=sub<GL(3,GF(151))| [32,0,0,0,32,0,0,0,32],[1,0,0,0,32,0,0,0,32],[1,0,0,0,48,73,0,78,40],[150,0,0,0,48,15,0,78,103] >;
C32×D25 in GAP, Magma, Sage, TeX
C_3^2\times D_{25} % in TeX
G:=Group("C3^2xD25"); // GroupNames label
G:=SmallGroup(450,5);
// by ID
G=gap.SmallGroup(450,5);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,-5,3243,418,9004]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^25=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