direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9×D25, C25⋊C18, C75.C6, C225⋊2C2, C45.2D5, C5.(C9×D5), C3.(C3×D25), (C3×D25).C3, C15.2(C3×D5), SmallGroup(450,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C9×D25 |
Generators and relations for C9×D25
G = < a,b,c | a9=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C9×D25
►On 225 points
Generators in S225
(1 210 139 72 181 123 35 164 94)(2 211 140 73 182 124 36 165 95)(3 212 141 74 183 125 37 166 96)(4 213 142 75 184 101 38 167 97)(5 214 143 51 185 102 39 168 98)(6 215 144 52 186 103 40 169 99)(7 216 145 53 187 104 41 170 100)(8 217 146 54 188 105 42 171 76)(9 218 147 55 189 106 43 172 77)(10 219 148 56 190 107 44 173 78)(11 220 149 57 191 108 45 174 79)(12 221 150 58 192 109 46 175 80)(13 222 126 59 193 110 47 151 81)(14 223 127 60 194 111 48 152 82)(15 224 128 61 195 112 49 153 83)(16 225 129 62 196 113 50 154 84)(17 201 130 63 197 114 26 155 85)(18 202 131 64 198 115 27 156 86)(19 203 132 65 199 116 28 157 87)(20 204 133 66 200 117 29 158 88)(21 205 134 67 176 118 30 159 89)(22 206 135 68 177 119 31 160 90)(23 207 136 69 178 120 32 161 91)(24 208 137 70 179 121 33 162 92)(25 209 138 71 180 122 34 163 93)
(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 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(44 50)(45 49)(46 48)(51 67)(52 66)(53 65)(54 64)(55 63)(56 62)(57 61)(58 60)(68 75)(69 74)(70 73)(71 72)(76 86)(77 85)(78 84)(79 83)(80 82)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(120 125)(121 124)(122 123)(127 150)(128 149)(129 148)(130 147)(131 146)(132 145)(133 144)(134 143)(135 142)(136 141)(137 140)(138 139)(152 175)(153 174)(154 173)(155 172)(156 171)(157 170)(158 169)(159 168)(160 167)(161 166)(162 165)(163 164)(176 185)(177 184)(178 183)(179 182)(180 181)(186 200)(187 199)(188 198)(189 197)(190 196)(191 195)(192 194)(201 218)(202 217)(203 216)(204 215)(205 214)(206 213)(207 212)(208 211)(209 210)(219 225)(220 224)(221 223)
G:=sub<Sym(225)| (1,210,139,72,181,123,35,164,94)(2,211,140,73,182,124,36,165,95)(3,212,141,74,183,125,37,166,96)(4,213,142,75,184,101,38,167,97)(5,214,143,51,185,102,39,168,98)(6,215,144,52,186,103,40,169,99)(7,216,145,53,187,104,41,170,100)(8,217,146,54,188,105,42,171,76)(9,218,147,55,189,106,43,172,77)(10,219,148,56,190,107,44,173,78)(11,220,149,57,191,108,45,174,79)(12,221,150,58,192,109,46,175,80)(13,222,126,59,193,110,47,151,81)(14,223,127,60,194,111,48,152,82)(15,224,128,61,195,112,49,153,83)(16,225,129,62,196,113,50,154,84)(17,201,130,63,197,114,26,155,85)(18,202,131,64,198,115,27,156,86)(19,203,132,65,199,116,28,157,87)(20,204,133,66,200,117,29,158,88)(21,205,134,67,176,118,30,159,89)(22,206,135,68,177,119,31,160,90)(23,207,136,69,178,120,32,161,91)(24,208,137,70,179,121,33,162,92)(25,209,138,71,180,122,34,163,93), (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)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(44,50)(45,49)(46,48)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(57,61)(58,60)(68,75)(69,74)(70,73)(71,72)(76,86)(77,85)(78,84)(79,83)(80,82)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(120,125)(121,124)(122,123)(127,150)(128,149)(129,148)(130,147)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,140)(138,139)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)(176,185)(177,184)(178,183)(179,182)(180,181)(186,200)(187,199)(188,198)(189,197)(190,196)(191,195)(192,194)(201,218)(202,217)(203,216)(204,215)(205,214)(206,213)(207,212)(208,211)(209,210)(219,225)(220,224)(221,223)>;
G:=Group( (1,210,139,72,181,123,35,164,94)(2,211,140,73,182,124,36,165,95)(3,212,141,74,183,125,37,166,96)(4,213,142,75,184,101,38,167,97)(5,214,143,51,185,102,39,168,98)(6,215,144,52,186,103,40,169,99)(7,216,145,53,187,104,41,170,100)(8,217,146,54,188,105,42,171,76)(9,218,147,55,189,106,43,172,77)(10,219,148,56,190,107,44,173,78)(11,220,149,57,191,108,45,174,79)(12,221,150,58,192,109,46,175,80)(13,222,126,59,193,110,47,151,81)(14,223,127,60,194,111,48,152,82)(15,224,128,61,195,112,49,153,83)(16,225,129,62,196,113,50,154,84)(17,201,130,63,197,114,26,155,85)(18,202,131,64,198,115,27,156,86)(19,203,132,65,199,116,28,157,87)(20,204,133,66,200,117,29,158,88)(21,205,134,67,176,118,30,159,89)(22,206,135,68,177,119,31,160,90)(23,207,136,69,178,120,32,161,91)(24,208,137,70,179,121,33,162,92)(25,209,138,71,180,122,34,163,93), (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)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(44,50)(45,49)(46,48)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(57,61)(58,60)(68,75)(69,74)(70,73)(71,72)(76,86)(77,85)(78,84)(79,83)(80,82)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(120,125)(121,124)(122,123)(127,150)(128,149)(129,148)(130,147)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,140)(138,139)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)(176,185)(177,184)(178,183)(179,182)(180,181)(186,200)(187,199)(188,198)(189,197)(190,196)(191,195)(192,194)(201,218)(202,217)(203,216)(204,215)(205,214)(206,213)(207,212)(208,211)(209,210)(219,225)(220,224)(221,223) );
G=PermutationGroup([[(1,210,139,72,181,123,35,164,94),(2,211,140,73,182,124,36,165,95),(3,212,141,74,183,125,37,166,96),(4,213,142,75,184,101,38,167,97),(5,214,143,51,185,102,39,168,98),(6,215,144,52,186,103,40,169,99),(7,216,145,53,187,104,41,170,100),(8,217,146,54,188,105,42,171,76),(9,218,147,55,189,106,43,172,77),(10,219,148,56,190,107,44,173,78),(11,220,149,57,191,108,45,174,79),(12,221,150,58,192,109,46,175,80),(13,222,126,59,193,110,47,151,81),(14,223,127,60,194,111,48,152,82),(15,224,128,61,195,112,49,153,83),(16,225,129,62,196,113,50,154,84),(17,201,130,63,197,114,26,155,85),(18,202,131,64,198,115,27,156,86),(19,203,132,65,199,116,28,157,87),(20,204,133,66,200,117,29,158,88),(21,205,134,67,176,118,30,159,89),(22,206,135,68,177,119,31,160,90),(23,207,136,69,178,120,32,161,91),(24,208,137,70,179,121,33,162,92),(25,209,138,71,180,122,34,163,93)], [(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),(26,43),(27,42),(28,41),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(44,50),(45,49),(46,48),(51,67),(52,66),(53,65),(54,64),(55,63),(56,62),(57,61),(58,60),(68,75),(69,74),(70,73),(71,72),(76,86),(77,85),(78,84),(79,83),(80,82),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,112),(109,111),(120,125),(121,124),(122,123),(127,150),(128,149),(129,148),(130,147),(131,146),(132,145),(133,144),(134,143),(135,142),(136,141),(137,140),(138,139),(152,175),(153,174),(154,173),(155,172),(156,171),(157,170),(158,169),(159,168),(160,167),(161,166),(162,165),(163,164),(176,185),(177,184),(178,183),(179,182),(180,181),(186,200),(187,199),(188,198),(189,197),(190,196),(191,195),(192,194),(201,218),(202,217),(203,216),(204,215),(205,214),(206,213),(207,212),(208,211),(209,210),(219,225),(220,224),(221,223)]])
126 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 9A | ··· | 9F | 15A | 15B | 15C | 15D | 18A | ··· | 18F | 25A | ··· | 25J | 45A | ··· | 45L | 75A | ··· | 75T | 225A | ··· | 225BH |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 9 | ··· | 9 | 15 | 15 | 15 | 15 | 18 | ··· | 18 | 25 | ··· | 25 | 45 | ··· | 45 | 75 | ··· | 75 | 225 | ··· | 225 |
| size | 1 | 25 | 1 | 1 | 2 | 2 | 25 | 25 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 25 | ··· | 25 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | D5 | C3×D5 | D25 | C9×D5 | C3×D25 | C9×D25 |
| kernel | C9×D25 | C225 | C3×D25 | C75 | D25 | C25 | C45 | C15 | C9 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 4 | 10 | 12 | 20 | 60 |
Matrix representation of C9×D25 ►in GL2(𝔽1801) generated by
| 925 | 0 |
| 0 | 925 |
| 656 | 40 |
| 890 | 1567 |
| 1734 | 696 |
| 863 | 67 |
G:=sub<GL(2,GF(1801))| [925,0,0,925],[656,890,40,1567],[1734,863,696,67] >;
C9×D25 in GAP, Magma, Sage, TeX
C_9\times D_{25} % in TeX
G:=Group("C9xD25"); // GroupNames label
G:=SmallGroup(450,2);
// by ID
G=gap.SmallGroup(450,2);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,-5,36,3243,418,9004]);
// Polycyclic
G:=Group<a,b,c|a^9=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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