Copied to
clipboard

## G = C9×D25order 450 = 2·32·52

### Direct product of C9 and D25

Aliases: C9×D25, C25⋊C18, C75.C6, C2252C2, 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

 Derived series C1 — C25 — C9×D25
 Chief series C1 — C5 — C25 — C75 — C225 — C9×D25
 Lower central C25 — C9×D25
 Upper central C1 — C9

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
(1 210 127 60 197 110 39 164 85)(2 211 128 61 198 111 40 165 86)(3 212 129 62 199 112 41 166 87)(4 213 130 63 200 113 42 167 88)(5 214 131 64 176 114 43 168 89)(6 215 132 65 177 115 44 169 90)(7 216 133 66 178 116 45 170 91)(8 217 134 67 179 117 46 171 92)(9 218 135 68 180 118 47 172 93)(10 219 136 69 181 119 48 173 94)(11 220 137 70 182 120 49 174 95)(12 221 138 71 183 121 50 175 96)(13 222 139 72 184 122 26 151 97)(14 223 140 73 185 123 27 152 98)(15 224 141 74 186 124 28 153 99)(16 225 142 75 187 125 29 154 100)(17 201 143 51 188 101 30 155 76)(18 202 144 52 189 102 31 156 77)(19 203 145 53 190 103 32 157 78)(20 204 146 54 191 104 33 158 79)(21 205 147 55 192 105 34 159 80)(22 206 148 56 193 106 35 160 81)(23 207 149 57 194 107 36 161 82)(24 208 150 58 195 108 37 162 83)(25 209 126 59 196 109 38 163 84)
(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)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 60)(69 75)(70 74)(71 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)(101 118)(102 117)(103 116)(104 115)(105 114)(106 113)(107 112)(108 111)(109 110)(119 125)(120 124)(121 123)(126 127)(128 150)(129 149)(130 148)(131 147)(132 146)(133 145)(134 144)(135 143)(136 142)(137 141)(138 140)(152 175)(153 174)(154 173)(155 172)(156 171)(157 170)(158 169)(159 168)(160 167)(161 166)(162 165)(163 164)(176 192)(177 191)(178 190)(179 189)(180 188)(181 187)(182 186)(183 185)(193 200)(194 199)(195 198)(196 197)(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)| (1,210,127,60,197,110,39,164,85)(2,211,128,61,198,111,40,165,86)(3,212,129,62,199,112,41,166,87)(4,213,130,63,200,113,42,167,88)(5,214,131,64,176,114,43,168,89)(6,215,132,65,177,115,44,169,90)(7,216,133,66,178,116,45,170,91)(8,217,134,67,179,117,46,171,92)(9,218,135,68,180,118,47,172,93)(10,219,136,69,181,119,48,173,94)(11,220,137,70,182,120,49,174,95)(12,221,138,71,183,121,50,175,96)(13,222,139,72,184,122,26,151,97)(14,223,140,73,185,123,27,152,98)(15,224,141,74,186,124,28,153,99)(16,225,142,75,187,125,29,154,100)(17,201,143,51,188,101,30,155,76)(18,202,144,52,189,102,31,156,77)(19,203,145,53,190,103,32,157,78)(20,204,146,54,191,104,33,158,79)(21,205,147,55,192,105,34,159,80)(22,206,148,56,193,106,35,160,81)(23,207,149,57,194,107,36,161,82)(24,208,150,58,195,108,37,162,83)(25,209,126,59,196,109,38,163,84), (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)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(69,75)(70,74)(71,73)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(94,100)(95,99)(96,98)(101,118)(102,117)(103,116)(104,115)(105,114)(106,113)(107,112)(108,111)(109,110)(119,125)(120,124)(121,123)(126,127)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,140)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)(176,192)(177,191)(178,190)(179,189)(180,188)(181,187)(182,186)(183,185)(193,200)(194,199)(195,198)(196,197)(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( (1,210,127,60,197,110,39,164,85)(2,211,128,61,198,111,40,165,86)(3,212,129,62,199,112,41,166,87)(4,213,130,63,200,113,42,167,88)(5,214,131,64,176,114,43,168,89)(6,215,132,65,177,115,44,169,90)(7,216,133,66,178,116,45,170,91)(8,217,134,67,179,117,46,171,92)(9,218,135,68,180,118,47,172,93)(10,219,136,69,181,119,48,173,94)(11,220,137,70,182,120,49,174,95)(12,221,138,71,183,121,50,175,96)(13,222,139,72,184,122,26,151,97)(14,223,140,73,185,123,27,152,98)(15,224,141,74,186,124,28,153,99)(16,225,142,75,187,125,29,154,100)(17,201,143,51,188,101,30,155,76)(18,202,144,52,189,102,31,156,77)(19,203,145,53,190,103,32,157,78)(20,204,146,54,191,104,33,158,79)(21,205,147,55,192,105,34,159,80)(22,206,148,56,193,106,35,160,81)(23,207,149,57,194,107,36,161,82)(24,208,150,58,195,108,37,162,83)(25,209,126,59,196,109,38,163,84), (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)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(69,75)(70,74)(71,73)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(94,100)(95,99)(96,98)(101,118)(102,117)(103,116)(104,115)(105,114)(106,113)(107,112)(108,111)(109,110)(119,125)(120,124)(121,123)(126,127)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,140)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)(176,192)(177,191)(178,190)(179,189)(180,188)(181,187)(182,186)(183,185)(193,200)(194,199)(195,198)(196,197)(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([(1,210,127,60,197,110,39,164,85),(2,211,128,61,198,111,40,165,86),(3,212,129,62,199,112,41,166,87),(4,213,130,63,200,113,42,167,88),(5,214,131,64,176,114,43,168,89),(6,215,132,65,177,115,44,169,90),(7,216,133,66,178,116,45,170,91),(8,217,134,67,179,117,46,171,92),(9,218,135,68,180,118,47,172,93),(10,219,136,69,181,119,48,173,94),(11,220,137,70,182,120,49,174,95),(12,221,138,71,183,121,50,175,96),(13,222,139,72,184,122,26,151,97),(14,223,140,73,185,123,27,152,98),(15,224,141,74,186,124,28,153,99),(16,225,142,75,187,125,29,154,100),(17,201,143,51,188,101,30,155,76),(18,202,144,52,189,102,31,156,77),(19,203,145,53,190,103,32,157,78),(20,204,146,54,191,104,33,158,79),(21,205,147,55,192,105,34,159,80),(22,206,148,56,193,106,35,160,81),(23,207,149,57,194,107,36,161,82),(24,208,150,58,195,108,37,162,83),(25,209,126,59,196,109,38,163,84)], [(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),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(51,68),(52,67),(53,66),(54,65),(55,64),(56,63),(57,62),(58,61),(59,60),(69,75),(70,74),(71,73),(76,93),(77,92),(78,91),(79,90),(80,89),(81,88),(82,87),(83,86),(84,85),(94,100),(95,99),(96,98),(101,118),(102,117),(103,116),(104,115),(105,114),(106,113),(107,112),(108,111),(109,110),(119,125),(120,124),(121,123),(126,127),(128,150),(129,149),(130,148),(131,147),(132,146),(133,145),(134,144),(135,143),(136,142),(137,141),(138,140),(152,175),(153,174),(154,173),(155,172),(156,171),(157,170),(158,169),(159,168),(160,167),(161,166),(162,165),(163,164),(176,192),(177,191),(178,190),(179,189),(180,188),(181,187),(182,186),(183,185),(193,200),(194,199),(195,198),(196,197),(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

Export

׿
×
𝔽