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G = D9×C25order 450 = 2·32·52

Direct product of C25 and D9

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D9×C25, C9⋊C50, C2253C2, C45.C10, C75.2S3, C5.(C5×D9), C3.(S3×C25), (C5×D9).C5, C15.2(C5×S3), SmallGroup(450,1)

Series: Derived Chief Lower central Upper central

C1C9 — D9×C25
C1C3C9C45C225 — D9×C25
C9 — D9×C25
C1C25

Generators and relations for D9×C25
 G = < a,b,c | a25=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
3S3
9C10
3C5×S3
9C50
3S3×C25

Smallest permutation representation of D9×C25
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)
(1 121 179 60 139 152 95 215 45)(2 122 180 61 140 153 96 216 46)(3 123 181 62 141 154 97 217 47)(4 124 182 63 142 155 98 218 48)(5 125 183 64 143 156 99 219 49)(6 101 184 65 144 157 100 220 50)(7 102 185 66 145 158 76 221 26)(8 103 186 67 146 159 77 222 27)(9 104 187 68 147 160 78 223 28)(10 105 188 69 148 161 79 224 29)(11 106 189 70 149 162 80 225 30)(12 107 190 71 150 163 81 201 31)(13 108 191 72 126 164 82 202 32)(14 109 192 73 127 165 83 203 33)(15 110 193 74 128 166 84 204 34)(16 111 194 75 129 167 85 205 35)(17 112 195 51 130 168 86 206 36)(18 113 196 52 131 169 87 207 37)(19 114 197 53 132 170 88 208 38)(20 115 198 54 133 171 89 209 39)(21 116 199 55 134 172 90 210 40)(22 117 200 56 135 173 91 211 41)(23 118 176 57 136 174 92 212 42)(24 119 177 58 137 175 93 213 43)(25 120 178 59 138 151 94 214 44)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(51 168)(52 169)(53 170)(54 171)(55 172)(56 173)(57 174)(58 175)(59 151)(60 152)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 161)(70 162)(71 163)(72 164)(73 165)(74 166)(75 167)(76 185)(77 186)(78 187)(79 188)(80 189)(81 190)(82 191)(83 192)(84 193)(85 194)(86 195)(87 196)(88 197)(89 198)(90 199)(91 200)(92 176)(93 177)(94 178)(95 179)(96 180)(97 181)(98 182)(99 183)(100 184)(101 220)(102 221)(103 222)(104 223)(105 224)(106 225)(107 201)(108 202)(109 203)(110 204)(111 205)(112 206)(113 207)(114 208)(115 209)(116 210)(117 211)(118 212)(119 213)(120 214)(121 215)(122 216)(123 217)(124 218)(125 219)

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

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

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

150 conjugacy classes

class 1  2  3 5A5B5C5D9A9B9C10A10B10C10D15A15B15C15D25A···25T45A···45L50A···50T75A···75T225A···225BH
order1235555999101010101515151525···2545···4550···5075···75225···225
size1921111222999922221···12···29···92···22···2

150 irreducible representations

dim111111222222
type++++
imageC1C2C5C10C25C50S3D9C5×S3C5×D9S3×C25D9×C25
kernelD9×C25C225C5×D9C45D9C9C75C25C15C5C3C1
# reps11442020134122060

Matrix representation of D9×C25 in GL2(𝔽1801) generated by

1280
0128
,
1067298
15031365
,
15031365
1067298
G:=sub<GL(2,GF(1801))| [128,0,0,128],[1067,1503,298,1365],[1503,1067,1365,298] >;

D9×C25 in GAP, Magma, Sage, TeX

D_9\times C_{25}
% in TeX

G:=Group("D9xC25");
// GroupNames label

G:=SmallGroup(450,1);
// by ID

G=gap.SmallGroup(450,1);
# by ID

G:=PCGroup([5,-2,-5,-5,-3,-3,56,5003,138,7504]);
// Polycyclic

G:=Group<a,b,c|a^25=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D9×C25 in TeX

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