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G = C32×D25order 450 = 2·32·52

Direct product of C32 and D25

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C32×D25, C752C6, 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

C1C25 — C32×D25
C1C5C25C75C3×C75 — C32×D25
C25 — C32×D25
C1C32

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 >

25C2
25C6
25C6
25C6
25C6
5D5
25C3×C6
5C3×D5
5C3×D5
5C3×D5
5C3×D5
5C32×D5

Smallest permutation representation of C32×D25
On 225 points
Generators in S225
(1 218 102)(2 219 103)(3 220 104)(4 221 105)(5 222 106)(6 223 107)(7 224 108)(8 225 109)(9 201 110)(10 202 111)(11 203 112)(12 204 113)(13 205 114)(14 206 115)(15 207 116)(16 208 117)(17 209 118)(18 210 119)(19 211 120)(20 212 121)(21 213 122)(22 214 123)(23 215 124)(24 216 125)(25 217 101)(26 171 147)(27 172 148)(28 173 149)(29 174 150)(30 175 126)(31 151 127)(32 152 128)(33 153 129)(34 154 130)(35 155 131)(36 156 132)(37 157 133)(38 158 134)(39 159 135)(40 160 136)(41 161 137)(42 162 138)(43 163 139)(44 164 140)(45 165 141)(46 166 142)(47 167 143)(48 168 144)(49 169 145)(50 170 146)(51 197 84)(52 198 85)(53 199 86)(54 200 87)(55 176 88)(56 177 89)(57 178 90)(58 179 91)(59 180 92)(60 181 93)(61 182 94)(62 183 95)(63 184 96)(64 185 97)(65 186 98)(66 187 99)(67 188 100)(68 189 76)(69 190 77)(70 191 78)(71 192 79)(72 193 80)(73 194 81)(74 195 82)(75 196 83)
(1 64 27)(2 65 28)(3 66 29)(4 67 30)(5 68 31)(6 69 32)(7 70 33)(8 71 34)(9 72 35)(10 73 36)(11 74 37)(12 75 38)(13 51 39)(14 52 40)(15 53 41)(16 54 42)(17 55 43)(18 56 44)(19 57 45)(20 58 46)(21 59 47)(22 60 48)(23 61 49)(24 62 50)(25 63 26)(76 127 106)(77 128 107)(78 129 108)(79 130 109)(80 131 110)(81 132 111)(82 133 112)(83 134 113)(84 135 114)(85 136 115)(86 137 116)(87 138 117)(88 139 118)(89 140 119)(90 141 120)(91 142 121)(92 143 122)(93 144 123)(94 145 124)(95 146 125)(96 147 101)(97 148 102)(98 149 103)(99 150 104)(100 126 105)(151 222 189)(152 223 190)(153 224 191)(154 225 192)(155 201 193)(156 202 194)(157 203 195)(158 204 196)(159 205 197)(160 206 198)(161 207 199)(162 208 200)(163 209 176)(164 210 177)(165 211 178)(166 212 179)(167 213 180)(168 214 181)(169 215 182)(170 216 183)(171 217 184)(172 218 185)(173 219 186)(174 220 187)(175 221 188)
(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 27)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(52 75)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(76 92)(77 91)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(93 100)(94 99)(95 98)(96 97)(101 102)(103 125)(104 124)(105 123)(106 122)(107 121)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(126 144)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(145 150)(146 149)(147 148)(151 167)(152 166)(153 165)(154 164)(155 163)(156 162)(157 161)(158 160)(168 175)(169 174)(170 173)(171 172)(176 193)(177 192)(178 191)(179 190)(180 189)(181 188)(182 187)(183 186)(184 185)(194 200)(195 199)(196 198)(201 209)(202 208)(203 207)(204 206)(210 225)(211 224)(212 223)(213 222)(214 221)(215 220)(216 219)(217 218)

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

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

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

126 conjugacy classes

class 1  2 3A···3H5A5B6A···6H15A···15P25A···25J75A···75CB
order123···3556···615···1525···2575···75
size1251···12225···252···22···22···2

126 irreducible representations

dim11112222
type++++
imageC1C2C3C6D5C3×D5D25C3×D25
kernelC32×D25C3×C75C3×D25C75C3×C15C15C32C3
# reps11882161080

Matrix representation of C32×D25 in GL3(𝔽151) generated by

3200
0320
0032
,
100
0320
0032
,
100
04878
07340
,
15000
04878
015103
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

Subgroup lattice of C32×D25 in TeX

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