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## G = C3⋊S3×C25order 450 = 2·32·52

### Direct product of C25 and C3⋊S3

Aliases: C3⋊S3×C25, C753S3, C322C50, C3⋊(S3×C25), (C3×C75)⋊5C2, C15.3(C5×S3), (C3×C15).2C10, C5.(C5×C3⋊S3), (C5×C3⋊S3).C5, SmallGroup(450,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3×C25
 Chief series C1 — C3 — C32 — C3×C15 — C3×C75 — C3⋊S3×C25
 Lower central C32 — C3⋊S3×C25
 Upper central C1 — C25

Generators and relations for C3⋊S3×C25
G = < a,b,c,d | a25=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Smallest permutation representation of C3⋊S3×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)
(1 214 59)(2 215 60)(3 216 61)(4 217 62)(5 218 63)(6 219 64)(7 220 65)(8 221 66)(9 222 67)(10 223 68)(11 224 69)(12 225 70)(13 201 71)(14 202 72)(15 203 73)(16 204 74)(17 205 75)(18 206 51)(19 207 52)(20 208 53)(21 209 54)(22 210 55)(23 211 56)(24 212 57)(25 213 58)(26 78 183)(27 79 184)(28 80 185)(29 81 186)(30 82 187)(31 83 188)(32 84 189)(33 85 190)(34 86 191)(35 87 192)(36 88 193)(37 89 194)(38 90 195)(39 91 196)(40 92 197)(41 93 198)(42 94 199)(43 95 200)(44 96 176)(45 97 177)(46 98 178)(47 99 179)(48 100 180)(49 76 181)(50 77 182)(101 155 140)(102 156 141)(103 157 142)(104 158 143)(105 159 144)(106 160 145)(107 161 146)(108 162 147)(109 163 148)(110 164 149)(111 165 150)(112 166 126)(113 167 127)(114 168 128)(115 169 129)(116 170 130)(117 171 131)(118 172 132)(119 173 133)(120 174 134)(121 175 135)(122 151 136)(123 152 137)(124 153 138)(125 154 139)
(1 174 29)(2 175 30)(3 151 31)(4 152 32)(5 153 33)(6 154 34)(7 155 35)(8 156 36)(9 157 37)(10 158 38)(11 159 39)(12 160 40)(13 161 41)(14 162 42)(15 163 43)(16 164 44)(17 165 45)(18 166 46)(19 167 47)(20 168 48)(21 169 49)(22 170 50)(23 171 26)(24 172 27)(25 173 28)(51 112 178)(52 113 179)(53 114 180)(54 115 181)(55 116 182)(56 117 183)(57 118 184)(58 119 185)(59 120 186)(60 121 187)(61 122 188)(62 123 189)(63 124 190)(64 125 191)(65 101 192)(66 102 193)(67 103 194)(68 104 195)(69 105 196)(70 106 197)(71 107 198)(72 108 199)(73 109 200)(74 110 176)(75 111 177)(76 209 129)(77 210 130)(78 211 131)(79 212 132)(80 213 133)(81 214 134)(82 215 135)(83 216 136)(84 217 137)(85 218 138)(86 219 139)(87 220 140)(88 221 141)(89 222 142)(90 223 143)(91 224 144)(92 225 145)(93 201 146)(94 202 147)(95 203 148)(96 204 149)(97 205 150)(98 206 126)(99 207 127)(100 208 128)
(26 171)(27 172)(28 173)(29 174)(30 175)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(41 161)(42 162)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 169)(50 170)(51 206)(52 207)(53 208)(54 209)(55 210)(56 211)(57 212)(58 213)(59 214)(60 215)(61 216)(62 217)(63 218)(64 219)(65 220)(66 221)(67 222)(68 223)(69 224)(70 225)(71 201)(72 202)(73 203)(74 204)(75 205)(76 115)(77 116)(78 117)(79 118)(80 119)(81 120)(82 121)(83 122)(84 123)(85 124)(86 125)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(99 113)(100 114)(126 178)(127 179)(128 180)(129 181)(130 182)(131 183)(132 184)(133 185)(134 186)(135 187)(136 188)(137 189)(138 190)(139 191)(140 192)(141 193)(142 194)(143 195)(144 196)(145 197)(146 198)(147 199)(148 200)(149 176)(150 177)

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

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

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

150 conjugacy classes

 class 1 2 3A 3B 3C 3D 5A 5B 5C 5D 10A 10B 10C 10D 15A ··· 15P 25A ··· 25T 50A ··· 50T 75A ··· 75CB order 1 2 3 3 3 3 5 5 5 5 10 10 10 10 15 ··· 15 25 ··· 25 50 ··· 50 75 ··· 75 size 1 9 2 2 2 2 1 1 1 1 9 9 9 9 2 ··· 2 1 ··· 1 9 ··· 9 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + + image C1 C2 C5 C10 C25 C50 S3 C5×S3 S3×C25 kernel C3⋊S3×C25 C3×C75 C5×C3⋊S3 C3×C15 C3⋊S3 C32 C75 C15 C3 # reps 1 1 4 4 20 20 4 16 80

Matrix representation of C3⋊S3×C25 in GL4(𝔽151) generated by

 68 0 0 0 0 68 0 0 0 0 8 0 0 0 0 8
,
 0 1 0 0 150 150 0 0 0 0 0 1 0 0 150 150
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 150 150
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(151))| [68,0,0,0,0,68,0,0,0,0,8,0,0,0,0,8],[0,150,0,0,1,150,0,0,0,0,0,150,0,0,1,150],[1,0,0,0,0,1,0,0,0,0,0,150,0,0,1,150],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3⋊S3×C25 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_{25}
% in TeX

G:=Group("C3:S3xC25");
// GroupNames label

G:=SmallGroup(450,8);
// by ID

G=gap.SmallGroup(450,8);
# by ID

G:=PCGroup([5,-2,-5,-5,-3,-3,56,2003,7504]);
// Polycyclic

G:=Group<a,b,c,d|a^25=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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