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G = C3×Dic25order 300 = 22·3·52

Direct product of C3 and Dic25

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

Aliases: C3×Dic25, C754C4, C50.C6, C252C12, C30.4D5, C6.2D25, C150.2C2, C15.2Dic5, C2.(C3×D25), C5.(C3×Dic5), C10.1(C3×D5), SmallGroup(300,2)

Series: Derived Chief Lower central Upper central

C1C25 — C3×Dic25
C1C5C25C50C150 — C3×Dic25
C25 — C3×Dic25
C1C6

Generators and relations for C3×Dic25
 G = < a,b,c | a3=b50=1, c2=b25, ab=ba, ac=ca, cbc-1=b-1 >

25C4
25C12
5Dic5
5C3×Dic5

Smallest permutation representation of C3×Dic25
Regular action on 300 points
Generators in S300
(1 143 98)(2 144 99)(3 145 100)(4 146 51)(5 147 52)(6 148 53)(7 149 54)(8 150 55)(9 101 56)(10 102 57)(11 103 58)(12 104 59)(13 105 60)(14 106 61)(15 107 62)(16 108 63)(17 109 64)(18 110 65)(19 111 66)(20 112 67)(21 113 68)(22 114 69)(23 115 70)(24 116 71)(25 117 72)(26 118 73)(27 119 74)(28 120 75)(29 121 76)(30 122 77)(31 123 78)(32 124 79)(33 125 80)(34 126 81)(35 127 82)(36 128 83)(37 129 84)(38 130 85)(39 131 86)(40 132 87)(41 133 88)(42 134 89)(43 135 90)(44 136 91)(45 137 92)(46 138 93)(47 139 94)(48 140 95)(49 141 96)(50 142 97)(151 251 215)(152 252 216)(153 253 217)(154 254 218)(155 255 219)(156 256 220)(157 257 221)(158 258 222)(159 259 223)(160 260 224)(161 261 225)(162 262 226)(163 263 227)(164 264 228)(165 265 229)(166 266 230)(167 267 231)(168 268 232)(169 269 233)(170 270 234)(171 271 235)(172 272 236)(173 273 237)(174 274 238)(175 275 239)(176 276 240)(177 277 241)(178 278 242)(179 279 243)(180 280 244)(181 281 245)(182 282 246)(183 283 247)(184 284 248)(185 285 249)(186 286 250)(187 287 201)(188 288 202)(189 289 203)(190 290 204)(191 291 205)(192 292 206)(193 293 207)(194 294 208)(195 295 209)(196 296 210)(197 297 211)(198 298 212)(199 299 213)(200 300 214)
(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)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250)(251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300)
(1 155 26 180)(2 154 27 179)(3 153 28 178)(4 152 29 177)(5 151 30 176)(6 200 31 175)(7 199 32 174)(8 198 33 173)(9 197 34 172)(10 196 35 171)(11 195 36 170)(12 194 37 169)(13 193 38 168)(14 192 39 167)(15 191 40 166)(16 190 41 165)(17 189 42 164)(18 188 43 163)(19 187 44 162)(20 186 45 161)(21 185 46 160)(22 184 47 159)(23 183 48 158)(24 182 49 157)(25 181 50 156)(51 216 76 241)(52 215 77 240)(53 214 78 239)(54 213 79 238)(55 212 80 237)(56 211 81 236)(57 210 82 235)(58 209 83 234)(59 208 84 233)(60 207 85 232)(61 206 86 231)(62 205 87 230)(63 204 88 229)(64 203 89 228)(65 202 90 227)(66 201 91 226)(67 250 92 225)(68 249 93 224)(69 248 94 223)(70 247 95 222)(71 246 96 221)(72 245 97 220)(73 244 98 219)(74 243 99 218)(75 242 100 217)(101 297 126 272)(102 296 127 271)(103 295 128 270)(104 294 129 269)(105 293 130 268)(106 292 131 267)(107 291 132 266)(108 290 133 265)(109 289 134 264)(110 288 135 263)(111 287 136 262)(112 286 137 261)(113 285 138 260)(114 284 139 259)(115 283 140 258)(116 282 141 257)(117 281 142 256)(118 280 143 255)(119 279 144 254)(120 278 145 253)(121 277 146 252)(122 276 147 251)(123 275 148 300)(124 274 149 299)(125 273 150 298)

G:=sub<Sym(300)| (1,143,98)(2,144,99)(3,145,100)(4,146,51)(5,147,52)(6,148,53)(7,149,54)(8,150,55)(9,101,56)(10,102,57)(11,103,58)(12,104,59)(13,105,60)(14,106,61)(15,107,62)(16,108,63)(17,109,64)(18,110,65)(19,111,66)(20,112,67)(21,113,68)(22,114,69)(23,115,70)(24,116,71)(25,117,72)(26,118,73)(27,119,74)(28,120,75)(29,121,76)(30,122,77)(31,123,78)(32,124,79)(33,125,80)(34,126,81)(35,127,82)(36,128,83)(37,129,84)(38,130,85)(39,131,86)(40,132,87)(41,133,88)(42,134,89)(43,135,90)(44,136,91)(45,137,92)(46,138,93)(47,139,94)(48,140,95)(49,141,96)(50,142,97)(151,251,215)(152,252,216)(153,253,217)(154,254,218)(155,255,219)(156,256,220)(157,257,221)(158,258,222)(159,259,223)(160,260,224)(161,261,225)(162,262,226)(163,263,227)(164,264,228)(165,265,229)(166,266,230)(167,267,231)(168,268,232)(169,269,233)(170,270,234)(171,271,235)(172,272,236)(173,273,237)(174,274,238)(175,275,239)(176,276,240)(177,277,241)(178,278,242)(179,279,243)(180,280,244)(181,281,245)(182,282,246)(183,283,247)(184,284,248)(185,285,249)(186,286,250)(187,287,201)(188,288,202)(189,289,203)(190,290,204)(191,291,205)(192,292,206)(193,293,207)(194,294,208)(195,295,209)(196,296,210)(197,297,211)(198,298,212)(199,299,213)(200,300,214), (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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250)(251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300), (1,155,26,180)(2,154,27,179)(3,153,28,178)(4,152,29,177)(5,151,30,176)(6,200,31,175)(7,199,32,174)(8,198,33,173)(9,197,34,172)(10,196,35,171)(11,195,36,170)(12,194,37,169)(13,193,38,168)(14,192,39,167)(15,191,40,166)(16,190,41,165)(17,189,42,164)(18,188,43,163)(19,187,44,162)(20,186,45,161)(21,185,46,160)(22,184,47,159)(23,183,48,158)(24,182,49,157)(25,181,50,156)(51,216,76,241)(52,215,77,240)(53,214,78,239)(54,213,79,238)(55,212,80,237)(56,211,81,236)(57,210,82,235)(58,209,83,234)(59,208,84,233)(60,207,85,232)(61,206,86,231)(62,205,87,230)(63,204,88,229)(64,203,89,228)(65,202,90,227)(66,201,91,226)(67,250,92,225)(68,249,93,224)(69,248,94,223)(70,247,95,222)(71,246,96,221)(72,245,97,220)(73,244,98,219)(74,243,99,218)(75,242,100,217)(101,297,126,272)(102,296,127,271)(103,295,128,270)(104,294,129,269)(105,293,130,268)(106,292,131,267)(107,291,132,266)(108,290,133,265)(109,289,134,264)(110,288,135,263)(111,287,136,262)(112,286,137,261)(113,285,138,260)(114,284,139,259)(115,283,140,258)(116,282,141,257)(117,281,142,256)(118,280,143,255)(119,279,144,254)(120,278,145,253)(121,277,146,252)(122,276,147,251)(123,275,148,300)(124,274,149,299)(125,273,150,298)>;

G:=Group( (1,143,98)(2,144,99)(3,145,100)(4,146,51)(5,147,52)(6,148,53)(7,149,54)(8,150,55)(9,101,56)(10,102,57)(11,103,58)(12,104,59)(13,105,60)(14,106,61)(15,107,62)(16,108,63)(17,109,64)(18,110,65)(19,111,66)(20,112,67)(21,113,68)(22,114,69)(23,115,70)(24,116,71)(25,117,72)(26,118,73)(27,119,74)(28,120,75)(29,121,76)(30,122,77)(31,123,78)(32,124,79)(33,125,80)(34,126,81)(35,127,82)(36,128,83)(37,129,84)(38,130,85)(39,131,86)(40,132,87)(41,133,88)(42,134,89)(43,135,90)(44,136,91)(45,137,92)(46,138,93)(47,139,94)(48,140,95)(49,141,96)(50,142,97)(151,251,215)(152,252,216)(153,253,217)(154,254,218)(155,255,219)(156,256,220)(157,257,221)(158,258,222)(159,259,223)(160,260,224)(161,261,225)(162,262,226)(163,263,227)(164,264,228)(165,265,229)(166,266,230)(167,267,231)(168,268,232)(169,269,233)(170,270,234)(171,271,235)(172,272,236)(173,273,237)(174,274,238)(175,275,239)(176,276,240)(177,277,241)(178,278,242)(179,279,243)(180,280,244)(181,281,245)(182,282,246)(183,283,247)(184,284,248)(185,285,249)(186,286,250)(187,287,201)(188,288,202)(189,289,203)(190,290,204)(191,291,205)(192,292,206)(193,293,207)(194,294,208)(195,295,209)(196,296,210)(197,297,211)(198,298,212)(199,299,213)(200,300,214), (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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250)(251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300), (1,155,26,180)(2,154,27,179)(3,153,28,178)(4,152,29,177)(5,151,30,176)(6,200,31,175)(7,199,32,174)(8,198,33,173)(9,197,34,172)(10,196,35,171)(11,195,36,170)(12,194,37,169)(13,193,38,168)(14,192,39,167)(15,191,40,166)(16,190,41,165)(17,189,42,164)(18,188,43,163)(19,187,44,162)(20,186,45,161)(21,185,46,160)(22,184,47,159)(23,183,48,158)(24,182,49,157)(25,181,50,156)(51,216,76,241)(52,215,77,240)(53,214,78,239)(54,213,79,238)(55,212,80,237)(56,211,81,236)(57,210,82,235)(58,209,83,234)(59,208,84,233)(60,207,85,232)(61,206,86,231)(62,205,87,230)(63,204,88,229)(64,203,89,228)(65,202,90,227)(66,201,91,226)(67,250,92,225)(68,249,93,224)(69,248,94,223)(70,247,95,222)(71,246,96,221)(72,245,97,220)(73,244,98,219)(74,243,99,218)(75,242,100,217)(101,297,126,272)(102,296,127,271)(103,295,128,270)(104,294,129,269)(105,293,130,268)(106,292,131,267)(107,291,132,266)(108,290,133,265)(109,289,134,264)(110,288,135,263)(111,287,136,262)(112,286,137,261)(113,285,138,260)(114,284,139,259)(115,283,140,258)(116,282,141,257)(117,281,142,256)(118,280,143,255)(119,279,144,254)(120,278,145,253)(121,277,146,252)(122,276,147,251)(123,275,148,300)(124,274,149,299)(125,273,150,298) );

G=PermutationGroup([(1,143,98),(2,144,99),(3,145,100),(4,146,51),(5,147,52),(6,148,53),(7,149,54),(8,150,55),(9,101,56),(10,102,57),(11,103,58),(12,104,59),(13,105,60),(14,106,61),(15,107,62),(16,108,63),(17,109,64),(18,110,65),(19,111,66),(20,112,67),(21,113,68),(22,114,69),(23,115,70),(24,116,71),(25,117,72),(26,118,73),(27,119,74),(28,120,75),(29,121,76),(30,122,77),(31,123,78),(32,124,79),(33,125,80),(34,126,81),(35,127,82),(36,128,83),(37,129,84),(38,130,85),(39,131,86),(40,132,87),(41,133,88),(42,134,89),(43,135,90),(44,136,91),(45,137,92),(46,138,93),(47,139,94),(48,140,95),(49,141,96),(50,142,97),(151,251,215),(152,252,216),(153,253,217),(154,254,218),(155,255,219),(156,256,220),(157,257,221),(158,258,222),(159,259,223),(160,260,224),(161,261,225),(162,262,226),(163,263,227),(164,264,228),(165,265,229),(166,266,230),(167,267,231),(168,268,232),(169,269,233),(170,270,234),(171,271,235),(172,272,236),(173,273,237),(174,274,238),(175,275,239),(176,276,240),(177,277,241),(178,278,242),(179,279,243),(180,280,244),(181,281,245),(182,282,246),(183,283,247),(184,284,248),(185,285,249),(186,286,250),(187,287,201),(188,288,202),(189,289,203),(190,290,204),(191,291,205),(192,292,206),(193,293,207),(194,294,208),(195,295,209),(196,296,210),(197,297,211),(198,298,212),(199,299,213),(200,300,214)], [(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),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250),(251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)], [(1,155,26,180),(2,154,27,179),(3,153,28,178),(4,152,29,177),(5,151,30,176),(6,200,31,175),(7,199,32,174),(8,198,33,173),(9,197,34,172),(10,196,35,171),(11,195,36,170),(12,194,37,169),(13,193,38,168),(14,192,39,167),(15,191,40,166),(16,190,41,165),(17,189,42,164),(18,188,43,163),(19,187,44,162),(20,186,45,161),(21,185,46,160),(22,184,47,159),(23,183,48,158),(24,182,49,157),(25,181,50,156),(51,216,76,241),(52,215,77,240),(53,214,78,239),(54,213,79,238),(55,212,80,237),(56,211,81,236),(57,210,82,235),(58,209,83,234),(59,208,84,233),(60,207,85,232),(61,206,86,231),(62,205,87,230),(63,204,88,229),(64,203,89,228),(65,202,90,227),(66,201,91,226),(67,250,92,225),(68,249,93,224),(69,248,94,223),(70,247,95,222),(71,246,96,221),(72,245,97,220),(73,244,98,219),(74,243,99,218),(75,242,100,217),(101,297,126,272),(102,296,127,271),(103,295,128,270),(104,294,129,269),(105,293,130,268),(106,292,131,267),(107,291,132,266),(108,290,133,265),(109,289,134,264),(110,288,135,263),(111,287,136,262),(112,286,137,261),(113,285,138,260),(114,284,139,259),(115,283,140,258),(116,282,141,257),(117,281,142,256),(118,280,143,255),(119,279,144,254),(120,278,145,253),(121,277,146,252),(122,276,147,251),(123,275,148,300),(124,274,149,299),(125,273,150,298)])

84 conjugacy classes

class 1  2 3A3B4A4B5A5B6A6B10A10B12A12B12C12D15A15B15C15D25A···25J30A30B30C30D50A···50J75A···75T150A···150T
order12334455661010121212121515151525···253030303050···5075···75150···150
size111125252211222525252522222···222222···22···22···2

84 irreducible representations

dim11111122222222
type+++-+-
imageC1C2C3C4C6C12D5Dic5C3×D5D25C3×Dic5Dic25C3×D25C3×Dic25
kernelC3×Dic25C150Dic25C75C50C25C30C15C10C6C5C3C2C1
# reps112224224104102020

Matrix representation of C3×Dic25 in GL2(𝔽601) generated by

240
024
,
469373
228453
,
2177
205580
G:=sub<GL(2,GF(601))| [24,0,0,24],[469,228,373,453],[21,205,77,580] >;

C3×Dic25 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{25}
% in TeX

G:=Group("C3xDic25");
// GroupNames label

G:=SmallGroup(300,2);
// by ID

G=gap.SmallGroup(300,2);
# by ID

G:=PCGroup([5,-2,-3,-2,-5,-5,30,2163,418,6004]);
// Polycyclic

G:=Group<a,b,c|a^3=b^50=1,c^2=b^25,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic25 in TeX

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