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G = Dic3×C25order 300 = 22·3·52

Direct product of C25 and Dic3

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

Aliases: Dic3×C25, C3⋊C100, C755C4, C6.C50, C15.C20, C50.2S3, C150.3C2, C30.3C10, C2.(S3×C25), C5.(C5×Dic3), C10.2(C5×S3), (C5×Dic3).C5, SmallGroup(300,1)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C25
C1C3C15C30C150 — Dic3×C25
C3 — Dic3×C25
C1C50

Generators and relations for Dic3×C25
 G = < a,b,c | a25=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C20
3C100

Smallest permutation representation of Dic3×C25
Regular action on 300 points
Generators in S300
(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)
(1 102 188 40 290 229)(2 103 189 41 291 230)(3 104 190 42 292 231)(4 105 191 43 293 232)(5 106 192 44 294 233)(6 107 193 45 295 234)(7 108 194 46 296 235)(8 109 195 47 297 236)(9 110 196 48 298 237)(10 111 197 49 299 238)(11 112 198 50 300 239)(12 113 199 26 276 240)(13 114 200 27 277 241)(14 115 176 28 278 242)(15 116 177 29 279 243)(16 117 178 30 280 244)(17 118 179 31 281 245)(18 119 180 32 282 246)(19 120 181 33 283 247)(20 121 182 34 284 248)(21 122 183 35 285 249)(22 123 184 36 286 250)(23 124 185 37 287 226)(24 125 186 38 288 227)(25 101 187 39 289 228)(51 140 275 218 169 77)(52 141 251 219 170 78)(53 142 252 220 171 79)(54 143 253 221 172 80)(55 144 254 222 173 81)(56 145 255 223 174 82)(57 146 256 224 175 83)(58 147 257 225 151 84)(59 148 258 201 152 85)(60 149 259 202 153 86)(61 150 260 203 154 87)(62 126 261 204 155 88)(63 127 262 205 156 89)(64 128 263 206 157 90)(65 129 264 207 158 91)(66 130 265 208 159 92)(67 131 266 209 160 93)(68 132 267 210 161 94)(69 133 268 211 162 95)(70 134 269 212 163 96)(71 135 270 213 164 97)(72 136 271 214 165 98)(73 137 272 215 166 99)(74 138 273 216 167 100)(75 139 274 217 168 76)
(1 56 40 223)(2 57 41 224)(3 58 42 225)(4 59 43 201)(5 60 44 202)(6 61 45 203)(7 62 46 204)(8 63 47 205)(9 64 48 206)(10 65 49 207)(11 66 50 208)(12 67 26 209)(13 68 27 210)(14 69 28 211)(15 70 29 212)(16 71 30 213)(17 72 31 214)(18 73 32 215)(19 74 33 216)(20 75 34 217)(21 51 35 218)(22 52 36 219)(23 53 37 220)(24 54 38 221)(25 55 39 222)(76 284 274 121)(77 285 275 122)(78 286 251 123)(79 287 252 124)(80 288 253 125)(81 289 254 101)(82 290 255 102)(83 291 256 103)(84 292 257 104)(85 293 258 105)(86 294 259 106)(87 295 260 107)(88 296 261 108)(89 297 262 109)(90 298 263 110)(91 299 264 111)(92 300 265 112)(93 276 266 113)(94 277 267 114)(95 278 268 115)(96 279 269 116)(97 280 270 117)(98 281 271 118)(99 282 272 119)(100 283 273 120)(126 194 155 235)(127 195 156 236)(128 196 157 237)(129 197 158 238)(130 198 159 239)(131 199 160 240)(132 200 161 241)(133 176 162 242)(134 177 163 243)(135 178 164 244)(136 179 165 245)(137 180 166 246)(138 181 167 247)(139 182 168 248)(140 183 169 249)(141 184 170 250)(142 185 171 226)(143 186 172 227)(144 187 173 228)(145 188 174 229)(146 189 175 230)(147 190 151 231)(148 191 152 232)(149 192 153 233)(150 193 154 234)

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

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

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

150 conjugacy classes

class 1  2  3 4A4B5A5B5C5D 6 10A10B10C10D15A15B15C15D20A···20H25A···25T30A30B30C30D50A···50T75A···75T100A···100AN150A···150T
order1234455556101010101515151520···2025···253030303050···5075···75100···100150···150
size1123311112111122223···31···122221···12···23···32···2

150 irreducible representations

dim111111111222222
type+++-
imageC1C2C4C5C10C20C25C50C100S3Dic3C5×S3C5×Dic3S3×C25Dic3×C25
kernelDic3×C25C150C75C5×Dic3C30C15Dic3C6C3C50C25C10C5C2C1
# reps11244820204011442020

Matrix representation of Dic3×C25 in GL2(𝔽601) generated by

2450
0245
,
0600
11
,
542438
49759
G:=sub<GL(2,GF(601))| [245,0,0,245],[0,1,600,1],[542,497,438,59] >;

Dic3×C25 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{25}
% in TeX

G:=Group("Dic3xC25");
// GroupNames label

G:=SmallGroup(300,1);
// by ID

G=gap.SmallGroup(300,1);
# by ID

G:=PCGroup([5,-2,-5,-2,-5,-3,50,106,5004]);
// Polycyclic

G:=Group<a,b,c|a^25=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Dic3×C25 in TeX

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