direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C52⋊6C4, C52⋊9C12, C30.6D5, C15⋊4Dic5, (C5×C15)⋊12C4, (C5×C10).4C6, (C5×C30).4C2, C6.2(C5⋊D5), C5⋊2(C3×Dic5), C10.3(C3×D5), C2.(C3×C5⋊D5), SmallGroup(300,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C3×C52⋊6C4 |
Generators and relations for C3×C52⋊6C4
G = < a,b,c,d | a3=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Smallest permutation representation of C3×C52⋊6C4
►Regular action on 300 points
Generators in S300
(1 123 73)(2 124 74)(3 125 75)(4 121 71)(5 122 72)(6 270 220)(7 266 216)(8 267 217)(9 268 218)(10 269 219)(11 265 215)(12 261 211)(13 262 212)(14 263 213)(15 264 214)(16 260 210)(17 256 206)(18 257 207)(19 258 208)(20 259 209)(21 255 205)(22 251 201)(23 252 202)(24 253 203)(25 254 204)(26 126 76)(27 127 77)(28 128 78)(29 129 79)(30 130 80)(31 131 81)(32 132 82)(33 133 83)(34 134 84)(35 135 85)(36 136 86)(37 137 87)(38 138 88)(39 139 89)(40 140 90)(41 141 91)(42 142 92)(43 143 93)(44 144 94)(45 145 95)(46 146 96)(47 147 97)(48 148 98)(49 149 99)(50 150 100)(51 166 101)(52 167 102)(53 168 103)(54 169 104)(55 170 105)(56 161 106)(57 162 107)(58 163 108)(59 164 109)(60 165 110)(61 156 111)(62 157 112)(63 158 113)(64 159 114)(65 160 115)(66 151 116)(67 152 117)(68 153 118)(69 154 119)(70 155 120)(171 271 221)(172 272 222)(173 273 223)(174 274 224)(175 275 225)(176 276 226)(177 277 227)(178 278 228)(179 279 229)(180 280 230)(181 281 231)(182 282 232)(183 283 233)(184 284 234)(185 285 235)(186 286 236)(187 287 237)(188 288 238)(189 289 239)(190 290 240)(191 291 241)(192 292 242)(193 293 243)(194 294 244)(195 295 245)(196 296 246)(197 297 247)(198 298 248)(199 299 249)(200 300 250)
(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 44 33 28 38)(2 45 34 29 39)(3 41 35 30 40)(4 42 31 26 36)(5 43 32 27 37)(6 16 300 25 11)(7 17 296 21 12)(8 18 297 22 13)(9 19 298 23 14)(10 20 299 24 15)(46 67 56 51 61)(47 68 57 52 62)(48 69 58 53 63)(49 70 59 54 64)(50 66 60 55 65)(71 92 81 76 86)(72 93 82 77 87)(73 94 83 78 88)(74 95 84 79 89)(75 91 85 80 90)(96 117 106 101 111)(97 118 107 102 112)(98 119 108 103 113)(99 120 109 104 114)(100 116 110 105 115)(121 142 131 126 136)(122 143 132 127 137)(123 144 133 128 138)(124 145 134 129 139)(125 141 135 130 140)(146 152 161 166 156)(147 153 162 167 157)(148 154 163 168 158)(149 155 164 169 159)(150 151 165 170 160)(171 180 186 191 181)(172 176 187 192 182)(173 177 188 193 183)(174 178 189 194 184)(175 179 190 195 185)(196 205 211 216 206)(197 201 212 217 207)(198 202 213 218 208)(199 203 214 219 209)(200 204 215 220 210)(221 230 236 241 231)(222 226 237 242 232)(223 227 238 243 233)(224 228 239 244 234)(225 229 240 245 235)(246 255 261 266 256)(247 251 262 267 257)(248 252 263 268 258)(249 253 264 269 259)(250 254 265 270 260)(271 280 286 291 281)(272 276 287 292 282)(273 277 288 293 283)(274 278 289 294 284)(275 279 290 295 285)
(1 212 48 187)(2 211 49 186)(3 215 50 190)(4 214 46 189)(5 213 47 188)(6 160 295 140)(7 159 291 139)(8 158 292 138)(9 157 293 137)(10 156 294 136)(11 150 290 125)(12 149 286 124)(13 148 287 123)(14 147 288 122)(15 146 289 121)(16 170 285 130)(17 169 281 129)(18 168 282 128)(19 167 283 127)(20 166 284 126)(21 155 280 145)(22 154 276 144)(23 153 277 143)(24 152 278 142)(25 151 279 141)(26 209 51 184)(27 208 52 183)(28 207 53 182)(29 206 54 181)(30 210 55 185)(31 199 56 174)(32 198 57 173)(33 197 58 172)(34 196 59 171)(35 200 60 175)(36 219 61 194)(37 218 62 193)(38 217 63 192)(39 216 64 191)(40 220 65 195)(41 204 66 179)(42 203 67 178)(43 202 68 177)(44 201 69 176)(45 205 70 180)(71 264 96 239)(72 263 97 238)(73 262 98 237)(74 261 99 236)(75 265 100 240)(76 259 101 234)(77 258 102 233)(78 257 103 232)(79 256 104 231)(80 260 105 235)(81 249 106 224)(82 248 107 223)(83 247 108 222)(84 246 109 221)(85 250 110 225)(86 269 111 244)(87 268 112 243)(88 267 113 242)(89 266 114 241)(90 270 115 245)(91 254 116 229)(92 253 117 228)(93 252 118 227)(94 251 119 226)(95 255 120 230)(131 299 161 274)(132 298 162 273)(133 297 163 272)(134 296 164 271)(135 300 165 275)
G:=sub<Sym(300)| (1,123,73)(2,124,74)(3,125,75)(4,121,71)(5,122,72)(6,270,220)(7,266,216)(8,267,217)(9,268,218)(10,269,219)(11,265,215)(12,261,211)(13,262,212)(14,263,213)(15,264,214)(16,260,210)(17,256,206)(18,257,207)(19,258,208)(20,259,209)(21,255,205)(22,251,201)(23,252,202)(24,253,203)(25,254,204)(26,126,76)(27,127,77)(28,128,78)(29,129,79)(30,130,80)(31,131,81)(32,132,82)(33,133,83)(34,134,84)(35,135,85)(36,136,86)(37,137,87)(38,138,88)(39,139,89)(40,140,90)(41,141,91)(42,142,92)(43,143,93)(44,144,94)(45,145,95)(46,146,96)(47,147,97)(48,148,98)(49,149,99)(50,150,100)(51,166,101)(52,167,102)(53,168,103)(54,169,104)(55,170,105)(56,161,106)(57,162,107)(58,163,108)(59,164,109)(60,165,110)(61,156,111)(62,157,112)(63,158,113)(64,159,114)(65,160,115)(66,151,116)(67,152,117)(68,153,118)(69,154,119)(70,155,120)(171,271,221)(172,272,222)(173,273,223)(174,274,224)(175,275,225)(176,276,226)(177,277,227)(178,278,228)(179,279,229)(180,280,230)(181,281,231)(182,282,232)(183,283,233)(184,284,234)(185,285,235)(186,286,236)(187,287,237)(188,288,238)(189,289,239)(190,290,240)(191,291,241)(192,292,242)(193,293,243)(194,294,244)(195,295,245)(196,296,246)(197,297,247)(198,298,248)(199,299,249)(200,300,250), (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), (1,44,33,28,38)(2,45,34,29,39)(3,41,35,30,40)(4,42,31,26,36)(5,43,32,27,37)(6,16,300,25,11)(7,17,296,21,12)(8,18,297,22,13)(9,19,298,23,14)(10,20,299,24,15)(46,67,56,51,61)(47,68,57,52,62)(48,69,58,53,63)(49,70,59,54,64)(50,66,60,55,65)(71,92,81,76,86)(72,93,82,77,87)(73,94,83,78,88)(74,95,84,79,89)(75,91,85,80,90)(96,117,106,101,111)(97,118,107,102,112)(98,119,108,103,113)(99,120,109,104,114)(100,116,110,105,115)(121,142,131,126,136)(122,143,132,127,137)(123,144,133,128,138)(124,145,134,129,139)(125,141,135,130,140)(146,152,161,166,156)(147,153,162,167,157)(148,154,163,168,158)(149,155,164,169,159)(150,151,165,170,160)(171,180,186,191,181)(172,176,187,192,182)(173,177,188,193,183)(174,178,189,194,184)(175,179,190,195,185)(196,205,211,216,206)(197,201,212,217,207)(198,202,213,218,208)(199,203,214,219,209)(200,204,215,220,210)(221,230,236,241,231)(222,226,237,242,232)(223,227,238,243,233)(224,228,239,244,234)(225,229,240,245,235)(246,255,261,266,256)(247,251,262,267,257)(248,252,263,268,258)(249,253,264,269,259)(250,254,265,270,260)(271,280,286,291,281)(272,276,287,292,282)(273,277,288,293,283)(274,278,289,294,284)(275,279,290,295,285), (1,212,48,187)(2,211,49,186)(3,215,50,190)(4,214,46,189)(5,213,47,188)(6,160,295,140)(7,159,291,139)(8,158,292,138)(9,157,293,137)(10,156,294,136)(11,150,290,125)(12,149,286,124)(13,148,287,123)(14,147,288,122)(15,146,289,121)(16,170,285,130)(17,169,281,129)(18,168,282,128)(19,167,283,127)(20,166,284,126)(21,155,280,145)(22,154,276,144)(23,153,277,143)(24,152,278,142)(25,151,279,141)(26,209,51,184)(27,208,52,183)(28,207,53,182)(29,206,54,181)(30,210,55,185)(31,199,56,174)(32,198,57,173)(33,197,58,172)(34,196,59,171)(35,200,60,175)(36,219,61,194)(37,218,62,193)(38,217,63,192)(39,216,64,191)(40,220,65,195)(41,204,66,179)(42,203,67,178)(43,202,68,177)(44,201,69,176)(45,205,70,180)(71,264,96,239)(72,263,97,238)(73,262,98,237)(74,261,99,236)(75,265,100,240)(76,259,101,234)(77,258,102,233)(78,257,103,232)(79,256,104,231)(80,260,105,235)(81,249,106,224)(82,248,107,223)(83,247,108,222)(84,246,109,221)(85,250,110,225)(86,269,111,244)(87,268,112,243)(88,267,113,242)(89,266,114,241)(90,270,115,245)(91,254,116,229)(92,253,117,228)(93,252,118,227)(94,251,119,226)(95,255,120,230)(131,299,161,274)(132,298,162,273)(133,297,163,272)(134,296,164,271)(135,300,165,275)>;
G:=Group( (1,123,73)(2,124,74)(3,125,75)(4,121,71)(5,122,72)(6,270,220)(7,266,216)(8,267,217)(9,268,218)(10,269,219)(11,265,215)(12,261,211)(13,262,212)(14,263,213)(15,264,214)(16,260,210)(17,256,206)(18,257,207)(19,258,208)(20,259,209)(21,255,205)(22,251,201)(23,252,202)(24,253,203)(25,254,204)(26,126,76)(27,127,77)(28,128,78)(29,129,79)(30,130,80)(31,131,81)(32,132,82)(33,133,83)(34,134,84)(35,135,85)(36,136,86)(37,137,87)(38,138,88)(39,139,89)(40,140,90)(41,141,91)(42,142,92)(43,143,93)(44,144,94)(45,145,95)(46,146,96)(47,147,97)(48,148,98)(49,149,99)(50,150,100)(51,166,101)(52,167,102)(53,168,103)(54,169,104)(55,170,105)(56,161,106)(57,162,107)(58,163,108)(59,164,109)(60,165,110)(61,156,111)(62,157,112)(63,158,113)(64,159,114)(65,160,115)(66,151,116)(67,152,117)(68,153,118)(69,154,119)(70,155,120)(171,271,221)(172,272,222)(173,273,223)(174,274,224)(175,275,225)(176,276,226)(177,277,227)(178,278,228)(179,279,229)(180,280,230)(181,281,231)(182,282,232)(183,283,233)(184,284,234)(185,285,235)(186,286,236)(187,287,237)(188,288,238)(189,289,239)(190,290,240)(191,291,241)(192,292,242)(193,293,243)(194,294,244)(195,295,245)(196,296,246)(197,297,247)(198,298,248)(199,299,249)(200,300,250), (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), (1,44,33,28,38)(2,45,34,29,39)(3,41,35,30,40)(4,42,31,26,36)(5,43,32,27,37)(6,16,300,25,11)(7,17,296,21,12)(8,18,297,22,13)(9,19,298,23,14)(10,20,299,24,15)(46,67,56,51,61)(47,68,57,52,62)(48,69,58,53,63)(49,70,59,54,64)(50,66,60,55,65)(71,92,81,76,86)(72,93,82,77,87)(73,94,83,78,88)(74,95,84,79,89)(75,91,85,80,90)(96,117,106,101,111)(97,118,107,102,112)(98,119,108,103,113)(99,120,109,104,114)(100,116,110,105,115)(121,142,131,126,136)(122,143,132,127,137)(123,144,133,128,138)(124,145,134,129,139)(125,141,135,130,140)(146,152,161,166,156)(147,153,162,167,157)(148,154,163,168,158)(149,155,164,169,159)(150,151,165,170,160)(171,180,186,191,181)(172,176,187,192,182)(173,177,188,193,183)(174,178,189,194,184)(175,179,190,195,185)(196,205,211,216,206)(197,201,212,217,207)(198,202,213,218,208)(199,203,214,219,209)(200,204,215,220,210)(221,230,236,241,231)(222,226,237,242,232)(223,227,238,243,233)(224,228,239,244,234)(225,229,240,245,235)(246,255,261,266,256)(247,251,262,267,257)(248,252,263,268,258)(249,253,264,269,259)(250,254,265,270,260)(271,280,286,291,281)(272,276,287,292,282)(273,277,288,293,283)(274,278,289,294,284)(275,279,290,295,285), (1,212,48,187)(2,211,49,186)(3,215,50,190)(4,214,46,189)(5,213,47,188)(6,160,295,140)(7,159,291,139)(8,158,292,138)(9,157,293,137)(10,156,294,136)(11,150,290,125)(12,149,286,124)(13,148,287,123)(14,147,288,122)(15,146,289,121)(16,170,285,130)(17,169,281,129)(18,168,282,128)(19,167,283,127)(20,166,284,126)(21,155,280,145)(22,154,276,144)(23,153,277,143)(24,152,278,142)(25,151,279,141)(26,209,51,184)(27,208,52,183)(28,207,53,182)(29,206,54,181)(30,210,55,185)(31,199,56,174)(32,198,57,173)(33,197,58,172)(34,196,59,171)(35,200,60,175)(36,219,61,194)(37,218,62,193)(38,217,63,192)(39,216,64,191)(40,220,65,195)(41,204,66,179)(42,203,67,178)(43,202,68,177)(44,201,69,176)(45,205,70,180)(71,264,96,239)(72,263,97,238)(73,262,98,237)(74,261,99,236)(75,265,100,240)(76,259,101,234)(77,258,102,233)(78,257,103,232)(79,256,104,231)(80,260,105,235)(81,249,106,224)(82,248,107,223)(83,247,108,222)(84,246,109,221)(85,250,110,225)(86,269,111,244)(87,268,112,243)(88,267,113,242)(89,266,114,241)(90,270,115,245)(91,254,116,229)(92,253,117,228)(93,252,118,227)(94,251,119,226)(95,255,120,230)(131,299,161,274)(132,298,162,273)(133,297,163,272)(134,296,164,271)(135,300,165,275) );
G=PermutationGroup([[(1,123,73),(2,124,74),(3,125,75),(4,121,71),(5,122,72),(6,270,220),(7,266,216),(8,267,217),(9,268,218),(10,269,219),(11,265,215),(12,261,211),(13,262,212),(14,263,213),(15,264,214),(16,260,210),(17,256,206),(18,257,207),(19,258,208),(20,259,209),(21,255,205),(22,251,201),(23,252,202),(24,253,203),(25,254,204),(26,126,76),(27,127,77),(28,128,78),(29,129,79),(30,130,80),(31,131,81),(32,132,82),(33,133,83),(34,134,84),(35,135,85),(36,136,86),(37,137,87),(38,138,88),(39,139,89),(40,140,90),(41,141,91),(42,142,92),(43,143,93),(44,144,94),(45,145,95),(46,146,96),(47,147,97),(48,148,98),(49,149,99),(50,150,100),(51,166,101),(52,167,102),(53,168,103),(54,169,104),(55,170,105),(56,161,106),(57,162,107),(58,163,108),(59,164,109),(60,165,110),(61,156,111),(62,157,112),(63,158,113),(64,159,114),(65,160,115),(66,151,116),(67,152,117),(68,153,118),(69,154,119),(70,155,120),(171,271,221),(172,272,222),(173,273,223),(174,274,224),(175,275,225),(176,276,226),(177,277,227),(178,278,228),(179,279,229),(180,280,230),(181,281,231),(182,282,232),(183,283,233),(184,284,234),(185,285,235),(186,286,236),(187,287,237),(188,288,238),(189,289,239),(190,290,240),(191,291,241),(192,292,242),(193,293,243),(194,294,244),(195,295,245),(196,296,246),(197,297,247),(198,298,248),(199,299,249),(200,300,250)], [(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)], [(1,44,33,28,38),(2,45,34,29,39),(3,41,35,30,40),(4,42,31,26,36),(5,43,32,27,37),(6,16,300,25,11),(7,17,296,21,12),(8,18,297,22,13),(9,19,298,23,14),(10,20,299,24,15),(46,67,56,51,61),(47,68,57,52,62),(48,69,58,53,63),(49,70,59,54,64),(50,66,60,55,65),(71,92,81,76,86),(72,93,82,77,87),(73,94,83,78,88),(74,95,84,79,89),(75,91,85,80,90),(96,117,106,101,111),(97,118,107,102,112),(98,119,108,103,113),(99,120,109,104,114),(100,116,110,105,115),(121,142,131,126,136),(122,143,132,127,137),(123,144,133,128,138),(124,145,134,129,139),(125,141,135,130,140),(146,152,161,166,156),(147,153,162,167,157),(148,154,163,168,158),(149,155,164,169,159),(150,151,165,170,160),(171,180,186,191,181),(172,176,187,192,182),(173,177,188,193,183),(174,178,189,194,184),(175,179,190,195,185),(196,205,211,216,206),(197,201,212,217,207),(198,202,213,218,208),(199,203,214,219,209),(200,204,215,220,210),(221,230,236,241,231),(222,226,237,242,232),(223,227,238,243,233),(224,228,239,244,234),(225,229,240,245,235),(246,255,261,266,256),(247,251,262,267,257),(248,252,263,268,258),(249,253,264,269,259),(250,254,265,270,260),(271,280,286,291,281),(272,276,287,292,282),(273,277,288,293,283),(274,278,289,294,284),(275,279,290,295,285)], [(1,212,48,187),(2,211,49,186),(3,215,50,190),(4,214,46,189),(5,213,47,188),(6,160,295,140),(7,159,291,139),(8,158,292,138),(9,157,293,137),(10,156,294,136),(11,150,290,125),(12,149,286,124),(13,148,287,123),(14,147,288,122),(15,146,289,121),(16,170,285,130),(17,169,281,129),(18,168,282,128),(19,167,283,127),(20,166,284,126),(21,155,280,145),(22,154,276,144),(23,153,277,143),(24,152,278,142),(25,151,279,141),(26,209,51,184),(27,208,52,183),(28,207,53,182),(29,206,54,181),(30,210,55,185),(31,199,56,174),(32,198,57,173),(33,197,58,172),(34,196,59,171),(35,200,60,175),(36,219,61,194),(37,218,62,193),(38,217,63,192),(39,216,64,191),(40,220,65,195),(41,204,66,179),(42,203,67,178),(43,202,68,177),(44,201,69,176),(45,205,70,180),(71,264,96,239),(72,263,97,238),(73,262,98,237),(74,261,99,236),(75,265,100,240),(76,259,101,234),(77,258,102,233),(78,257,103,232),(79,256,104,231),(80,260,105,235),(81,249,106,224),(82,248,107,223),(83,247,108,222),(84,246,109,221),(85,250,110,225),(86,269,111,244),(87,268,112,243),(88,267,113,242),(89,266,114,241),(90,270,115,245),(91,254,116,229),(92,253,117,228),(93,252,118,227),(94,251,119,226),(95,255,120,230),(131,299,161,274),(132,298,162,273),(133,297,163,272),(134,296,164,271),(135,300,165,275)]])
84 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | ··· | 5L | 6A | 6B | 10A | ··· | 10L | 12A | 12B | 12C | 12D | 15A | ··· | 15X | 30A | ··· | 30X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | ··· | 5 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 25 | 25 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 25 | 25 | 25 | 25 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D5 | Dic5 | C3×D5 | C3×Dic5 |
| kernel | C3×C52⋊6C4 | C5×C30 | C52⋊6C4 | C5×C15 | C5×C10 | C52 | C30 | C15 | C10 | C5 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 12 | 12 | 24 | 24 |
Matrix representation of C3×C52⋊6C4 ►in GL5(𝔽61)
| 47 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 | 0 |
| 0 | 0 | 58 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 42 | 18 |
| 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 2 |
| 0 | 0 | 0 | 2 | 34 |
G:=sub<GL(5,GF(61))| [47,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,20,0,0,0,0,0,58,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,42,0,0,0,1,18],[50,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,27,2,0,0,0,2,34] >;
C3×C52⋊6C4 in GAP, Magma, Sage, TeX
C_3\times C_5^2\rtimes_6C_4
% in TeX
G:=Group("C3xC5^2:6C4"); // GroupNames label
G:=SmallGroup(300,17);
// by ID
G=gap.SmallGroup(300,17);
# by ID
G:=PCGroup([5,-2,-3,-2,-5,-5,30,963,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export