metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C39⋊Q8, C26.7D6, C6.7D26, C3⋊1Dic26, C13⋊1Dic6, Dic3.D13, C78.7C22, Dic39.2C2, Dic13.1S3, C2.7(S3×D13), (Dic3×C13).1C2, (C3×Dic13).1C2, SmallGroup(312,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C39⋊Q8
G = < a,b,c | a39=b4=1, c2=b2, bab-1=a14, cac-1=a25, cbc-1=b-1 >
Smallest permutation representation of C39⋊Q8
►Regular action on 312 points
Generators in S312
(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)(274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312)
(1 142 50 103)(2 156 51 117)(3 131 52 92)(4 145 53 106)(5 120 54 81)(6 134 55 95)(7 148 56 109)(8 123 57 84)(9 137 58 98)(10 151 59 112)(11 126 60 87)(12 140 61 101)(13 154 62 115)(14 129 63 90)(15 143 64 104)(16 118 65 79)(17 132 66 93)(18 146 67 107)(19 121 68 82)(20 135 69 96)(21 149 70 110)(22 124 71 85)(23 138 72 99)(24 152 73 113)(25 127 74 88)(26 141 75 102)(27 155 76 116)(28 130 77 91)(29 144 78 105)(30 119 40 80)(31 133 41 94)(32 147 42 108)(33 122 43 83)(34 136 44 97)(35 150 45 111)(36 125 46 86)(37 139 47 100)(38 153 48 114)(39 128 49 89)(157 241 229 298)(158 255 230 312)(159 269 231 287)(160 244 232 301)(161 258 233 276)(162 272 234 290)(163 247 196 304)(164 261 197 279)(165 236 198 293)(166 250 199 307)(167 264 200 282)(168 239 201 296)(169 253 202 310)(170 267 203 285)(171 242 204 299)(172 256 205 274)(173 270 206 288)(174 245 207 302)(175 259 208 277)(176 273 209 291)(177 248 210 305)(178 262 211 280)(179 237 212 294)(180 251 213 308)(181 265 214 283)(182 240 215 297)(183 254 216 311)(184 268 217 286)(185 243 218 300)(186 257 219 275)(187 271 220 289)(188 246 221 303)(189 260 222 278)(190 235 223 292)(191 249 224 306)(192 263 225 281)(193 238 226 295)(194 252 227 309)(195 266 228 284)
(1 205 50 172)(2 230 51 158)(3 216 52 183)(4 202 53 169)(5 227 54 194)(6 213 55 180)(7 199 56 166)(8 224 57 191)(9 210 58 177)(10 196 59 163)(11 221 60 188)(12 207 61 174)(13 232 62 160)(14 218 63 185)(15 204 64 171)(16 229 65 157)(17 215 66 182)(18 201 67 168)(19 226 68 193)(20 212 69 179)(21 198 70 165)(22 223 71 190)(23 209 72 176)(24 234 73 162)(25 220 74 187)(26 206 75 173)(27 231 76 159)(28 217 77 184)(29 203 78 170)(30 228 40 195)(31 214 41 181)(32 200 42 167)(33 225 43 192)(34 211 44 178)(35 197 45 164)(36 222 46 189)(37 208 47 175)(38 233 48 161)(39 219 49 186)(79 298 118 241)(80 284 119 266)(81 309 120 252)(82 295 121 238)(83 281 122 263)(84 306 123 249)(85 292 124 235)(86 278 125 260)(87 303 126 246)(88 289 127 271)(89 275 128 257)(90 300 129 243)(91 286 130 268)(92 311 131 254)(93 297 132 240)(94 283 133 265)(95 308 134 251)(96 294 135 237)(97 280 136 262)(98 305 137 248)(99 291 138 273)(100 277 139 259)(101 302 140 245)(102 288 141 270)(103 274 142 256)(104 299 143 242)(105 285 144 267)(106 310 145 253)(107 296 146 239)(108 282 147 264)(109 307 148 250)(110 293 149 236)(111 279 150 261)(112 304 151 247)(113 290 152 272)(114 276 153 258)(115 301 154 244)(116 287 155 269)(117 312 156 255)
G:=sub<Sym(312)| (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)(274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312), (1,142,50,103)(2,156,51,117)(3,131,52,92)(4,145,53,106)(5,120,54,81)(6,134,55,95)(7,148,56,109)(8,123,57,84)(9,137,58,98)(10,151,59,112)(11,126,60,87)(12,140,61,101)(13,154,62,115)(14,129,63,90)(15,143,64,104)(16,118,65,79)(17,132,66,93)(18,146,67,107)(19,121,68,82)(20,135,69,96)(21,149,70,110)(22,124,71,85)(23,138,72,99)(24,152,73,113)(25,127,74,88)(26,141,75,102)(27,155,76,116)(28,130,77,91)(29,144,78,105)(30,119,40,80)(31,133,41,94)(32,147,42,108)(33,122,43,83)(34,136,44,97)(35,150,45,111)(36,125,46,86)(37,139,47,100)(38,153,48,114)(39,128,49,89)(157,241,229,298)(158,255,230,312)(159,269,231,287)(160,244,232,301)(161,258,233,276)(162,272,234,290)(163,247,196,304)(164,261,197,279)(165,236,198,293)(166,250,199,307)(167,264,200,282)(168,239,201,296)(169,253,202,310)(170,267,203,285)(171,242,204,299)(172,256,205,274)(173,270,206,288)(174,245,207,302)(175,259,208,277)(176,273,209,291)(177,248,210,305)(178,262,211,280)(179,237,212,294)(180,251,213,308)(181,265,214,283)(182,240,215,297)(183,254,216,311)(184,268,217,286)(185,243,218,300)(186,257,219,275)(187,271,220,289)(188,246,221,303)(189,260,222,278)(190,235,223,292)(191,249,224,306)(192,263,225,281)(193,238,226,295)(194,252,227,309)(195,266,228,284), (1,205,50,172)(2,230,51,158)(3,216,52,183)(4,202,53,169)(5,227,54,194)(6,213,55,180)(7,199,56,166)(8,224,57,191)(9,210,58,177)(10,196,59,163)(11,221,60,188)(12,207,61,174)(13,232,62,160)(14,218,63,185)(15,204,64,171)(16,229,65,157)(17,215,66,182)(18,201,67,168)(19,226,68,193)(20,212,69,179)(21,198,70,165)(22,223,71,190)(23,209,72,176)(24,234,73,162)(25,220,74,187)(26,206,75,173)(27,231,76,159)(28,217,77,184)(29,203,78,170)(30,228,40,195)(31,214,41,181)(32,200,42,167)(33,225,43,192)(34,211,44,178)(35,197,45,164)(36,222,46,189)(37,208,47,175)(38,233,48,161)(39,219,49,186)(79,298,118,241)(80,284,119,266)(81,309,120,252)(82,295,121,238)(83,281,122,263)(84,306,123,249)(85,292,124,235)(86,278,125,260)(87,303,126,246)(88,289,127,271)(89,275,128,257)(90,300,129,243)(91,286,130,268)(92,311,131,254)(93,297,132,240)(94,283,133,265)(95,308,134,251)(96,294,135,237)(97,280,136,262)(98,305,137,248)(99,291,138,273)(100,277,139,259)(101,302,140,245)(102,288,141,270)(103,274,142,256)(104,299,143,242)(105,285,144,267)(106,310,145,253)(107,296,146,239)(108,282,147,264)(109,307,148,250)(110,293,149,236)(111,279,150,261)(112,304,151,247)(113,290,152,272)(114,276,153,258)(115,301,154,244)(116,287,155,269)(117,312,156,255)>;
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)(274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312), (1,142,50,103)(2,156,51,117)(3,131,52,92)(4,145,53,106)(5,120,54,81)(6,134,55,95)(7,148,56,109)(8,123,57,84)(9,137,58,98)(10,151,59,112)(11,126,60,87)(12,140,61,101)(13,154,62,115)(14,129,63,90)(15,143,64,104)(16,118,65,79)(17,132,66,93)(18,146,67,107)(19,121,68,82)(20,135,69,96)(21,149,70,110)(22,124,71,85)(23,138,72,99)(24,152,73,113)(25,127,74,88)(26,141,75,102)(27,155,76,116)(28,130,77,91)(29,144,78,105)(30,119,40,80)(31,133,41,94)(32,147,42,108)(33,122,43,83)(34,136,44,97)(35,150,45,111)(36,125,46,86)(37,139,47,100)(38,153,48,114)(39,128,49,89)(157,241,229,298)(158,255,230,312)(159,269,231,287)(160,244,232,301)(161,258,233,276)(162,272,234,290)(163,247,196,304)(164,261,197,279)(165,236,198,293)(166,250,199,307)(167,264,200,282)(168,239,201,296)(169,253,202,310)(170,267,203,285)(171,242,204,299)(172,256,205,274)(173,270,206,288)(174,245,207,302)(175,259,208,277)(176,273,209,291)(177,248,210,305)(178,262,211,280)(179,237,212,294)(180,251,213,308)(181,265,214,283)(182,240,215,297)(183,254,216,311)(184,268,217,286)(185,243,218,300)(186,257,219,275)(187,271,220,289)(188,246,221,303)(189,260,222,278)(190,235,223,292)(191,249,224,306)(192,263,225,281)(193,238,226,295)(194,252,227,309)(195,266,228,284), (1,205,50,172)(2,230,51,158)(3,216,52,183)(4,202,53,169)(5,227,54,194)(6,213,55,180)(7,199,56,166)(8,224,57,191)(9,210,58,177)(10,196,59,163)(11,221,60,188)(12,207,61,174)(13,232,62,160)(14,218,63,185)(15,204,64,171)(16,229,65,157)(17,215,66,182)(18,201,67,168)(19,226,68,193)(20,212,69,179)(21,198,70,165)(22,223,71,190)(23,209,72,176)(24,234,73,162)(25,220,74,187)(26,206,75,173)(27,231,76,159)(28,217,77,184)(29,203,78,170)(30,228,40,195)(31,214,41,181)(32,200,42,167)(33,225,43,192)(34,211,44,178)(35,197,45,164)(36,222,46,189)(37,208,47,175)(38,233,48,161)(39,219,49,186)(79,298,118,241)(80,284,119,266)(81,309,120,252)(82,295,121,238)(83,281,122,263)(84,306,123,249)(85,292,124,235)(86,278,125,260)(87,303,126,246)(88,289,127,271)(89,275,128,257)(90,300,129,243)(91,286,130,268)(92,311,131,254)(93,297,132,240)(94,283,133,265)(95,308,134,251)(96,294,135,237)(97,280,136,262)(98,305,137,248)(99,291,138,273)(100,277,139,259)(101,302,140,245)(102,288,141,270)(103,274,142,256)(104,299,143,242)(105,285,144,267)(106,310,145,253)(107,296,146,239)(108,282,147,264)(109,307,148,250)(110,293,149,236)(111,279,150,261)(112,304,151,247)(113,290,152,272)(114,276,153,258)(115,301,154,244)(116,287,155,269)(117,312,156,255) );
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),(274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312)], [(1,142,50,103),(2,156,51,117),(3,131,52,92),(4,145,53,106),(5,120,54,81),(6,134,55,95),(7,148,56,109),(8,123,57,84),(9,137,58,98),(10,151,59,112),(11,126,60,87),(12,140,61,101),(13,154,62,115),(14,129,63,90),(15,143,64,104),(16,118,65,79),(17,132,66,93),(18,146,67,107),(19,121,68,82),(20,135,69,96),(21,149,70,110),(22,124,71,85),(23,138,72,99),(24,152,73,113),(25,127,74,88),(26,141,75,102),(27,155,76,116),(28,130,77,91),(29,144,78,105),(30,119,40,80),(31,133,41,94),(32,147,42,108),(33,122,43,83),(34,136,44,97),(35,150,45,111),(36,125,46,86),(37,139,47,100),(38,153,48,114),(39,128,49,89),(157,241,229,298),(158,255,230,312),(159,269,231,287),(160,244,232,301),(161,258,233,276),(162,272,234,290),(163,247,196,304),(164,261,197,279),(165,236,198,293),(166,250,199,307),(167,264,200,282),(168,239,201,296),(169,253,202,310),(170,267,203,285),(171,242,204,299),(172,256,205,274),(173,270,206,288),(174,245,207,302),(175,259,208,277),(176,273,209,291),(177,248,210,305),(178,262,211,280),(179,237,212,294),(180,251,213,308),(181,265,214,283),(182,240,215,297),(183,254,216,311),(184,268,217,286),(185,243,218,300),(186,257,219,275),(187,271,220,289),(188,246,221,303),(189,260,222,278),(190,235,223,292),(191,249,224,306),(192,263,225,281),(193,238,226,295),(194,252,227,309),(195,266,228,284)], [(1,205,50,172),(2,230,51,158),(3,216,52,183),(4,202,53,169),(5,227,54,194),(6,213,55,180),(7,199,56,166),(8,224,57,191),(9,210,58,177),(10,196,59,163),(11,221,60,188),(12,207,61,174),(13,232,62,160),(14,218,63,185),(15,204,64,171),(16,229,65,157),(17,215,66,182),(18,201,67,168),(19,226,68,193),(20,212,69,179),(21,198,70,165),(22,223,71,190),(23,209,72,176),(24,234,73,162),(25,220,74,187),(26,206,75,173),(27,231,76,159),(28,217,77,184),(29,203,78,170),(30,228,40,195),(31,214,41,181),(32,200,42,167),(33,225,43,192),(34,211,44,178),(35,197,45,164),(36,222,46,189),(37,208,47,175),(38,233,48,161),(39,219,49,186),(79,298,118,241),(80,284,119,266),(81,309,120,252),(82,295,121,238),(83,281,122,263),(84,306,123,249),(85,292,124,235),(86,278,125,260),(87,303,126,246),(88,289,127,271),(89,275,128,257),(90,300,129,243),(91,286,130,268),(92,311,131,254),(93,297,132,240),(94,283,133,265),(95,308,134,251),(96,294,135,237),(97,280,136,262),(98,305,137,248),(99,291,138,273),(100,277,139,259),(101,302,140,245),(102,288,141,270),(103,274,142,256),(104,299,143,242),(105,285,144,267),(106,310,145,253),(107,296,146,239),(108,282,147,264),(109,307,148,250),(110,293,149,236),(111,279,150,261),(112,304,151,247),(113,290,152,272),(114,276,153,258),(115,301,154,244),(116,287,155,269),(117,312,156,255)]])
45 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 12A | 12B | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39F | 52A | ··· | 52L | 78A | ··· | 78F |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 |
| size | 1 | 1 | 2 | 6 | 26 | 78 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D6 | Dic6 | D13 | D26 | Dic26 | S3×D13 | C39⋊Q8 |
| kernel | C39⋊Q8 | Dic3×C13 | C3×Dic13 | Dic39 | Dic13 | C39 | C26 | C13 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 6 | 6 |
Matrix representation of C39⋊Q8 ►in GL4(𝔽157) generated by
| 156 | 1 | 0 | 0 |
| 156 | 0 | 0 | 0 |
| 0 | 0 | 57 | 56 |
| 0 | 0 | 126 | 88 |
| 79 | 136 | 0 | 0 |
| 58 | 78 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 24 | 109 | 0 | 0 |
| 48 | 133 | 0 | 0 |
| 0 | 0 | 3 | 122 |
| 0 | 0 | 72 | 154 |
G:=sub<GL(4,GF(157))| [156,156,0,0,1,0,0,0,0,0,57,126,0,0,56,88],[79,58,0,0,136,78,0,0,0,0,1,0,0,0,0,1],[24,48,0,0,109,133,0,0,0,0,3,72,0,0,122,154] >;
C39⋊Q8 in GAP, Magma, Sage, TeX
C_{39}\rtimes Q_8 % in TeX
G:=Group("C39:Q8"); // GroupNames label
G:=SmallGroup(312,21);
// by ID
G=gap.SmallGroup(312,21);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,20,61,26,168,7204]);
// Polycyclic
G:=Group<a,b,c|a^39=b^4=1,c^2=b^2,b*a*b^-1=a^14,c*a*c^-1=a^25,c*b*c^-1=b^-1>;
// generators/relations
Export