direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C39, C4.C78, C52.7C6, C156.7C2, C12.3C26, C78.24C22, C2.2(C2×C78), C6.7(C2×C26), C26.15(C2×C6), SmallGroup(312,44)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C39
G = < a,b,c | a39=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C39
►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)
(1 246 89 175)(2 247 90 176)(3 248 91 177)(4 249 92 178)(5 250 93 179)(6 251 94 180)(7 252 95 181)(8 253 96 182)(9 254 97 183)(10 255 98 184)(11 256 99 185)(12 257 100 186)(13 258 101 187)(14 259 102 188)(15 260 103 189)(16 261 104 190)(17 262 105 191)(18 263 106 192)(19 264 107 193)(20 265 108 194)(21 266 109 195)(22 267 110 157)(23 268 111 158)(24 269 112 159)(25 270 113 160)(26 271 114 161)(27 272 115 162)(28 273 116 163)(29 235 117 164)(30 236 79 165)(31 237 80 166)(32 238 81 167)(33 239 82 168)(34 240 83 169)(35 241 84 170)(36 242 85 171)(37 243 86 172)(38 244 87 173)(39 245 88 174)(40 292 203 148)(41 293 204 149)(42 294 205 150)(43 295 206 151)(44 296 207 152)(45 297 208 153)(46 298 209 154)(47 299 210 155)(48 300 211 156)(49 301 212 118)(50 302 213 119)(51 303 214 120)(52 304 215 121)(53 305 216 122)(54 306 217 123)(55 307 218 124)(56 308 219 125)(57 309 220 126)(58 310 221 127)(59 311 222 128)(60 312 223 129)(61 274 224 130)(62 275 225 131)(63 276 226 132)(64 277 227 133)(65 278 228 134)(66 279 229 135)(67 280 230 136)(68 281 231 137)(69 282 232 138)(70 283 233 139)(71 284 234 140)(72 285 196 141)(73 286 197 142)(74 287 198 143)(75 288 199 144)(76 289 200 145)(77 290 201 146)(78 291 202 147)
(1 43 89 206)(2 44 90 207)(3 45 91 208)(4 46 92 209)(5 47 93 210)(6 48 94 211)(7 49 95 212)(8 50 96 213)(9 51 97 214)(10 52 98 215)(11 53 99 216)(12 54 100 217)(13 55 101 218)(14 56 102 219)(15 57 103 220)(16 58 104 221)(17 59 105 222)(18 60 106 223)(19 61 107 224)(20 62 108 225)(21 63 109 226)(22 64 110 227)(23 65 111 228)(24 66 112 229)(25 67 113 230)(26 68 114 231)(27 69 115 232)(28 70 116 233)(29 71 117 234)(30 72 79 196)(31 73 80 197)(32 74 81 198)(33 75 82 199)(34 76 83 200)(35 77 84 201)(36 78 85 202)(37 40 86 203)(38 41 87 204)(39 42 88 205)(118 181 301 252)(119 182 302 253)(120 183 303 254)(121 184 304 255)(122 185 305 256)(123 186 306 257)(124 187 307 258)(125 188 308 259)(126 189 309 260)(127 190 310 261)(128 191 311 262)(129 192 312 263)(130 193 274 264)(131 194 275 265)(132 195 276 266)(133 157 277 267)(134 158 278 268)(135 159 279 269)(136 160 280 270)(137 161 281 271)(138 162 282 272)(139 163 283 273)(140 164 284 235)(141 165 285 236)(142 166 286 237)(143 167 287 238)(144 168 288 239)(145 169 289 240)(146 170 290 241)(147 171 291 242)(148 172 292 243)(149 173 293 244)(150 174 294 245)(151 175 295 246)(152 176 296 247)(153 177 297 248)(154 178 298 249)(155 179 299 250)(156 180 300 251)
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), (1,246,89,175)(2,247,90,176)(3,248,91,177)(4,249,92,178)(5,250,93,179)(6,251,94,180)(7,252,95,181)(8,253,96,182)(9,254,97,183)(10,255,98,184)(11,256,99,185)(12,257,100,186)(13,258,101,187)(14,259,102,188)(15,260,103,189)(16,261,104,190)(17,262,105,191)(18,263,106,192)(19,264,107,193)(20,265,108,194)(21,266,109,195)(22,267,110,157)(23,268,111,158)(24,269,112,159)(25,270,113,160)(26,271,114,161)(27,272,115,162)(28,273,116,163)(29,235,117,164)(30,236,79,165)(31,237,80,166)(32,238,81,167)(33,239,82,168)(34,240,83,169)(35,241,84,170)(36,242,85,171)(37,243,86,172)(38,244,87,173)(39,245,88,174)(40,292,203,148)(41,293,204,149)(42,294,205,150)(43,295,206,151)(44,296,207,152)(45,297,208,153)(46,298,209,154)(47,299,210,155)(48,300,211,156)(49,301,212,118)(50,302,213,119)(51,303,214,120)(52,304,215,121)(53,305,216,122)(54,306,217,123)(55,307,218,124)(56,308,219,125)(57,309,220,126)(58,310,221,127)(59,311,222,128)(60,312,223,129)(61,274,224,130)(62,275,225,131)(63,276,226,132)(64,277,227,133)(65,278,228,134)(66,279,229,135)(67,280,230,136)(68,281,231,137)(69,282,232,138)(70,283,233,139)(71,284,234,140)(72,285,196,141)(73,286,197,142)(74,287,198,143)(75,288,199,144)(76,289,200,145)(77,290,201,146)(78,291,202,147), (1,43,89,206)(2,44,90,207)(3,45,91,208)(4,46,92,209)(5,47,93,210)(6,48,94,211)(7,49,95,212)(8,50,96,213)(9,51,97,214)(10,52,98,215)(11,53,99,216)(12,54,100,217)(13,55,101,218)(14,56,102,219)(15,57,103,220)(16,58,104,221)(17,59,105,222)(18,60,106,223)(19,61,107,224)(20,62,108,225)(21,63,109,226)(22,64,110,227)(23,65,111,228)(24,66,112,229)(25,67,113,230)(26,68,114,231)(27,69,115,232)(28,70,116,233)(29,71,117,234)(30,72,79,196)(31,73,80,197)(32,74,81,198)(33,75,82,199)(34,76,83,200)(35,77,84,201)(36,78,85,202)(37,40,86,203)(38,41,87,204)(39,42,88,205)(118,181,301,252)(119,182,302,253)(120,183,303,254)(121,184,304,255)(122,185,305,256)(123,186,306,257)(124,187,307,258)(125,188,308,259)(126,189,309,260)(127,190,310,261)(128,191,311,262)(129,192,312,263)(130,193,274,264)(131,194,275,265)(132,195,276,266)(133,157,277,267)(134,158,278,268)(135,159,279,269)(136,160,280,270)(137,161,281,271)(138,162,282,272)(139,163,283,273)(140,164,284,235)(141,165,285,236)(142,166,286,237)(143,167,287,238)(144,168,288,239)(145,169,289,240)(146,170,290,241)(147,171,291,242)(148,172,292,243)(149,173,293,244)(150,174,294,245)(151,175,295,246)(152,176,296,247)(153,177,297,248)(154,178,298,249)(155,179,299,250)(156,180,300,251)>;
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), (1,246,89,175)(2,247,90,176)(3,248,91,177)(4,249,92,178)(5,250,93,179)(6,251,94,180)(7,252,95,181)(8,253,96,182)(9,254,97,183)(10,255,98,184)(11,256,99,185)(12,257,100,186)(13,258,101,187)(14,259,102,188)(15,260,103,189)(16,261,104,190)(17,262,105,191)(18,263,106,192)(19,264,107,193)(20,265,108,194)(21,266,109,195)(22,267,110,157)(23,268,111,158)(24,269,112,159)(25,270,113,160)(26,271,114,161)(27,272,115,162)(28,273,116,163)(29,235,117,164)(30,236,79,165)(31,237,80,166)(32,238,81,167)(33,239,82,168)(34,240,83,169)(35,241,84,170)(36,242,85,171)(37,243,86,172)(38,244,87,173)(39,245,88,174)(40,292,203,148)(41,293,204,149)(42,294,205,150)(43,295,206,151)(44,296,207,152)(45,297,208,153)(46,298,209,154)(47,299,210,155)(48,300,211,156)(49,301,212,118)(50,302,213,119)(51,303,214,120)(52,304,215,121)(53,305,216,122)(54,306,217,123)(55,307,218,124)(56,308,219,125)(57,309,220,126)(58,310,221,127)(59,311,222,128)(60,312,223,129)(61,274,224,130)(62,275,225,131)(63,276,226,132)(64,277,227,133)(65,278,228,134)(66,279,229,135)(67,280,230,136)(68,281,231,137)(69,282,232,138)(70,283,233,139)(71,284,234,140)(72,285,196,141)(73,286,197,142)(74,287,198,143)(75,288,199,144)(76,289,200,145)(77,290,201,146)(78,291,202,147), (1,43,89,206)(2,44,90,207)(3,45,91,208)(4,46,92,209)(5,47,93,210)(6,48,94,211)(7,49,95,212)(8,50,96,213)(9,51,97,214)(10,52,98,215)(11,53,99,216)(12,54,100,217)(13,55,101,218)(14,56,102,219)(15,57,103,220)(16,58,104,221)(17,59,105,222)(18,60,106,223)(19,61,107,224)(20,62,108,225)(21,63,109,226)(22,64,110,227)(23,65,111,228)(24,66,112,229)(25,67,113,230)(26,68,114,231)(27,69,115,232)(28,70,116,233)(29,71,117,234)(30,72,79,196)(31,73,80,197)(32,74,81,198)(33,75,82,199)(34,76,83,200)(35,77,84,201)(36,78,85,202)(37,40,86,203)(38,41,87,204)(39,42,88,205)(118,181,301,252)(119,182,302,253)(120,183,303,254)(121,184,304,255)(122,185,305,256)(123,186,306,257)(124,187,307,258)(125,188,308,259)(126,189,309,260)(127,190,310,261)(128,191,311,262)(129,192,312,263)(130,193,274,264)(131,194,275,265)(132,195,276,266)(133,157,277,267)(134,158,278,268)(135,159,279,269)(136,160,280,270)(137,161,281,271)(138,162,282,272)(139,163,283,273)(140,164,284,235)(141,165,285,236)(142,166,286,237)(143,167,287,238)(144,168,288,239)(145,169,289,240)(146,170,290,241)(147,171,291,242)(148,172,292,243)(149,173,293,244)(150,174,294,245)(151,175,295,246)(152,176,296,247)(153,177,297,248)(154,178,298,249)(155,179,299,250)(156,180,300,251) );
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)], [(1,246,89,175),(2,247,90,176),(3,248,91,177),(4,249,92,178),(5,250,93,179),(6,251,94,180),(7,252,95,181),(8,253,96,182),(9,254,97,183),(10,255,98,184),(11,256,99,185),(12,257,100,186),(13,258,101,187),(14,259,102,188),(15,260,103,189),(16,261,104,190),(17,262,105,191),(18,263,106,192),(19,264,107,193),(20,265,108,194),(21,266,109,195),(22,267,110,157),(23,268,111,158),(24,269,112,159),(25,270,113,160),(26,271,114,161),(27,272,115,162),(28,273,116,163),(29,235,117,164),(30,236,79,165),(31,237,80,166),(32,238,81,167),(33,239,82,168),(34,240,83,169),(35,241,84,170),(36,242,85,171),(37,243,86,172),(38,244,87,173),(39,245,88,174),(40,292,203,148),(41,293,204,149),(42,294,205,150),(43,295,206,151),(44,296,207,152),(45,297,208,153),(46,298,209,154),(47,299,210,155),(48,300,211,156),(49,301,212,118),(50,302,213,119),(51,303,214,120),(52,304,215,121),(53,305,216,122),(54,306,217,123),(55,307,218,124),(56,308,219,125),(57,309,220,126),(58,310,221,127),(59,311,222,128),(60,312,223,129),(61,274,224,130),(62,275,225,131),(63,276,226,132),(64,277,227,133),(65,278,228,134),(66,279,229,135),(67,280,230,136),(68,281,231,137),(69,282,232,138),(70,283,233,139),(71,284,234,140),(72,285,196,141),(73,286,197,142),(74,287,198,143),(75,288,199,144),(76,289,200,145),(77,290,201,146),(78,291,202,147)], [(1,43,89,206),(2,44,90,207),(3,45,91,208),(4,46,92,209),(5,47,93,210),(6,48,94,211),(7,49,95,212),(8,50,96,213),(9,51,97,214),(10,52,98,215),(11,53,99,216),(12,54,100,217),(13,55,101,218),(14,56,102,219),(15,57,103,220),(16,58,104,221),(17,59,105,222),(18,60,106,223),(19,61,107,224),(20,62,108,225),(21,63,109,226),(22,64,110,227),(23,65,111,228),(24,66,112,229),(25,67,113,230),(26,68,114,231),(27,69,115,232),(28,70,116,233),(29,71,117,234),(30,72,79,196),(31,73,80,197),(32,74,81,198),(33,75,82,199),(34,76,83,200),(35,77,84,201),(36,78,85,202),(37,40,86,203),(38,41,87,204),(39,42,88,205),(118,181,301,252),(119,182,302,253),(120,183,303,254),(121,184,304,255),(122,185,305,256),(123,186,306,257),(124,187,307,258),(125,188,308,259),(126,189,309,260),(127,190,310,261),(128,191,311,262),(129,192,312,263),(130,193,274,264),(131,194,275,265),(132,195,276,266),(133,157,277,267),(134,158,278,268),(135,159,279,269),(136,160,280,270),(137,161,281,271),(138,162,282,272),(139,163,283,273),(140,164,284,235),(141,165,285,236),(142,166,286,237),(143,167,287,238),(144,168,288,239),(145,169,289,240),(146,170,290,241),(147,171,291,242),(148,172,292,243),(149,173,293,244),(150,174,294,245),(151,175,295,246),(152,176,296,247),(153,177,297,248),(154,178,298,249),(155,179,299,250),(156,180,300,251)]])
195 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 12A | ··· | 12F | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39X | 52A | ··· | 52AJ | 78A | ··· | 78X | 156A | ··· | 156BT |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
195 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | - | |||||||||
| image | C1 | C2 | C3 | C6 | C13 | C26 | C39 | C78 | Q8 | C3×Q8 | Q8×C13 | Q8×C39 |
| kernel | Q8×C39 | C156 | Q8×C13 | C52 | C3×Q8 | C12 | Q8 | C4 | C39 | C13 | C3 | C1 |
| # reps | 1 | 3 | 2 | 6 | 12 | 36 | 24 | 72 | 1 | 2 | 12 | 24 |
Matrix representation of Q8×C39 ►in GL3(𝔽157) generated by
| 12 | 0 | 0 |
| 0 | 75 | 0 |
| 0 | 0 | 75 |
| 1 | 0 | 0 |
| 0 | 49 | 155 |
| 0 | 102 | 108 |
| 1 | 0 | 0 |
| 0 | 92 | 149 |
| 0 | 18 | 65 |
G:=sub<GL(3,GF(157))| [12,0,0,0,75,0,0,0,75],[1,0,0,0,49,102,0,155,108],[1,0,0,0,92,18,0,149,65] >;
Q8×C39 in GAP, Magma, Sage, TeX
Q_8\times C_{39} % in TeX
G:=Group("Q8xC39"); // GroupNames label
G:=SmallGroup(312,44);
// by ID
G=gap.SmallGroup(312,44);
# by ID
G:=PCGroup([5,-2,-2,-3,-13,-2,780,1581,786]);
// Polycyclic
G:=Group<a,b,c|a^39=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export