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G = Q8×C39order 312 = 23·3·13

Direct product of C39 and Q8

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

C1C2 — Q8×C39
C1C2C26C78C156 — Q8×C39
C1C2 — Q8×C39
C1C78 — Q8×C39

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 84 287 129)(2 85 288 130)(3 86 289 131)(4 87 290 132)(5 88 291 133)(6 89 292 134)(7 90 293 135)(8 91 294 136)(9 92 295 137)(10 93 296 138)(11 94 297 139)(12 95 298 140)(13 96 299 141)(14 97 300 142)(15 98 301 143)(16 99 302 144)(17 100 303 145)(18 101 304 146)(19 102 305 147)(20 103 306 148)(21 104 307 149)(22 105 308 150)(23 106 309 151)(24 107 310 152)(25 108 311 153)(26 109 312 154)(27 110 274 155)(28 111 275 156)(29 112 276 118)(30 113 277 119)(31 114 278 120)(32 115 279 121)(33 116 280 122)(34 117 281 123)(35 79 282 124)(36 80 283 125)(37 81 284 126)(38 82 285 127)(39 83 286 128)(40 191 237 209)(41 192 238 210)(42 193 239 211)(43 194 240 212)(44 195 241 213)(45 157 242 214)(46 158 243 215)(47 159 244 216)(48 160 245 217)(49 161 246 218)(50 162 247 219)(51 163 248 220)(52 164 249 221)(53 165 250 222)(54 166 251 223)(55 167 252 224)(56 168 253 225)(57 169 254 226)(58 170 255 227)(59 171 256 228)(60 172 257 229)(61 173 258 230)(62 174 259 231)(63 175 260 232)(64 176 261 233)(65 177 262 234)(66 178 263 196)(67 179 264 197)(68 180 265 198)(69 181 266 199)(70 182 267 200)(71 183 268 201)(72 184 269 202)(73 185 270 203)(74 186 271 204)(75 187 272 205)(76 188 273 206)(77 189 235 207)(78 190 236 208)
(1 77 287 235)(2 78 288 236)(3 40 289 237)(4 41 290 238)(5 42 291 239)(6 43 292 240)(7 44 293 241)(8 45 294 242)(9 46 295 243)(10 47 296 244)(11 48 297 245)(12 49 298 246)(13 50 299 247)(14 51 300 248)(15 52 301 249)(16 53 302 250)(17 54 303 251)(18 55 304 252)(19 56 305 253)(20 57 306 254)(21 58 307 255)(22 59 308 256)(23 60 309 257)(24 61 310 258)(25 62 311 259)(26 63 312 260)(27 64 274 261)(28 65 275 262)(29 66 276 263)(30 67 277 264)(31 68 278 265)(32 69 279 266)(33 70 280 267)(34 71 281 268)(35 72 282 269)(36 73 283 270)(37 74 284 271)(38 75 285 272)(39 76 286 273)(79 202 124 184)(80 203 125 185)(81 204 126 186)(82 205 127 187)(83 206 128 188)(84 207 129 189)(85 208 130 190)(86 209 131 191)(87 210 132 192)(88 211 133 193)(89 212 134 194)(90 213 135 195)(91 214 136 157)(92 215 137 158)(93 216 138 159)(94 217 139 160)(95 218 140 161)(96 219 141 162)(97 220 142 163)(98 221 143 164)(99 222 144 165)(100 223 145 166)(101 224 146 167)(102 225 147 168)(103 226 148 169)(104 227 149 170)(105 228 150 171)(106 229 151 172)(107 230 152 173)(108 231 153 174)(109 232 154 175)(110 233 155 176)(111 234 156 177)(112 196 118 178)(113 197 119 179)(114 198 120 180)(115 199 121 181)(116 200 122 182)(117 201 123 183)

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), (1,84,287,129)(2,85,288,130)(3,86,289,131)(4,87,290,132)(5,88,291,133)(6,89,292,134)(7,90,293,135)(8,91,294,136)(9,92,295,137)(10,93,296,138)(11,94,297,139)(12,95,298,140)(13,96,299,141)(14,97,300,142)(15,98,301,143)(16,99,302,144)(17,100,303,145)(18,101,304,146)(19,102,305,147)(20,103,306,148)(21,104,307,149)(22,105,308,150)(23,106,309,151)(24,107,310,152)(25,108,311,153)(26,109,312,154)(27,110,274,155)(28,111,275,156)(29,112,276,118)(30,113,277,119)(31,114,278,120)(32,115,279,121)(33,116,280,122)(34,117,281,123)(35,79,282,124)(36,80,283,125)(37,81,284,126)(38,82,285,127)(39,83,286,128)(40,191,237,209)(41,192,238,210)(42,193,239,211)(43,194,240,212)(44,195,241,213)(45,157,242,214)(46,158,243,215)(47,159,244,216)(48,160,245,217)(49,161,246,218)(50,162,247,219)(51,163,248,220)(52,164,249,221)(53,165,250,222)(54,166,251,223)(55,167,252,224)(56,168,253,225)(57,169,254,226)(58,170,255,227)(59,171,256,228)(60,172,257,229)(61,173,258,230)(62,174,259,231)(63,175,260,232)(64,176,261,233)(65,177,262,234)(66,178,263,196)(67,179,264,197)(68,180,265,198)(69,181,266,199)(70,182,267,200)(71,183,268,201)(72,184,269,202)(73,185,270,203)(74,186,271,204)(75,187,272,205)(76,188,273,206)(77,189,235,207)(78,190,236,208), (1,77,287,235)(2,78,288,236)(3,40,289,237)(4,41,290,238)(5,42,291,239)(6,43,292,240)(7,44,293,241)(8,45,294,242)(9,46,295,243)(10,47,296,244)(11,48,297,245)(12,49,298,246)(13,50,299,247)(14,51,300,248)(15,52,301,249)(16,53,302,250)(17,54,303,251)(18,55,304,252)(19,56,305,253)(20,57,306,254)(21,58,307,255)(22,59,308,256)(23,60,309,257)(24,61,310,258)(25,62,311,259)(26,63,312,260)(27,64,274,261)(28,65,275,262)(29,66,276,263)(30,67,277,264)(31,68,278,265)(32,69,279,266)(33,70,280,267)(34,71,281,268)(35,72,282,269)(36,73,283,270)(37,74,284,271)(38,75,285,272)(39,76,286,273)(79,202,124,184)(80,203,125,185)(81,204,126,186)(82,205,127,187)(83,206,128,188)(84,207,129,189)(85,208,130,190)(86,209,131,191)(87,210,132,192)(88,211,133,193)(89,212,134,194)(90,213,135,195)(91,214,136,157)(92,215,137,158)(93,216,138,159)(94,217,139,160)(95,218,140,161)(96,219,141,162)(97,220,142,163)(98,221,143,164)(99,222,144,165)(100,223,145,166)(101,224,146,167)(102,225,147,168)(103,226,148,169)(104,227,149,170)(105,228,150,171)(106,229,151,172)(107,230,152,173)(108,231,153,174)(109,232,154,175)(110,233,155,176)(111,234,156,177)(112,196,118,178)(113,197,119,179)(114,198,120,180)(115,199,121,181)(116,200,122,182)(117,201,123,183)>;

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), (1,84,287,129)(2,85,288,130)(3,86,289,131)(4,87,290,132)(5,88,291,133)(6,89,292,134)(7,90,293,135)(8,91,294,136)(9,92,295,137)(10,93,296,138)(11,94,297,139)(12,95,298,140)(13,96,299,141)(14,97,300,142)(15,98,301,143)(16,99,302,144)(17,100,303,145)(18,101,304,146)(19,102,305,147)(20,103,306,148)(21,104,307,149)(22,105,308,150)(23,106,309,151)(24,107,310,152)(25,108,311,153)(26,109,312,154)(27,110,274,155)(28,111,275,156)(29,112,276,118)(30,113,277,119)(31,114,278,120)(32,115,279,121)(33,116,280,122)(34,117,281,123)(35,79,282,124)(36,80,283,125)(37,81,284,126)(38,82,285,127)(39,83,286,128)(40,191,237,209)(41,192,238,210)(42,193,239,211)(43,194,240,212)(44,195,241,213)(45,157,242,214)(46,158,243,215)(47,159,244,216)(48,160,245,217)(49,161,246,218)(50,162,247,219)(51,163,248,220)(52,164,249,221)(53,165,250,222)(54,166,251,223)(55,167,252,224)(56,168,253,225)(57,169,254,226)(58,170,255,227)(59,171,256,228)(60,172,257,229)(61,173,258,230)(62,174,259,231)(63,175,260,232)(64,176,261,233)(65,177,262,234)(66,178,263,196)(67,179,264,197)(68,180,265,198)(69,181,266,199)(70,182,267,200)(71,183,268,201)(72,184,269,202)(73,185,270,203)(74,186,271,204)(75,187,272,205)(76,188,273,206)(77,189,235,207)(78,190,236,208), (1,77,287,235)(2,78,288,236)(3,40,289,237)(4,41,290,238)(5,42,291,239)(6,43,292,240)(7,44,293,241)(8,45,294,242)(9,46,295,243)(10,47,296,244)(11,48,297,245)(12,49,298,246)(13,50,299,247)(14,51,300,248)(15,52,301,249)(16,53,302,250)(17,54,303,251)(18,55,304,252)(19,56,305,253)(20,57,306,254)(21,58,307,255)(22,59,308,256)(23,60,309,257)(24,61,310,258)(25,62,311,259)(26,63,312,260)(27,64,274,261)(28,65,275,262)(29,66,276,263)(30,67,277,264)(31,68,278,265)(32,69,279,266)(33,70,280,267)(34,71,281,268)(35,72,282,269)(36,73,283,270)(37,74,284,271)(38,75,285,272)(39,76,286,273)(79,202,124,184)(80,203,125,185)(81,204,126,186)(82,205,127,187)(83,206,128,188)(84,207,129,189)(85,208,130,190)(86,209,131,191)(87,210,132,192)(88,211,133,193)(89,212,134,194)(90,213,135,195)(91,214,136,157)(92,215,137,158)(93,216,138,159)(94,217,139,160)(95,218,140,161)(96,219,141,162)(97,220,142,163)(98,221,143,164)(99,222,144,165)(100,223,145,166)(101,224,146,167)(102,225,147,168)(103,226,148,169)(104,227,149,170)(105,228,150,171)(106,229,151,172)(107,230,152,173)(108,231,153,174)(109,232,154,175)(110,233,155,176)(111,234,156,177)(112,196,118,178)(113,197,119,179)(114,198,120,180)(115,199,121,181)(116,200,122,182)(117,201,123,183) );

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)], [(1,84,287,129),(2,85,288,130),(3,86,289,131),(4,87,290,132),(5,88,291,133),(6,89,292,134),(7,90,293,135),(8,91,294,136),(9,92,295,137),(10,93,296,138),(11,94,297,139),(12,95,298,140),(13,96,299,141),(14,97,300,142),(15,98,301,143),(16,99,302,144),(17,100,303,145),(18,101,304,146),(19,102,305,147),(20,103,306,148),(21,104,307,149),(22,105,308,150),(23,106,309,151),(24,107,310,152),(25,108,311,153),(26,109,312,154),(27,110,274,155),(28,111,275,156),(29,112,276,118),(30,113,277,119),(31,114,278,120),(32,115,279,121),(33,116,280,122),(34,117,281,123),(35,79,282,124),(36,80,283,125),(37,81,284,126),(38,82,285,127),(39,83,286,128),(40,191,237,209),(41,192,238,210),(42,193,239,211),(43,194,240,212),(44,195,241,213),(45,157,242,214),(46,158,243,215),(47,159,244,216),(48,160,245,217),(49,161,246,218),(50,162,247,219),(51,163,248,220),(52,164,249,221),(53,165,250,222),(54,166,251,223),(55,167,252,224),(56,168,253,225),(57,169,254,226),(58,170,255,227),(59,171,256,228),(60,172,257,229),(61,173,258,230),(62,174,259,231),(63,175,260,232),(64,176,261,233),(65,177,262,234),(66,178,263,196),(67,179,264,197),(68,180,265,198),(69,181,266,199),(70,182,267,200),(71,183,268,201),(72,184,269,202),(73,185,270,203),(74,186,271,204),(75,187,272,205),(76,188,273,206),(77,189,235,207),(78,190,236,208)], [(1,77,287,235),(2,78,288,236),(3,40,289,237),(4,41,290,238),(5,42,291,239),(6,43,292,240),(7,44,293,241),(8,45,294,242),(9,46,295,243),(10,47,296,244),(11,48,297,245),(12,49,298,246),(13,50,299,247),(14,51,300,248),(15,52,301,249),(16,53,302,250),(17,54,303,251),(18,55,304,252),(19,56,305,253),(20,57,306,254),(21,58,307,255),(22,59,308,256),(23,60,309,257),(24,61,310,258),(25,62,311,259),(26,63,312,260),(27,64,274,261),(28,65,275,262),(29,66,276,263),(30,67,277,264),(31,68,278,265),(32,69,279,266),(33,70,280,267),(34,71,281,268),(35,72,282,269),(36,73,283,270),(37,74,284,271),(38,75,285,272),(39,76,286,273),(79,202,124,184),(80,203,125,185),(81,204,126,186),(82,205,127,187),(83,206,128,188),(84,207,129,189),(85,208,130,190),(86,209,131,191),(87,210,132,192),(88,211,133,193),(89,212,134,194),(90,213,135,195),(91,214,136,157),(92,215,137,158),(93,216,138,159),(94,217,139,160),(95,218,140,161),(96,219,141,162),(97,220,142,163),(98,221,143,164),(99,222,144,165),(100,223,145,166),(101,224,146,167),(102,225,147,168),(103,226,148,169),(104,227,149,170),(105,228,150,171),(106,229,151,172),(107,230,152,173),(108,231,153,174),(109,232,154,175),(110,233,155,176),(111,234,156,177),(112,196,118,178),(113,197,119,179),(114,198,120,180),(115,199,121,181),(116,200,122,182),(117,201,123,183)])

195 conjugacy classes

class 1  2 3A3B4A4B4C6A6B12A···12F13A···13L26A···26L39A···39X52A···52AJ78A···78X156A···156BT
order12334446612···1213···1326···2639···3952···5278···78156···156
size1111222112···21···11···11···12···21···12···2

195 irreducible representations

dim111111112222
type++-
imageC1C2C3C6C13C26C39C78Q8C3×Q8Q8×C13Q8×C39
kernelQ8×C39C156Q8×C13C52C3×Q8C12Q8C4C39C13C3C1
# reps132612362472121224

Matrix representation of Q8×C39 in GL3(𝔽157) generated by

1200
0750
0075
,
100
049155
0102108
,
100
092149
01865
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

Subgroup lattice of Q8×C39 in TeX

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