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G = Q8×C43order 344 = 23·43

Direct product of C43 and Q8

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C43, C4.C86, C172.3C2, C86.7C22, C2.2(C2×C86), SmallGroup(344,10)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C43
C1C2C86C172 — Q8×C43
C1C2 — Q8×C43
C1C86 — Q8×C43

Generators and relations for Q8×C43
 G = < a,b,c | a43=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C43
Regular action on 344 points
Generators in S344
(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)(302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344)
(1 328 269 214)(2 329 270 215)(3 330 271 173)(4 331 272 174)(5 332 273 175)(6 333 274 176)(7 334 275 177)(8 335 276 178)(9 336 277 179)(10 337 278 180)(11 338 279 181)(12 339 280 182)(13 340 281 183)(14 341 282 184)(15 342 283 185)(16 343 284 186)(17 344 285 187)(18 302 286 188)(19 303 287 189)(20 304 288 190)(21 305 289 191)(22 306 290 192)(23 307 291 193)(24 308 292 194)(25 309 293 195)(26 310 294 196)(27 311 295 197)(28 312 296 198)(29 313 297 199)(30 314 298 200)(31 315 299 201)(32 316 300 202)(33 317 301 203)(34 318 259 204)(35 319 260 205)(36 320 261 206)(37 321 262 207)(38 322 263 208)(39 323 264 209)(40 324 265 210)(41 325 266 211)(42 326 267 212)(43 327 268 213)(44 219 109 144)(45 220 110 145)(46 221 111 146)(47 222 112 147)(48 223 113 148)(49 224 114 149)(50 225 115 150)(51 226 116 151)(52 227 117 152)(53 228 118 153)(54 229 119 154)(55 230 120 155)(56 231 121 156)(57 232 122 157)(58 233 123 158)(59 234 124 159)(60 235 125 160)(61 236 126 161)(62 237 127 162)(63 238 128 163)(64 239 129 164)(65 240 87 165)(66 241 88 166)(67 242 89 167)(68 243 90 168)(69 244 91 169)(70 245 92 170)(71 246 93 171)(72 247 94 172)(73 248 95 130)(74 249 96 131)(75 250 97 132)(76 251 98 133)(77 252 99 134)(78 253 100 135)(79 254 101 136)(80 255 102 137)(81 256 103 138)(82 257 104 139)(83 258 105 140)(84 216 106 141)(85 217 107 142)(86 218 108 143)
(1 221 269 146)(2 222 270 147)(3 223 271 148)(4 224 272 149)(5 225 273 150)(6 226 274 151)(7 227 275 152)(8 228 276 153)(9 229 277 154)(10 230 278 155)(11 231 279 156)(12 232 280 157)(13 233 281 158)(14 234 282 159)(15 235 283 160)(16 236 284 161)(17 237 285 162)(18 238 286 163)(19 239 287 164)(20 240 288 165)(21 241 289 166)(22 242 290 167)(23 243 291 168)(24 244 292 169)(25 245 293 170)(26 246 294 171)(27 247 295 172)(28 248 296 130)(29 249 297 131)(30 250 298 132)(31 251 299 133)(32 252 300 134)(33 253 301 135)(34 254 259 136)(35 255 260 137)(36 256 261 138)(37 257 262 139)(38 258 263 140)(39 216 264 141)(40 217 265 142)(41 218 266 143)(42 219 267 144)(43 220 268 145)(44 212 109 326)(45 213 110 327)(46 214 111 328)(47 215 112 329)(48 173 113 330)(49 174 114 331)(50 175 115 332)(51 176 116 333)(52 177 117 334)(53 178 118 335)(54 179 119 336)(55 180 120 337)(56 181 121 338)(57 182 122 339)(58 183 123 340)(59 184 124 341)(60 185 125 342)(61 186 126 343)(62 187 127 344)(63 188 128 302)(64 189 129 303)(65 190 87 304)(66 191 88 305)(67 192 89 306)(68 193 90 307)(69 194 91 308)(70 195 92 309)(71 196 93 310)(72 197 94 311)(73 198 95 312)(74 199 96 313)(75 200 97 314)(76 201 98 315)(77 202 99 316)(78 203 100 317)(79 204 101 318)(80 205 102 319)(81 206 103 320)(82 207 104 321)(83 208 105 322)(84 209 106 323)(85 210 107 324)(86 211 108 325)

G:=sub<Sym(344)| (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)(302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344), (1,328,269,214)(2,329,270,215)(3,330,271,173)(4,331,272,174)(5,332,273,175)(6,333,274,176)(7,334,275,177)(8,335,276,178)(9,336,277,179)(10,337,278,180)(11,338,279,181)(12,339,280,182)(13,340,281,183)(14,341,282,184)(15,342,283,185)(16,343,284,186)(17,344,285,187)(18,302,286,188)(19,303,287,189)(20,304,288,190)(21,305,289,191)(22,306,290,192)(23,307,291,193)(24,308,292,194)(25,309,293,195)(26,310,294,196)(27,311,295,197)(28,312,296,198)(29,313,297,199)(30,314,298,200)(31,315,299,201)(32,316,300,202)(33,317,301,203)(34,318,259,204)(35,319,260,205)(36,320,261,206)(37,321,262,207)(38,322,263,208)(39,323,264,209)(40,324,265,210)(41,325,266,211)(42,326,267,212)(43,327,268,213)(44,219,109,144)(45,220,110,145)(46,221,111,146)(47,222,112,147)(48,223,113,148)(49,224,114,149)(50,225,115,150)(51,226,116,151)(52,227,117,152)(53,228,118,153)(54,229,119,154)(55,230,120,155)(56,231,121,156)(57,232,122,157)(58,233,123,158)(59,234,124,159)(60,235,125,160)(61,236,126,161)(62,237,127,162)(63,238,128,163)(64,239,129,164)(65,240,87,165)(66,241,88,166)(67,242,89,167)(68,243,90,168)(69,244,91,169)(70,245,92,170)(71,246,93,171)(72,247,94,172)(73,248,95,130)(74,249,96,131)(75,250,97,132)(76,251,98,133)(77,252,99,134)(78,253,100,135)(79,254,101,136)(80,255,102,137)(81,256,103,138)(82,257,104,139)(83,258,105,140)(84,216,106,141)(85,217,107,142)(86,218,108,143), (1,221,269,146)(2,222,270,147)(3,223,271,148)(4,224,272,149)(5,225,273,150)(6,226,274,151)(7,227,275,152)(8,228,276,153)(9,229,277,154)(10,230,278,155)(11,231,279,156)(12,232,280,157)(13,233,281,158)(14,234,282,159)(15,235,283,160)(16,236,284,161)(17,237,285,162)(18,238,286,163)(19,239,287,164)(20,240,288,165)(21,241,289,166)(22,242,290,167)(23,243,291,168)(24,244,292,169)(25,245,293,170)(26,246,294,171)(27,247,295,172)(28,248,296,130)(29,249,297,131)(30,250,298,132)(31,251,299,133)(32,252,300,134)(33,253,301,135)(34,254,259,136)(35,255,260,137)(36,256,261,138)(37,257,262,139)(38,258,263,140)(39,216,264,141)(40,217,265,142)(41,218,266,143)(42,219,267,144)(43,220,268,145)(44,212,109,326)(45,213,110,327)(46,214,111,328)(47,215,112,329)(48,173,113,330)(49,174,114,331)(50,175,115,332)(51,176,116,333)(52,177,117,334)(53,178,118,335)(54,179,119,336)(55,180,120,337)(56,181,121,338)(57,182,122,339)(58,183,123,340)(59,184,124,341)(60,185,125,342)(61,186,126,343)(62,187,127,344)(63,188,128,302)(64,189,129,303)(65,190,87,304)(66,191,88,305)(67,192,89,306)(68,193,90,307)(69,194,91,308)(70,195,92,309)(71,196,93,310)(72,197,94,311)(73,198,95,312)(74,199,96,313)(75,200,97,314)(76,201,98,315)(77,202,99,316)(78,203,100,317)(79,204,101,318)(80,205,102,319)(81,206,103,320)(82,207,104,321)(83,208,105,322)(84,209,106,323)(85,210,107,324)(86,211,108,325)>;

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)(302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344), (1,328,269,214)(2,329,270,215)(3,330,271,173)(4,331,272,174)(5,332,273,175)(6,333,274,176)(7,334,275,177)(8,335,276,178)(9,336,277,179)(10,337,278,180)(11,338,279,181)(12,339,280,182)(13,340,281,183)(14,341,282,184)(15,342,283,185)(16,343,284,186)(17,344,285,187)(18,302,286,188)(19,303,287,189)(20,304,288,190)(21,305,289,191)(22,306,290,192)(23,307,291,193)(24,308,292,194)(25,309,293,195)(26,310,294,196)(27,311,295,197)(28,312,296,198)(29,313,297,199)(30,314,298,200)(31,315,299,201)(32,316,300,202)(33,317,301,203)(34,318,259,204)(35,319,260,205)(36,320,261,206)(37,321,262,207)(38,322,263,208)(39,323,264,209)(40,324,265,210)(41,325,266,211)(42,326,267,212)(43,327,268,213)(44,219,109,144)(45,220,110,145)(46,221,111,146)(47,222,112,147)(48,223,113,148)(49,224,114,149)(50,225,115,150)(51,226,116,151)(52,227,117,152)(53,228,118,153)(54,229,119,154)(55,230,120,155)(56,231,121,156)(57,232,122,157)(58,233,123,158)(59,234,124,159)(60,235,125,160)(61,236,126,161)(62,237,127,162)(63,238,128,163)(64,239,129,164)(65,240,87,165)(66,241,88,166)(67,242,89,167)(68,243,90,168)(69,244,91,169)(70,245,92,170)(71,246,93,171)(72,247,94,172)(73,248,95,130)(74,249,96,131)(75,250,97,132)(76,251,98,133)(77,252,99,134)(78,253,100,135)(79,254,101,136)(80,255,102,137)(81,256,103,138)(82,257,104,139)(83,258,105,140)(84,216,106,141)(85,217,107,142)(86,218,108,143), (1,221,269,146)(2,222,270,147)(3,223,271,148)(4,224,272,149)(5,225,273,150)(6,226,274,151)(7,227,275,152)(8,228,276,153)(9,229,277,154)(10,230,278,155)(11,231,279,156)(12,232,280,157)(13,233,281,158)(14,234,282,159)(15,235,283,160)(16,236,284,161)(17,237,285,162)(18,238,286,163)(19,239,287,164)(20,240,288,165)(21,241,289,166)(22,242,290,167)(23,243,291,168)(24,244,292,169)(25,245,293,170)(26,246,294,171)(27,247,295,172)(28,248,296,130)(29,249,297,131)(30,250,298,132)(31,251,299,133)(32,252,300,134)(33,253,301,135)(34,254,259,136)(35,255,260,137)(36,256,261,138)(37,257,262,139)(38,258,263,140)(39,216,264,141)(40,217,265,142)(41,218,266,143)(42,219,267,144)(43,220,268,145)(44,212,109,326)(45,213,110,327)(46,214,111,328)(47,215,112,329)(48,173,113,330)(49,174,114,331)(50,175,115,332)(51,176,116,333)(52,177,117,334)(53,178,118,335)(54,179,119,336)(55,180,120,337)(56,181,121,338)(57,182,122,339)(58,183,123,340)(59,184,124,341)(60,185,125,342)(61,186,126,343)(62,187,127,344)(63,188,128,302)(64,189,129,303)(65,190,87,304)(66,191,88,305)(67,192,89,306)(68,193,90,307)(69,194,91,308)(70,195,92,309)(71,196,93,310)(72,197,94,311)(73,198,95,312)(74,199,96,313)(75,200,97,314)(76,201,98,315)(77,202,99,316)(78,203,100,317)(79,204,101,318)(80,205,102,319)(81,206,103,320)(82,207,104,321)(83,208,105,322)(84,209,106,323)(85,210,107,324)(86,211,108,325) );

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),(302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344)], [(1,328,269,214),(2,329,270,215),(3,330,271,173),(4,331,272,174),(5,332,273,175),(6,333,274,176),(7,334,275,177),(8,335,276,178),(9,336,277,179),(10,337,278,180),(11,338,279,181),(12,339,280,182),(13,340,281,183),(14,341,282,184),(15,342,283,185),(16,343,284,186),(17,344,285,187),(18,302,286,188),(19,303,287,189),(20,304,288,190),(21,305,289,191),(22,306,290,192),(23,307,291,193),(24,308,292,194),(25,309,293,195),(26,310,294,196),(27,311,295,197),(28,312,296,198),(29,313,297,199),(30,314,298,200),(31,315,299,201),(32,316,300,202),(33,317,301,203),(34,318,259,204),(35,319,260,205),(36,320,261,206),(37,321,262,207),(38,322,263,208),(39,323,264,209),(40,324,265,210),(41,325,266,211),(42,326,267,212),(43,327,268,213),(44,219,109,144),(45,220,110,145),(46,221,111,146),(47,222,112,147),(48,223,113,148),(49,224,114,149),(50,225,115,150),(51,226,116,151),(52,227,117,152),(53,228,118,153),(54,229,119,154),(55,230,120,155),(56,231,121,156),(57,232,122,157),(58,233,123,158),(59,234,124,159),(60,235,125,160),(61,236,126,161),(62,237,127,162),(63,238,128,163),(64,239,129,164),(65,240,87,165),(66,241,88,166),(67,242,89,167),(68,243,90,168),(69,244,91,169),(70,245,92,170),(71,246,93,171),(72,247,94,172),(73,248,95,130),(74,249,96,131),(75,250,97,132),(76,251,98,133),(77,252,99,134),(78,253,100,135),(79,254,101,136),(80,255,102,137),(81,256,103,138),(82,257,104,139),(83,258,105,140),(84,216,106,141),(85,217,107,142),(86,218,108,143)], [(1,221,269,146),(2,222,270,147),(3,223,271,148),(4,224,272,149),(5,225,273,150),(6,226,274,151),(7,227,275,152),(8,228,276,153),(9,229,277,154),(10,230,278,155),(11,231,279,156),(12,232,280,157),(13,233,281,158),(14,234,282,159),(15,235,283,160),(16,236,284,161),(17,237,285,162),(18,238,286,163),(19,239,287,164),(20,240,288,165),(21,241,289,166),(22,242,290,167),(23,243,291,168),(24,244,292,169),(25,245,293,170),(26,246,294,171),(27,247,295,172),(28,248,296,130),(29,249,297,131),(30,250,298,132),(31,251,299,133),(32,252,300,134),(33,253,301,135),(34,254,259,136),(35,255,260,137),(36,256,261,138),(37,257,262,139),(38,258,263,140),(39,216,264,141),(40,217,265,142),(41,218,266,143),(42,219,267,144),(43,220,268,145),(44,212,109,326),(45,213,110,327),(46,214,111,328),(47,215,112,329),(48,173,113,330),(49,174,114,331),(50,175,115,332),(51,176,116,333),(52,177,117,334),(53,178,118,335),(54,179,119,336),(55,180,120,337),(56,181,121,338),(57,182,122,339),(58,183,123,340),(59,184,124,341),(60,185,125,342),(61,186,126,343),(62,187,127,344),(63,188,128,302),(64,189,129,303),(65,190,87,304),(66,191,88,305),(67,192,89,306),(68,193,90,307),(69,194,91,308),(70,195,92,309),(71,196,93,310),(72,197,94,311),(73,198,95,312),(74,199,96,313),(75,200,97,314),(76,201,98,315),(77,202,99,316),(78,203,100,317),(79,204,101,318),(80,205,102,319),(81,206,103,320),(82,207,104,321),(83,208,105,322),(84,209,106,323),(85,210,107,324),(86,211,108,325)])

215 conjugacy classes

class 1  2 4A4B4C43A···43AP86A···86AP172A···172DV
order1244443···4386···86172···172
size112221···11···12···2

215 irreducible representations

dim111122
type++-
imageC1C2C43C86Q8Q8×C43
kernelQ8×C43C172Q8C4C43C1
# reps1342126142

Matrix representation of Q8×C43 in GL2(𝔽173) generated by

960
096
,
1171
1172
,
11651
5557
G:=sub<GL(2,GF(173))| [96,0,0,96],[1,1,171,172],[116,55,51,57] >;

Q8×C43 in GAP, Magma, Sage, TeX

Q_8\times C_{43}
% in TeX

G:=Group("Q8xC43");
// GroupNames label

G:=SmallGroup(344,10);
// by ID

G=gap.SmallGroup(344,10);
# by ID

G:=PCGroup([4,-2,-2,-43,-2,688,1393,693]);
// Polycyclic

G:=Group<a,b,c|a^43=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×C43 in TeX

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