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G = C5×Q64order 320 = 26·5

Direct product of C5 and Q64

direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary

Aliases: C5×Q64, C32.C10, Q32.C10, C160.2C2, C40.70D4, C20.41D8, C10.17D16, C80.21C22, C8.7(C5×D4), C4.3(C5×D8), C2.5(C5×D16), C16.4(C2×C10), (C5×Q32).2C2, SmallGroup(320,178)

Series: Derived Chief Lower central Upper central

C1C16 — C5×Q64
C1C2C4C8C16C80C5×Q32 — C5×Q64
C1C2C4C8C16 — C5×Q64
C1C10C20C40C80 — C5×Q64

Generators and relations for C5×Q64
 G = < a,b,c | a5=b32=1, c2=b16, ab=ba, ac=ca, cbc-1=b-1 >

8C4
8C4
4Q8
4Q8
8C20
8C20
2Q16
2Q16
4C5×Q8
4C5×Q8
2C5×Q16
2C5×Q16

Smallest permutation representation of C5×Q64
Regular action on 320 points
Generators in S320
(1 248 140 120 38)(2 249 141 121 39)(3 250 142 122 40)(4 251 143 123 41)(5 252 144 124 42)(6 253 145 125 43)(7 254 146 126 44)(8 255 147 127 45)(9 256 148 128 46)(10 225 149 97 47)(11 226 150 98 48)(12 227 151 99 49)(13 228 152 100 50)(14 229 153 101 51)(15 230 154 102 52)(16 231 155 103 53)(17 232 156 104 54)(18 233 157 105 55)(19 234 158 106 56)(20 235 159 107 57)(21 236 160 108 58)(22 237 129 109 59)(23 238 130 110 60)(24 239 131 111 61)(25 240 132 112 62)(26 241 133 113 63)(27 242 134 114 64)(28 243 135 115 33)(29 244 136 116 34)(30 245 137 117 35)(31 246 138 118 36)(32 247 139 119 37)(65 284 173 302 218)(66 285 174 303 219)(67 286 175 304 220)(68 287 176 305 221)(69 288 177 306 222)(70 257 178 307 223)(71 258 179 308 224)(72 259 180 309 193)(73 260 181 310 194)(74 261 182 311 195)(75 262 183 312 196)(76 263 184 313 197)(77 264 185 314 198)(78 265 186 315 199)(79 266 187 316 200)(80 267 188 317 201)(81 268 189 318 202)(82 269 190 319 203)(83 270 191 320 204)(84 271 192 289 205)(85 272 161 290 206)(86 273 162 291 207)(87 274 163 292 208)(88 275 164 293 209)(89 276 165 294 210)(90 277 166 295 211)(91 278 167 296 212)(92 279 168 297 213)(93 280 169 298 214)(94 281 170 299 215)(95 282 171 300 216)(96 283 172 301 217)
(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)
(1 277 17 261)(2 276 18 260)(3 275 19 259)(4 274 20 258)(5 273 21 257)(6 272 22 288)(7 271 23 287)(8 270 24 286)(9 269 25 285)(10 268 26 284)(11 267 27 283)(12 266 28 282)(13 265 29 281)(14 264 30 280)(15 263 31 279)(16 262 32 278)(33 95 49 79)(34 94 50 78)(35 93 51 77)(36 92 52 76)(37 91 53 75)(38 90 54 74)(39 89 55 73)(40 88 56 72)(41 87 57 71)(42 86 58 70)(43 85 59 69)(44 84 60 68)(45 83 61 67)(46 82 62 66)(47 81 63 65)(48 80 64 96)(97 202 113 218)(98 201 114 217)(99 200 115 216)(100 199 116 215)(101 198 117 214)(102 197 118 213)(103 196 119 212)(104 195 120 211)(105 194 121 210)(106 193 122 209)(107 224 123 208)(108 223 124 207)(109 222 125 206)(110 221 126 205)(111 220 127 204)(112 219 128 203)(129 306 145 290)(130 305 146 289)(131 304 147 320)(132 303 148 319)(133 302 149 318)(134 301 150 317)(135 300 151 316)(136 299 152 315)(137 298 153 314)(138 297 154 313)(139 296 155 312)(140 295 156 311)(141 294 157 310)(142 293 158 309)(143 292 159 308)(144 291 160 307)(161 237 177 253)(162 236 178 252)(163 235 179 251)(164 234 180 250)(165 233 181 249)(166 232 182 248)(167 231 183 247)(168 230 184 246)(169 229 185 245)(170 228 186 244)(171 227 187 243)(172 226 188 242)(173 225 189 241)(174 256 190 240)(175 255 191 239)(176 254 192 238)

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

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

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

95 conjugacy classes

class 1  2 4A4B4C5A5B5C5D8A8B10A10B10C10D16A16B16C16D20A20B20C20D20E···20L32A···32H40A···40H80A···80P160A···160AF
order1244455558810101010161616162020202020···2032···3240···4080···80160···160
size112161611112211112222222216···162···22···22···22···2

95 irreducible representations

dim11111122222222
type++++++-
imageC1C2C2C5C10C10D4D8D16C5×D4Q64C5×D8C5×D16C5×Q64
kernelC5×Q64C160C5×Q32Q64C32Q32C40C20C10C8C5C4C2C1
# reps1124481244881632

Matrix representation of C5×Q64 in GL2(𝔽31) generated by

20
02
,
1812
244
,
027
80
G:=sub<GL(2,GF(31))| [2,0,0,2],[18,24,12,4],[0,8,27,0] >;

C5×Q64 in GAP, Magma, Sage, TeX

C_5\times Q_{64}
% in TeX

G:=Group("C5xQ64");
// GroupNames label

G:=SmallGroup(320,178);
// by ID

G=gap.SmallGroup(320,178);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,1120,309,1128,1683,850,192,4204,2111,242,10085,5052,124]);
// Polycyclic

G:=Group<a,b,c|a^5=b^32=1,c^2=b^16,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×Q64 in TeX

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