direct product, metacyclic, nilpotent (class 4), monomial
Aliases: C32×Q32, C48.4C6, C8.4C62, C16.(C3×C6), Q16.(C3×C6), (C3×C48).3C2, C6.23(C3×D8), (C3×C6).46D8, C24.29(C2×C6), C12.47(C3×D4), (C3×Q16).4C6, C4.3(D4×C32), C2.5(C32×D8), (C3×C12).144D4, (C3×C24).62C22, (C32×Q16).3C2, SmallGroup(288,331)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×Q32
G = < a,b,c,d | a3=b3=c16=1, d2=c8, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 120 in 72 conjugacy classes, 48 normal (12 characteristic)
C1, C2, C3, C4, C4, C6, C8, Q8, C32, C12, C12, C16, Q16, C3×C6, C24, C3×Q8, Q32, C3×C12, C3×C12, C48, C3×Q16, C3×C24, Q8×C32, C3×Q32, C3×C48, C32×Q16, C32×Q32
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, D8, C3×C6, C3×D4, Q32, C62, C3×D8, D4×C32, C3×Q32, C32×D8, C32×Q32
►Regular action on 288 points
Generators in S288
(1 141 92)(2 142 93)(3 143 94)(4 144 95)(5 129 96)(6 130 81)(7 131 82)(8 132 83)(9 133 84)(10 134 85)(11 135 86)(12 136 87)(13 137 88)(14 138 89)(15 139 90)(16 140 91)(17 37 110)(18 38 111)(19 39 112)(20 40 97)(21 41 98)(22 42 99)(23 43 100)(24 44 101)(25 45 102)(26 46 103)(27 47 104)(28 48 105)(29 33 106)(30 34 107)(31 35 108)(32 36 109)(49 174 68)(50 175 69)(51 176 70)(52 161 71)(53 162 72)(54 163 73)(55 164 74)(56 165 75)(57 166 76)(58 167 77)(59 168 78)(60 169 79)(61 170 80)(62 171 65)(63 172 66)(64 173 67)(113 192 274)(114 177 275)(115 178 276)(116 179 277)(117 180 278)(118 181 279)(119 182 280)(120 183 281)(121 184 282)(122 185 283)(123 186 284)(124 187 285)(125 188 286)(126 189 287)(127 190 288)(128 191 273)(145 236 202)(146 237 203)(147 238 204)(148 239 205)(149 240 206)(150 225 207)(151 226 208)(152 227 193)(153 228 194)(154 229 195)(155 230 196)(156 231 197)(157 232 198)(158 233 199)(159 234 200)(160 235 201)(209 247 262)(210 248 263)(211 249 264)(212 250 265)(213 251 266)(214 252 267)(215 253 268)(216 254 269)(217 255 270)(218 256 271)(219 241 272)(220 242 257)(221 243 258)(222 244 259)(223 245 260)(224 246 261)
(1 114 109)(2 115 110)(3 116 111)(4 117 112)(5 118 97)(6 119 98)(7 120 99)(8 121 100)(9 122 101)(10 123 102)(11 124 103)(12 125 104)(13 126 105)(14 127 106)(15 128 107)(16 113 108)(17 142 178)(18 143 179)(19 144 180)(20 129 181)(21 130 182)(22 131 183)(23 132 184)(24 133 185)(25 134 186)(26 135 187)(27 136 188)(28 137 189)(29 138 190)(30 139 191)(31 140 192)(32 141 177)(33 89 288)(34 90 273)(35 91 274)(36 92 275)(37 93 276)(38 94 277)(39 95 278)(40 96 279)(41 81 280)(42 82 281)(43 83 282)(44 84 283)(45 85 284)(46 86 285)(47 87 286)(48 88 287)(49 247 148)(50 248 149)(51 249 150)(52 250 151)(53 251 152)(54 252 153)(55 253 154)(56 254 155)(57 255 156)(58 256 157)(59 241 158)(60 242 159)(61 243 160)(62 244 145)(63 245 146)(64 246 147)(65 222 202)(66 223 203)(67 224 204)(68 209 205)(69 210 206)(70 211 207)(71 212 208)(72 213 193)(73 214 194)(74 215 195)(75 216 196)(76 217 197)(77 218 198)(78 219 199)(79 220 200)(80 221 201)(161 265 226)(162 266 227)(163 267 228)(164 268 229)(165 269 230)(166 270 231)(167 271 232)(168 272 233)(169 257 234)(170 258 235)(171 259 236)(172 260 237)(173 261 238)(174 262 239)(175 263 240)(176 264 225)
(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)
(1 55 9 63)(2 54 10 62)(3 53 11 61)(4 52 12 60)(5 51 13 59)(6 50 14 58)(7 49 15 57)(8 64 16 56)(17 228 25 236)(18 227 26 235)(19 226 27 234)(20 225 28 233)(21 240 29 232)(22 239 30 231)(23 238 31 230)(24 237 32 229)(33 198 41 206)(34 197 42 205)(35 196 43 204)(36 195 44 203)(37 194 45 202)(38 193 46 201)(39 208 47 200)(40 207 48 199)(65 93 73 85)(66 92 74 84)(67 91 75 83)(68 90 76 82)(69 89 77 81)(70 88 78 96)(71 87 79 95)(72 86 80 94)(97 150 105 158)(98 149 106 157)(99 148 107 156)(100 147 108 155)(101 146 109 154)(102 145 110 153)(103 160 111 152)(104 159 112 151)(113 254 121 246)(114 253 122 245)(115 252 123 244)(116 251 124 243)(117 250 125 242)(118 249 126 241)(119 248 127 256)(120 247 128 255)(129 176 137 168)(130 175 138 167)(131 174 139 166)(132 173 140 165)(133 172 141 164)(134 171 142 163)(135 170 143 162)(136 169 144 161)(177 268 185 260)(178 267 186 259)(179 266 187 258)(180 265 188 257)(181 264 189 272)(182 263 190 271)(183 262 191 270)(184 261 192 269)(209 273 217 281)(210 288 218 280)(211 287 219 279)(212 286 220 278)(213 285 221 277)(214 284 222 276)(215 283 223 275)(216 282 224 274)
G:=sub<Sym(288)| (1,141,92)(2,142,93)(3,143,94)(4,144,95)(5,129,96)(6,130,81)(7,131,82)(8,132,83)(9,133,84)(10,134,85)(11,135,86)(12,136,87)(13,137,88)(14,138,89)(15,139,90)(16,140,91)(17,37,110)(18,38,111)(19,39,112)(20,40,97)(21,41,98)(22,42,99)(23,43,100)(24,44,101)(25,45,102)(26,46,103)(27,47,104)(28,48,105)(29,33,106)(30,34,107)(31,35,108)(32,36,109)(49,174,68)(50,175,69)(51,176,70)(52,161,71)(53,162,72)(54,163,73)(55,164,74)(56,165,75)(57,166,76)(58,167,77)(59,168,78)(60,169,79)(61,170,80)(62,171,65)(63,172,66)(64,173,67)(113,192,274)(114,177,275)(115,178,276)(116,179,277)(117,180,278)(118,181,279)(119,182,280)(120,183,281)(121,184,282)(122,185,283)(123,186,284)(124,187,285)(125,188,286)(126,189,287)(127,190,288)(128,191,273)(145,236,202)(146,237,203)(147,238,204)(148,239,205)(149,240,206)(150,225,207)(151,226,208)(152,227,193)(153,228,194)(154,229,195)(155,230,196)(156,231,197)(157,232,198)(158,233,199)(159,234,200)(160,235,201)(209,247,262)(210,248,263)(211,249,264)(212,250,265)(213,251,266)(214,252,267)(215,253,268)(216,254,269)(217,255,270)(218,256,271)(219,241,272)(220,242,257)(221,243,258)(222,244,259)(223,245,260)(224,246,261), (1,114,109)(2,115,110)(3,116,111)(4,117,112)(5,118,97)(6,119,98)(7,120,99)(8,121,100)(9,122,101)(10,123,102)(11,124,103)(12,125,104)(13,126,105)(14,127,106)(15,128,107)(16,113,108)(17,142,178)(18,143,179)(19,144,180)(20,129,181)(21,130,182)(22,131,183)(23,132,184)(24,133,185)(25,134,186)(26,135,187)(27,136,188)(28,137,189)(29,138,190)(30,139,191)(31,140,192)(32,141,177)(33,89,288)(34,90,273)(35,91,274)(36,92,275)(37,93,276)(38,94,277)(39,95,278)(40,96,279)(41,81,280)(42,82,281)(43,83,282)(44,84,283)(45,85,284)(46,86,285)(47,87,286)(48,88,287)(49,247,148)(50,248,149)(51,249,150)(52,250,151)(53,251,152)(54,252,153)(55,253,154)(56,254,155)(57,255,156)(58,256,157)(59,241,158)(60,242,159)(61,243,160)(62,244,145)(63,245,146)(64,246,147)(65,222,202)(66,223,203)(67,224,204)(68,209,205)(69,210,206)(70,211,207)(71,212,208)(72,213,193)(73,214,194)(74,215,195)(75,216,196)(76,217,197)(77,218,198)(78,219,199)(79,220,200)(80,221,201)(161,265,226)(162,266,227)(163,267,228)(164,268,229)(165,269,230)(166,270,231)(167,271,232)(168,272,233)(169,257,234)(170,258,235)(171,259,236)(172,260,237)(173,261,238)(174,262,239)(175,263,240)(176,264,225), (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), (1,55,9,63)(2,54,10,62)(3,53,11,61)(4,52,12,60)(5,51,13,59)(6,50,14,58)(7,49,15,57)(8,64,16,56)(17,228,25,236)(18,227,26,235)(19,226,27,234)(20,225,28,233)(21,240,29,232)(22,239,30,231)(23,238,31,230)(24,237,32,229)(33,198,41,206)(34,197,42,205)(35,196,43,204)(36,195,44,203)(37,194,45,202)(38,193,46,201)(39,208,47,200)(40,207,48,199)(65,93,73,85)(66,92,74,84)(67,91,75,83)(68,90,76,82)(69,89,77,81)(70,88,78,96)(71,87,79,95)(72,86,80,94)(97,150,105,158)(98,149,106,157)(99,148,107,156)(100,147,108,155)(101,146,109,154)(102,145,110,153)(103,160,111,152)(104,159,112,151)(113,254,121,246)(114,253,122,245)(115,252,123,244)(116,251,124,243)(117,250,125,242)(118,249,126,241)(119,248,127,256)(120,247,128,255)(129,176,137,168)(130,175,138,167)(131,174,139,166)(132,173,140,165)(133,172,141,164)(134,171,142,163)(135,170,143,162)(136,169,144,161)(177,268,185,260)(178,267,186,259)(179,266,187,258)(180,265,188,257)(181,264,189,272)(182,263,190,271)(183,262,191,270)(184,261,192,269)(209,273,217,281)(210,288,218,280)(211,287,219,279)(212,286,220,278)(213,285,221,277)(214,284,222,276)(215,283,223,275)(216,282,224,274)>;
G:=Group( (1,141,92)(2,142,93)(3,143,94)(4,144,95)(5,129,96)(6,130,81)(7,131,82)(8,132,83)(9,133,84)(10,134,85)(11,135,86)(12,136,87)(13,137,88)(14,138,89)(15,139,90)(16,140,91)(17,37,110)(18,38,111)(19,39,112)(20,40,97)(21,41,98)(22,42,99)(23,43,100)(24,44,101)(25,45,102)(26,46,103)(27,47,104)(28,48,105)(29,33,106)(30,34,107)(31,35,108)(32,36,109)(49,174,68)(50,175,69)(51,176,70)(52,161,71)(53,162,72)(54,163,73)(55,164,74)(56,165,75)(57,166,76)(58,167,77)(59,168,78)(60,169,79)(61,170,80)(62,171,65)(63,172,66)(64,173,67)(113,192,274)(114,177,275)(115,178,276)(116,179,277)(117,180,278)(118,181,279)(119,182,280)(120,183,281)(121,184,282)(122,185,283)(123,186,284)(124,187,285)(125,188,286)(126,189,287)(127,190,288)(128,191,273)(145,236,202)(146,237,203)(147,238,204)(148,239,205)(149,240,206)(150,225,207)(151,226,208)(152,227,193)(153,228,194)(154,229,195)(155,230,196)(156,231,197)(157,232,198)(158,233,199)(159,234,200)(160,235,201)(209,247,262)(210,248,263)(211,249,264)(212,250,265)(213,251,266)(214,252,267)(215,253,268)(216,254,269)(217,255,270)(218,256,271)(219,241,272)(220,242,257)(221,243,258)(222,244,259)(223,245,260)(224,246,261), (1,114,109)(2,115,110)(3,116,111)(4,117,112)(5,118,97)(6,119,98)(7,120,99)(8,121,100)(9,122,101)(10,123,102)(11,124,103)(12,125,104)(13,126,105)(14,127,106)(15,128,107)(16,113,108)(17,142,178)(18,143,179)(19,144,180)(20,129,181)(21,130,182)(22,131,183)(23,132,184)(24,133,185)(25,134,186)(26,135,187)(27,136,188)(28,137,189)(29,138,190)(30,139,191)(31,140,192)(32,141,177)(33,89,288)(34,90,273)(35,91,274)(36,92,275)(37,93,276)(38,94,277)(39,95,278)(40,96,279)(41,81,280)(42,82,281)(43,83,282)(44,84,283)(45,85,284)(46,86,285)(47,87,286)(48,88,287)(49,247,148)(50,248,149)(51,249,150)(52,250,151)(53,251,152)(54,252,153)(55,253,154)(56,254,155)(57,255,156)(58,256,157)(59,241,158)(60,242,159)(61,243,160)(62,244,145)(63,245,146)(64,246,147)(65,222,202)(66,223,203)(67,224,204)(68,209,205)(69,210,206)(70,211,207)(71,212,208)(72,213,193)(73,214,194)(74,215,195)(75,216,196)(76,217,197)(77,218,198)(78,219,199)(79,220,200)(80,221,201)(161,265,226)(162,266,227)(163,267,228)(164,268,229)(165,269,230)(166,270,231)(167,271,232)(168,272,233)(169,257,234)(170,258,235)(171,259,236)(172,260,237)(173,261,238)(174,262,239)(175,263,240)(176,264,225), (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), (1,55,9,63)(2,54,10,62)(3,53,11,61)(4,52,12,60)(5,51,13,59)(6,50,14,58)(7,49,15,57)(8,64,16,56)(17,228,25,236)(18,227,26,235)(19,226,27,234)(20,225,28,233)(21,240,29,232)(22,239,30,231)(23,238,31,230)(24,237,32,229)(33,198,41,206)(34,197,42,205)(35,196,43,204)(36,195,44,203)(37,194,45,202)(38,193,46,201)(39,208,47,200)(40,207,48,199)(65,93,73,85)(66,92,74,84)(67,91,75,83)(68,90,76,82)(69,89,77,81)(70,88,78,96)(71,87,79,95)(72,86,80,94)(97,150,105,158)(98,149,106,157)(99,148,107,156)(100,147,108,155)(101,146,109,154)(102,145,110,153)(103,160,111,152)(104,159,112,151)(113,254,121,246)(114,253,122,245)(115,252,123,244)(116,251,124,243)(117,250,125,242)(118,249,126,241)(119,248,127,256)(120,247,128,255)(129,176,137,168)(130,175,138,167)(131,174,139,166)(132,173,140,165)(133,172,141,164)(134,171,142,163)(135,170,143,162)(136,169,144,161)(177,268,185,260)(178,267,186,259)(179,266,187,258)(180,265,188,257)(181,264,189,272)(182,263,190,271)(183,262,191,270)(184,261,192,269)(209,273,217,281)(210,288,218,280)(211,287,219,279)(212,286,220,278)(213,285,221,277)(214,284,222,276)(215,283,223,275)(216,282,224,274) );
G=PermutationGroup([[(1,141,92),(2,142,93),(3,143,94),(4,144,95),(5,129,96),(6,130,81),(7,131,82),(8,132,83),(9,133,84),(10,134,85),(11,135,86),(12,136,87),(13,137,88),(14,138,89),(15,139,90),(16,140,91),(17,37,110),(18,38,111),(19,39,112),(20,40,97),(21,41,98),(22,42,99),(23,43,100),(24,44,101),(25,45,102),(26,46,103),(27,47,104),(28,48,105),(29,33,106),(30,34,107),(31,35,108),(32,36,109),(49,174,68),(50,175,69),(51,176,70),(52,161,71),(53,162,72),(54,163,73),(55,164,74),(56,165,75),(57,166,76),(58,167,77),(59,168,78),(60,169,79),(61,170,80),(62,171,65),(63,172,66),(64,173,67),(113,192,274),(114,177,275),(115,178,276),(116,179,277),(117,180,278),(118,181,279),(119,182,280),(120,183,281),(121,184,282),(122,185,283),(123,186,284),(124,187,285),(125,188,286),(126,189,287),(127,190,288),(128,191,273),(145,236,202),(146,237,203),(147,238,204),(148,239,205),(149,240,206),(150,225,207),(151,226,208),(152,227,193),(153,228,194),(154,229,195),(155,230,196),(156,231,197),(157,232,198),(158,233,199),(159,234,200),(160,235,201),(209,247,262),(210,248,263),(211,249,264),(212,250,265),(213,251,266),(214,252,267),(215,253,268),(216,254,269),(217,255,270),(218,256,271),(219,241,272),(220,242,257),(221,243,258),(222,244,259),(223,245,260),(224,246,261)], [(1,114,109),(2,115,110),(3,116,111),(4,117,112),(5,118,97),(6,119,98),(7,120,99),(8,121,100),(9,122,101),(10,123,102),(11,124,103),(12,125,104),(13,126,105),(14,127,106),(15,128,107),(16,113,108),(17,142,178),(18,143,179),(19,144,180),(20,129,181),(21,130,182),(22,131,183),(23,132,184),(24,133,185),(25,134,186),(26,135,187),(27,136,188),(28,137,189),(29,138,190),(30,139,191),(31,140,192),(32,141,177),(33,89,288),(34,90,273),(35,91,274),(36,92,275),(37,93,276),(38,94,277),(39,95,278),(40,96,279),(41,81,280),(42,82,281),(43,83,282),(44,84,283),(45,85,284),(46,86,285),(47,87,286),(48,88,287),(49,247,148),(50,248,149),(51,249,150),(52,250,151),(53,251,152),(54,252,153),(55,253,154),(56,254,155),(57,255,156),(58,256,157),(59,241,158),(60,242,159),(61,243,160),(62,244,145),(63,245,146),(64,246,147),(65,222,202),(66,223,203),(67,224,204),(68,209,205),(69,210,206),(70,211,207),(71,212,208),(72,213,193),(73,214,194),(74,215,195),(75,216,196),(76,217,197),(77,218,198),(78,219,199),(79,220,200),(80,221,201),(161,265,226),(162,266,227),(163,267,228),(164,268,229),(165,269,230),(166,270,231),(167,271,232),(168,272,233),(169,257,234),(170,258,235),(171,259,236),(172,260,237),(173,261,238),(174,262,239),(175,263,240),(176,264,225)], [(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)], [(1,55,9,63),(2,54,10,62),(3,53,11,61),(4,52,12,60),(5,51,13,59),(6,50,14,58),(7,49,15,57),(8,64,16,56),(17,228,25,236),(18,227,26,235),(19,226,27,234),(20,225,28,233),(21,240,29,232),(22,239,30,231),(23,238,31,230),(24,237,32,229),(33,198,41,206),(34,197,42,205),(35,196,43,204),(36,195,44,203),(37,194,45,202),(38,193,46,201),(39,208,47,200),(40,207,48,199),(65,93,73,85),(66,92,74,84),(67,91,75,83),(68,90,76,82),(69,89,77,81),(70,88,78,96),(71,87,79,95),(72,86,80,94),(97,150,105,158),(98,149,106,157),(99,148,107,156),(100,147,108,155),(101,146,109,154),(102,145,110,153),(103,160,111,152),(104,159,112,151),(113,254,121,246),(114,253,122,245),(115,252,123,244),(116,251,124,243),(117,250,125,242),(118,249,126,241),(119,248,127,256),(120,247,128,255),(129,176,137,168),(130,175,138,167),(131,174,139,166),(132,173,140,165),(133,172,141,164),(134,171,142,163),(135,170,143,162),(136,169,144,161),(177,268,185,260),(178,267,186,259),(179,266,187,258),(180,265,188,257),(181,264,189,272),(182,263,190,271),(183,262,191,270),(184,261,192,269),(209,273,217,281),(210,288,218,280),(211,287,219,279),(212,286,220,278),(213,285,221,277),(214,284,222,276),(215,283,223,275),(216,282,224,274)]])
99 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 4C | 6A | ··· | 6H | 8A | 8B | 12A | ··· | 12H | 12I | ··· | 12X | 16A | 16B | 16C | 16D | 24A | ··· | 24P | 48A | ··· | 48AF |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | ··· | 1 | 2 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D8 | C3×D4 | Q32 | C3×D8 | C3×Q32 |
| kernel | C32×Q32 | C3×C48 | C32×Q16 | C3×Q32 | C48 | C3×Q16 | C3×C12 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 8 | 8 | 16 | 1 | 2 | 8 | 4 | 16 | 32 |
Matrix representation of C32×Q32 ►in GL4(𝔽97) generated by
| 61 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 35 | 0 | 0 | 0 |
| 0 | 61 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 24 | 4 |
| 0 | 0 | 95 | 28 |
| 96 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 94 | 41 |
| 0 | 0 | 66 | 3 |
G:=sub<GL(4,GF(97))| [61,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[35,0,0,0,0,61,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,96,0,0,0,0,24,95,0,0,4,28],[96,0,0,0,0,1,0,0,0,0,94,66,0,0,41,3] >;
C32×Q32 in GAP, Magma, Sage, TeX
C_3^2\times Q_{32} % in TeX
G:=Group("C3^2xQ32"); // GroupNames label
G:=SmallGroup(288,331);
// by ID
G=gap.SmallGroup(288,331);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,1008,533,1016,3784,1901,242,9077,4548,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^16=1,d^2=c^8,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations