metabelian, supersoluble, monomial
Aliases: C48.3S3, C6.9D24, C32⋊5Q32, C3⋊1Dic24, C24.74D6, C12.45D12, C16.(C3⋊S3), (C3×C48).1C2, (C3×C6).25D8, (C3×C12).120D4, C4.3(C12⋊S3), C2.5(C32⋊5D8), C32⋊5Q16.1C2, (C3×C24).52C22, C8.15(C2×C3⋊S3), SmallGroup(288,276)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊5Q32
G = < a,b,c,d | a3=b3=c16=1, d2=c8, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 360 in 72 conjugacy classes, 33 normal (11 characteristic)
C1, C2, C3, C4, C4, C6, C8, Q8, C32, Dic3, C12, C16, Q16, C3×C6, C24, Dic6, Q32, C3⋊Dic3, C3×C12, C48, Dic12, C3×C24, C32⋊4Q8, Dic24, C3×C48, C32⋊5Q16, C32⋊5Q32
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, D12, Q32, C2×C3⋊S3, D24, C12⋊S3, Dic24, C32⋊5D8, C32⋊5Q32
►Regular action on 288 points
Generators in S288
(1 212 58)(2 213 59)(3 214 60)(4 215 61)(5 216 62)(6 217 63)(7 218 64)(8 219 49)(9 220 50)(10 221 51)(11 222 52)(12 223 53)(13 224 54)(14 209 55)(15 210 56)(16 211 57)(17 282 266)(18 283 267)(19 284 268)(20 285 269)(21 286 270)(22 287 271)(23 288 272)(24 273 257)(25 274 258)(26 275 259)(27 276 260)(28 277 261)(29 278 262)(30 279 263)(31 280 264)(32 281 265)(33 236 175)(34 237 176)(35 238 161)(36 239 162)(37 240 163)(38 225 164)(39 226 165)(40 227 166)(41 228 167)(42 229 168)(43 230 169)(44 231 170)(45 232 171)(46 233 172)(47 234 173)(48 235 174)(65 155 120)(66 156 121)(67 157 122)(68 158 123)(69 159 124)(70 160 125)(71 145 126)(72 146 127)(73 147 128)(74 148 113)(75 149 114)(76 150 115)(77 151 116)(78 152 117)(79 153 118)(80 154 119)(81 143 106)(82 144 107)(83 129 108)(84 130 109)(85 131 110)(86 132 111)(87 133 112)(88 134 97)(89 135 98)(90 136 99)(91 137 100)(92 138 101)(93 139 102)(94 140 103)(95 141 104)(96 142 105)(177 199 244)(178 200 245)(179 201 246)(180 202 247)(181 203 248)(182 204 249)(183 205 250)(184 206 251)(185 207 252)(186 208 253)(187 193 254)(188 194 255)(189 195 256)(190 196 241)(191 197 242)(192 198 243)
(1 117 99)(2 118 100)(3 119 101)(4 120 102)(5 121 103)(6 122 104)(7 123 105)(8 124 106)(9 125 107)(10 126 108)(11 127 109)(12 128 110)(13 113 111)(14 114 112)(15 115 97)(16 116 98)(17 182 174)(18 183 175)(19 184 176)(20 185 161)(21 186 162)(22 187 163)(23 188 164)(24 189 165)(25 190 166)(26 191 167)(27 192 168)(28 177 169)(29 178 170)(30 179 171)(31 180 172)(32 181 173)(33 283 205)(34 284 206)(35 285 207)(36 286 208)(37 287 193)(38 288 194)(39 273 195)(40 274 196)(41 275 197)(42 276 198)(43 277 199)(44 278 200)(45 279 201)(46 280 202)(47 281 203)(48 282 204)(49 159 143)(50 160 144)(51 145 129)(52 146 130)(53 147 131)(54 148 132)(55 149 133)(56 150 134)(57 151 135)(58 152 136)(59 153 137)(60 154 138)(61 155 139)(62 156 140)(63 157 141)(64 158 142)(65 93 215)(66 94 216)(67 95 217)(68 96 218)(69 81 219)(70 82 220)(71 83 221)(72 84 222)(73 85 223)(74 86 224)(75 87 209)(76 88 210)(77 89 211)(78 90 212)(79 91 213)(80 92 214)(225 272 255)(226 257 256)(227 258 241)(228 259 242)(229 260 243)(230 261 244)(231 262 245)(232 263 246)(233 264 247)(234 265 248)(235 266 249)(236 267 250)(237 268 251)(238 269 252)(239 270 253)(240 271 254)
(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)
(1 233 9 225)(2 232 10 240)(3 231 11 239)(4 230 12 238)(5 229 13 237)(6 228 14 236)(7 227 15 235)(8 226 16 234)(17 142 25 134)(18 141 26 133)(19 140 27 132)(20 139 28 131)(21 138 29 130)(22 137 30 129)(23 136 31 144)(24 135 32 143)(33 217 41 209)(34 216 42 224)(35 215 43 223)(36 214 44 222)(37 213 45 221)(38 212 46 220)(39 211 47 219)(40 210 48 218)(49 165 57 173)(50 164 58 172)(51 163 59 171)(52 162 60 170)(53 161 61 169)(54 176 62 168)(55 175 63 167)(56 174 64 166)(65 199 73 207)(66 198 74 206)(67 197 75 205)(68 196 76 204)(69 195 77 203)(70 194 78 202)(71 193 79 201)(72 208 80 200)(81 273 89 281)(82 288 90 280)(83 287 91 279)(84 286 92 278)(85 285 93 277)(86 284 94 276)(87 283 95 275)(88 282 96 274)(97 266 105 258)(98 265 106 257)(99 264 107 272)(100 263 108 271)(101 262 109 270)(102 261 110 269)(103 260 111 268)(104 259 112 267)(113 251 121 243)(114 250 122 242)(115 249 123 241)(116 248 124 256)(117 247 125 255)(118 246 126 254)(119 245 127 253)(120 244 128 252)(145 187 153 179)(146 186 154 178)(147 185 155 177)(148 184 156 192)(149 183 157 191)(150 182 158 190)(151 181 159 189)(152 180 160 188)
G:=sub<Sym(288)| (1,212,58)(2,213,59)(3,214,60)(4,215,61)(5,216,62)(6,217,63)(7,218,64)(8,219,49)(9,220,50)(10,221,51)(11,222,52)(12,223,53)(13,224,54)(14,209,55)(15,210,56)(16,211,57)(17,282,266)(18,283,267)(19,284,268)(20,285,269)(21,286,270)(22,287,271)(23,288,272)(24,273,257)(25,274,258)(26,275,259)(27,276,260)(28,277,261)(29,278,262)(30,279,263)(31,280,264)(32,281,265)(33,236,175)(34,237,176)(35,238,161)(36,239,162)(37,240,163)(38,225,164)(39,226,165)(40,227,166)(41,228,167)(42,229,168)(43,230,169)(44,231,170)(45,232,171)(46,233,172)(47,234,173)(48,235,174)(65,155,120)(66,156,121)(67,157,122)(68,158,123)(69,159,124)(70,160,125)(71,145,126)(72,146,127)(73,147,128)(74,148,113)(75,149,114)(76,150,115)(77,151,116)(78,152,117)(79,153,118)(80,154,119)(81,143,106)(82,144,107)(83,129,108)(84,130,109)(85,131,110)(86,132,111)(87,133,112)(88,134,97)(89,135,98)(90,136,99)(91,137,100)(92,138,101)(93,139,102)(94,140,103)(95,141,104)(96,142,105)(177,199,244)(178,200,245)(179,201,246)(180,202,247)(181,203,248)(182,204,249)(183,205,250)(184,206,251)(185,207,252)(186,208,253)(187,193,254)(188,194,255)(189,195,256)(190,196,241)(191,197,242)(192,198,243), (1,117,99)(2,118,100)(3,119,101)(4,120,102)(5,121,103)(6,122,104)(7,123,105)(8,124,106)(9,125,107)(10,126,108)(11,127,109)(12,128,110)(13,113,111)(14,114,112)(15,115,97)(16,116,98)(17,182,174)(18,183,175)(19,184,176)(20,185,161)(21,186,162)(22,187,163)(23,188,164)(24,189,165)(25,190,166)(26,191,167)(27,192,168)(28,177,169)(29,178,170)(30,179,171)(31,180,172)(32,181,173)(33,283,205)(34,284,206)(35,285,207)(36,286,208)(37,287,193)(38,288,194)(39,273,195)(40,274,196)(41,275,197)(42,276,198)(43,277,199)(44,278,200)(45,279,201)(46,280,202)(47,281,203)(48,282,204)(49,159,143)(50,160,144)(51,145,129)(52,146,130)(53,147,131)(54,148,132)(55,149,133)(56,150,134)(57,151,135)(58,152,136)(59,153,137)(60,154,138)(61,155,139)(62,156,140)(63,157,141)(64,158,142)(65,93,215)(66,94,216)(67,95,217)(68,96,218)(69,81,219)(70,82,220)(71,83,221)(72,84,222)(73,85,223)(74,86,224)(75,87,209)(76,88,210)(77,89,211)(78,90,212)(79,91,213)(80,92,214)(225,272,255)(226,257,256)(227,258,241)(228,259,242)(229,260,243)(230,261,244)(231,262,245)(232,263,246)(233,264,247)(234,265,248)(235,266,249)(236,267,250)(237,268,251)(238,269,252)(239,270,253)(240,271,254), (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), (1,233,9,225)(2,232,10,240)(3,231,11,239)(4,230,12,238)(5,229,13,237)(6,228,14,236)(7,227,15,235)(8,226,16,234)(17,142,25,134)(18,141,26,133)(19,140,27,132)(20,139,28,131)(21,138,29,130)(22,137,30,129)(23,136,31,144)(24,135,32,143)(33,217,41,209)(34,216,42,224)(35,215,43,223)(36,214,44,222)(37,213,45,221)(38,212,46,220)(39,211,47,219)(40,210,48,218)(49,165,57,173)(50,164,58,172)(51,163,59,171)(52,162,60,170)(53,161,61,169)(54,176,62,168)(55,175,63,167)(56,174,64,166)(65,199,73,207)(66,198,74,206)(67,197,75,205)(68,196,76,204)(69,195,77,203)(70,194,78,202)(71,193,79,201)(72,208,80,200)(81,273,89,281)(82,288,90,280)(83,287,91,279)(84,286,92,278)(85,285,93,277)(86,284,94,276)(87,283,95,275)(88,282,96,274)(97,266,105,258)(98,265,106,257)(99,264,107,272)(100,263,108,271)(101,262,109,270)(102,261,110,269)(103,260,111,268)(104,259,112,267)(113,251,121,243)(114,250,122,242)(115,249,123,241)(116,248,124,256)(117,247,125,255)(118,246,126,254)(119,245,127,253)(120,244,128,252)(145,187,153,179)(146,186,154,178)(147,185,155,177)(148,184,156,192)(149,183,157,191)(150,182,158,190)(151,181,159,189)(152,180,160,188)>;
G:=Group( (1,212,58)(2,213,59)(3,214,60)(4,215,61)(5,216,62)(6,217,63)(7,218,64)(8,219,49)(9,220,50)(10,221,51)(11,222,52)(12,223,53)(13,224,54)(14,209,55)(15,210,56)(16,211,57)(17,282,266)(18,283,267)(19,284,268)(20,285,269)(21,286,270)(22,287,271)(23,288,272)(24,273,257)(25,274,258)(26,275,259)(27,276,260)(28,277,261)(29,278,262)(30,279,263)(31,280,264)(32,281,265)(33,236,175)(34,237,176)(35,238,161)(36,239,162)(37,240,163)(38,225,164)(39,226,165)(40,227,166)(41,228,167)(42,229,168)(43,230,169)(44,231,170)(45,232,171)(46,233,172)(47,234,173)(48,235,174)(65,155,120)(66,156,121)(67,157,122)(68,158,123)(69,159,124)(70,160,125)(71,145,126)(72,146,127)(73,147,128)(74,148,113)(75,149,114)(76,150,115)(77,151,116)(78,152,117)(79,153,118)(80,154,119)(81,143,106)(82,144,107)(83,129,108)(84,130,109)(85,131,110)(86,132,111)(87,133,112)(88,134,97)(89,135,98)(90,136,99)(91,137,100)(92,138,101)(93,139,102)(94,140,103)(95,141,104)(96,142,105)(177,199,244)(178,200,245)(179,201,246)(180,202,247)(181,203,248)(182,204,249)(183,205,250)(184,206,251)(185,207,252)(186,208,253)(187,193,254)(188,194,255)(189,195,256)(190,196,241)(191,197,242)(192,198,243), (1,117,99)(2,118,100)(3,119,101)(4,120,102)(5,121,103)(6,122,104)(7,123,105)(8,124,106)(9,125,107)(10,126,108)(11,127,109)(12,128,110)(13,113,111)(14,114,112)(15,115,97)(16,116,98)(17,182,174)(18,183,175)(19,184,176)(20,185,161)(21,186,162)(22,187,163)(23,188,164)(24,189,165)(25,190,166)(26,191,167)(27,192,168)(28,177,169)(29,178,170)(30,179,171)(31,180,172)(32,181,173)(33,283,205)(34,284,206)(35,285,207)(36,286,208)(37,287,193)(38,288,194)(39,273,195)(40,274,196)(41,275,197)(42,276,198)(43,277,199)(44,278,200)(45,279,201)(46,280,202)(47,281,203)(48,282,204)(49,159,143)(50,160,144)(51,145,129)(52,146,130)(53,147,131)(54,148,132)(55,149,133)(56,150,134)(57,151,135)(58,152,136)(59,153,137)(60,154,138)(61,155,139)(62,156,140)(63,157,141)(64,158,142)(65,93,215)(66,94,216)(67,95,217)(68,96,218)(69,81,219)(70,82,220)(71,83,221)(72,84,222)(73,85,223)(74,86,224)(75,87,209)(76,88,210)(77,89,211)(78,90,212)(79,91,213)(80,92,214)(225,272,255)(226,257,256)(227,258,241)(228,259,242)(229,260,243)(230,261,244)(231,262,245)(232,263,246)(233,264,247)(234,265,248)(235,266,249)(236,267,250)(237,268,251)(238,269,252)(239,270,253)(240,271,254), (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), (1,233,9,225)(2,232,10,240)(3,231,11,239)(4,230,12,238)(5,229,13,237)(6,228,14,236)(7,227,15,235)(8,226,16,234)(17,142,25,134)(18,141,26,133)(19,140,27,132)(20,139,28,131)(21,138,29,130)(22,137,30,129)(23,136,31,144)(24,135,32,143)(33,217,41,209)(34,216,42,224)(35,215,43,223)(36,214,44,222)(37,213,45,221)(38,212,46,220)(39,211,47,219)(40,210,48,218)(49,165,57,173)(50,164,58,172)(51,163,59,171)(52,162,60,170)(53,161,61,169)(54,176,62,168)(55,175,63,167)(56,174,64,166)(65,199,73,207)(66,198,74,206)(67,197,75,205)(68,196,76,204)(69,195,77,203)(70,194,78,202)(71,193,79,201)(72,208,80,200)(81,273,89,281)(82,288,90,280)(83,287,91,279)(84,286,92,278)(85,285,93,277)(86,284,94,276)(87,283,95,275)(88,282,96,274)(97,266,105,258)(98,265,106,257)(99,264,107,272)(100,263,108,271)(101,262,109,270)(102,261,110,269)(103,260,111,268)(104,259,112,267)(113,251,121,243)(114,250,122,242)(115,249,123,241)(116,248,124,256)(117,247,125,255)(118,246,126,254)(119,245,127,253)(120,244,128,252)(145,187,153,179)(146,186,154,178)(147,185,155,177)(148,184,156,192)(149,183,157,191)(150,182,158,190)(151,181,159,189)(152,180,160,188) );
G=PermutationGroup([[(1,212,58),(2,213,59),(3,214,60),(4,215,61),(5,216,62),(6,217,63),(7,218,64),(8,219,49),(9,220,50),(10,221,51),(11,222,52),(12,223,53),(13,224,54),(14,209,55),(15,210,56),(16,211,57),(17,282,266),(18,283,267),(19,284,268),(20,285,269),(21,286,270),(22,287,271),(23,288,272),(24,273,257),(25,274,258),(26,275,259),(27,276,260),(28,277,261),(29,278,262),(30,279,263),(31,280,264),(32,281,265),(33,236,175),(34,237,176),(35,238,161),(36,239,162),(37,240,163),(38,225,164),(39,226,165),(40,227,166),(41,228,167),(42,229,168),(43,230,169),(44,231,170),(45,232,171),(46,233,172),(47,234,173),(48,235,174),(65,155,120),(66,156,121),(67,157,122),(68,158,123),(69,159,124),(70,160,125),(71,145,126),(72,146,127),(73,147,128),(74,148,113),(75,149,114),(76,150,115),(77,151,116),(78,152,117),(79,153,118),(80,154,119),(81,143,106),(82,144,107),(83,129,108),(84,130,109),(85,131,110),(86,132,111),(87,133,112),(88,134,97),(89,135,98),(90,136,99),(91,137,100),(92,138,101),(93,139,102),(94,140,103),(95,141,104),(96,142,105),(177,199,244),(178,200,245),(179,201,246),(180,202,247),(181,203,248),(182,204,249),(183,205,250),(184,206,251),(185,207,252),(186,208,253),(187,193,254),(188,194,255),(189,195,256),(190,196,241),(191,197,242),(192,198,243)], [(1,117,99),(2,118,100),(3,119,101),(4,120,102),(5,121,103),(6,122,104),(7,123,105),(8,124,106),(9,125,107),(10,126,108),(11,127,109),(12,128,110),(13,113,111),(14,114,112),(15,115,97),(16,116,98),(17,182,174),(18,183,175),(19,184,176),(20,185,161),(21,186,162),(22,187,163),(23,188,164),(24,189,165),(25,190,166),(26,191,167),(27,192,168),(28,177,169),(29,178,170),(30,179,171),(31,180,172),(32,181,173),(33,283,205),(34,284,206),(35,285,207),(36,286,208),(37,287,193),(38,288,194),(39,273,195),(40,274,196),(41,275,197),(42,276,198),(43,277,199),(44,278,200),(45,279,201),(46,280,202),(47,281,203),(48,282,204),(49,159,143),(50,160,144),(51,145,129),(52,146,130),(53,147,131),(54,148,132),(55,149,133),(56,150,134),(57,151,135),(58,152,136),(59,153,137),(60,154,138),(61,155,139),(62,156,140),(63,157,141),(64,158,142),(65,93,215),(66,94,216),(67,95,217),(68,96,218),(69,81,219),(70,82,220),(71,83,221),(72,84,222),(73,85,223),(74,86,224),(75,87,209),(76,88,210),(77,89,211),(78,90,212),(79,91,213),(80,92,214),(225,272,255),(226,257,256),(227,258,241),(228,259,242),(229,260,243),(230,261,244),(231,262,245),(232,263,246),(233,264,247),(234,265,248),(235,266,249),(236,267,250),(237,268,251),(238,269,252),(239,270,253),(240,271,254)], [(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)], [(1,233,9,225),(2,232,10,240),(3,231,11,239),(4,230,12,238),(5,229,13,237),(6,228,14,236),(7,227,15,235),(8,226,16,234),(17,142,25,134),(18,141,26,133),(19,140,27,132),(20,139,28,131),(21,138,29,130),(22,137,30,129),(23,136,31,144),(24,135,32,143),(33,217,41,209),(34,216,42,224),(35,215,43,223),(36,214,44,222),(37,213,45,221),(38,212,46,220),(39,211,47,219),(40,210,48,218),(49,165,57,173),(50,164,58,172),(51,163,59,171),(52,162,60,170),(53,161,61,169),(54,176,62,168),(55,175,63,167),(56,174,64,166),(65,199,73,207),(66,198,74,206),(67,197,75,205),(68,196,76,204),(69,195,77,203),(70,194,78,202),(71,193,79,201),(72,208,80,200),(81,273,89,281),(82,288,90,280),(83,287,91,279),(84,286,92,278),(85,285,93,277),(86,284,94,276),(87,283,95,275),(88,282,96,274),(97,266,105,258),(98,265,106,257),(99,264,107,272),(100,263,108,271),(101,262,109,270),(102,261,110,269),(103,260,111,268),(104,259,112,267),(113,251,121,243),(114,250,122,242),(115,249,123,241),(116,248,124,256),(117,247,125,255),(118,246,126,254),(119,245,127,253),(120,244,128,252),(145,187,153,179),(146,186,154,178),(147,185,155,177),(148,184,156,192),(149,183,157,191),(150,182,158,190),(151,181,159,189),(152,180,160,188)]])
75 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 12A | ··· | 12H | 16A | 16B | 16C | 16D | 24A | ··· | 24P | 48A | ··· | 48AF |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 72 | 72 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D8 | D12 | Q32 | D24 | Dic24 |
| kernel | C32⋊5Q32 | C3×C48 | C32⋊5Q16 | C48 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 1 | 4 | 2 | 8 | 4 | 16 | 32 |
Matrix representation of C32⋊5Q32 ►in GL4(𝔽97) generated by
| 0 | 1 | 0 | 0 |
| 96 | 96 | 0 | 0 |
| 0 | 0 | 96 | 96 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 96 | 96 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 96 | 96 |
| 32 | 19 | 0 | 0 |
| 78 | 13 | 0 | 0 |
| 0 | 0 | 2 | 18 |
| 0 | 0 | 79 | 81 |
| 50 | 33 | 0 | 0 |
| 80 | 47 | 0 | 0 |
| 0 | 0 | 93 | 36 |
| 0 | 0 | 40 | 4 |
G:=sub<GL(4,GF(97))| [0,96,0,0,1,96,0,0,0,0,96,1,0,0,96,0],[0,96,0,0,1,96,0,0,0,0,0,96,0,0,1,96],[32,78,0,0,19,13,0,0,0,0,2,79,0,0,18,81],[50,80,0,0,33,47,0,0,0,0,93,40,0,0,36,4] >;
C32⋊5Q32 in GAP, Magma, Sage, TeX
C_3^2\rtimes_5Q_{32} % in TeX
G:=Group("C3^2:5Q32"); // GroupNames label
G:=SmallGroup(288,276);
// by ID
G=gap.SmallGroup(288,276);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,92,254,142,675,80,2693,9414]);
// 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,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations