metabelian, supersoluble, monomial
Aliases: C32⋊7Q32, C24.21D6, (C3×C6).40D8, Q16.(C3⋊S3), (C3×C12).55D4, C3⋊3(C3⋊Q32), (C3×Q16).5S3, C6.26(D4⋊S3), C12.37(C3⋊D4), C32⋊5Q16.3C2, C24.S3.2C2, (C3×C24).20C22, C4.4(C32⋊7D4), C2.7(C32⋊7D8), (C32×Q16).2C2, C8.7(C2×C3⋊S3), SmallGroup(288,304)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊7Q32
G = < a,b,c,d | a3=b3=c16=1, d2=c8, ab=ba, cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 264 in 72 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C3, C4, C4, C6, C8, Q8, C32, Dic3, C12, C12, C16, Q16, Q16, C3×C6, C24, Dic6, C3×Q8, Q32, C3⋊Dic3, C3×C12, C3×C12, C3⋊C16, Dic12, C3×Q16, C3×C24, C32⋊4Q8, Q8×C32, C3⋊Q32, C24.S3, C32⋊5Q16, C32×Q16, C32⋊7Q32
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, C3⋊D4, Q32, C2×C3⋊S3, D4⋊S3, C32⋊7D4, C3⋊Q32, C32⋊7D8, C32⋊7Q32
►Regular action on 288 points
Generators in S288
(1 223 256)(2 241 224)(3 209 242)(4 243 210)(5 211 244)(6 245 212)(7 213 246)(8 247 214)(9 215 248)(10 249 216)(11 217 250)(12 251 218)(13 219 252)(14 253 220)(15 221 254)(16 255 222)(17 36 138)(18 139 37)(19 38 140)(20 141 39)(21 40 142)(22 143 41)(23 42 144)(24 129 43)(25 44 130)(26 131 45)(27 46 132)(28 133 47)(29 48 134)(30 135 33)(31 34 136)(32 137 35)(49 80 152)(50 153 65)(51 66 154)(52 155 67)(53 68 156)(54 157 69)(55 70 158)(56 159 71)(57 72 160)(58 145 73)(59 74 146)(60 147 75)(61 76 148)(62 149 77)(63 78 150)(64 151 79)(81 123 266)(82 267 124)(83 125 268)(84 269 126)(85 127 270)(86 271 128)(87 113 272)(88 257 114)(89 115 258)(90 259 116)(91 117 260)(92 261 118)(93 119 262)(94 263 120)(95 121 264)(96 265 122)(97 172 235)(98 236 173)(99 174 237)(100 238 175)(101 176 239)(102 240 161)(103 162 225)(104 226 163)(105 164 227)(106 228 165)(107 166 229)(108 230 167)(109 168 231)(110 232 169)(111 170 233)(112 234 171)(177 199 287)(178 288 200)(179 201 273)(180 274 202)(181 203 275)(182 276 204)(183 205 277)(184 278 206)(185 207 279)(186 280 208)(187 193 281)(188 282 194)(189 195 283)(190 284 196)(191 197 285)(192 286 198)
(1 276 37)(2 38 277)(3 278 39)(4 40 279)(5 280 41)(6 42 281)(7 282 43)(8 44 283)(9 284 45)(10 46 285)(11 286 47)(12 48 287)(13 288 33)(14 34 273)(15 274 35)(16 36 275)(17 203 222)(18 223 204)(19 205 224)(20 209 206)(21 207 210)(22 211 208)(23 193 212)(24 213 194)(25 195 214)(26 215 196)(27 197 216)(28 217 198)(29 199 218)(30 219 200)(31 201 220)(32 221 202)(49 228 272)(50 257 229)(51 230 258)(52 259 231)(53 232 260)(54 261 233)(55 234 262)(56 263 235)(57 236 264)(58 265 237)(59 238 266)(60 267 239)(61 240 268)(62 269 225)(63 226 270)(64 271 227)(65 88 166)(66 167 89)(67 90 168)(68 169 91)(69 92 170)(70 171 93)(71 94 172)(72 173 95)(73 96 174)(74 175 81)(75 82 176)(76 161 83)(77 84 162)(78 163 85)(79 86 164)(80 165 87)(97 159 120)(98 121 160)(99 145 122)(100 123 146)(101 147 124)(102 125 148)(103 149 126)(104 127 150)(105 151 128)(106 113 152)(107 153 114)(108 115 154)(109 155 116)(110 117 156)(111 157 118)(112 119 158)(129 246 188)(130 189 247)(131 248 190)(132 191 249)(133 250 192)(134 177 251)(135 252 178)(136 179 253)(137 254 180)(138 181 255)(139 256 182)(140 183 241)(141 242 184)(142 185 243)(143 244 186)(144 187 245)
(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)
(1 121 9 113)(2 120 10 128)(3 119 11 127)(4 118 12 126)(5 117 13 125)(6 116 14 124)(7 115 15 123)(8 114 16 122)(17 237 25 229)(18 236 26 228)(19 235 27 227)(20 234 28 226)(21 233 29 225)(22 232 30 240)(23 231 31 239)(24 230 32 238)(33 102 41 110)(34 101 42 109)(35 100 43 108)(36 99 44 107)(37 98 45 106)(38 97 46 105)(39 112 47 104)(40 111 48 103)(49 204 57 196)(50 203 58 195)(51 202 59 194)(52 201 60 193)(53 200 61 208)(54 199 62 207)(55 198 63 206)(56 197 64 205)(65 181 73 189)(66 180 74 188)(67 179 75 187)(68 178 76 186)(69 177 77 185)(70 192 78 184)(71 191 79 183)(72 190 80 182)(81 246 89 254)(82 245 90 253)(83 244 91 252)(84 243 92 251)(85 242 93 250)(86 241 94 249)(87 256 95 248)(88 255 96 247)(129 167 137 175)(130 166 138 174)(131 165 139 173)(132 164 140 172)(133 163 141 171)(134 162 142 170)(135 161 143 169)(136 176 144 168)(145 283 153 275)(146 282 154 274)(147 281 155 273)(148 280 156 288)(149 279 157 287)(150 278 158 286)(151 277 159 285)(152 276 160 284)(209 262 217 270)(210 261 218 269)(211 260 219 268)(212 259 220 267)(213 258 221 266)(214 257 222 265)(215 272 223 264)(216 271 224 263)
G:=sub<Sym(288)| (1,223,256)(2,241,224)(3,209,242)(4,243,210)(5,211,244)(6,245,212)(7,213,246)(8,247,214)(9,215,248)(10,249,216)(11,217,250)(12,251,218)(13,219,252)(14,253,220)(15,221,254)(16,255,222)(17,36,138)(18,139,37)(19,38,140)(20,141,39)(21,40,142)(22,143,41)(23,42,144)(24,129,43)(25,44,130)(26,131,45)(27,46,132)(28,133,47)(29,48,134)(30,135,33)(31,34,136)(32,137,35)(49,80,152)(50,153,65)(51,66,154)(52,155,67)(53,68,156)(54,157,69)(55,70,158)(56,159,71)(57,72,160)(58,145,73)(59,74,146)(60,147,75)(61,76,148)(62,149,77)(63,78,150)(64,151,79)(81,123,266)(82,267,124)(83,125,268)(84,269,126)(85,127,270)(86,271,128)(87,113,272)(88,257,114)(89,115,258)(90,259,116)(91,117,260)(92,261,118)(93,119,262)(94,263,120)(95,121,264)(96,265,122)(97,172,235)(98,236,173)(99,174,237)(100,238,175)(101,176,239)(102,240,161)(103,162,225)(104,226,163)(105,164,227)(106,228,165)(107,166,229)(108,230,167)(109,168,231)(110,232,169)(111,170,233)(112,234,171)(177,199,287)(178,288,200)(179,201,273)(180,274,202)(181,203,275)(182,276,204)(183,205,277)(184,278,206)(185,207,279)(186,280,208)(187,193,281)(188,282,194)(189,195,283)(190,284,196)(191,197,285)(192,286,198), (1,276,37)(2,38,277)(3,278,39)(4,40,279)(5,280,41)(6,42,281)(7,282,43)(8,44,283)(9,284,45)(10,46,285)(11,286,47)(12,48,287)(13,288,33)(14,34,273)(15,274,35)(16,36,275)(17,203,222)(18,223,204)(19,205,224)(20,209,206)(21,207,210)(22,211,208)(23,193,212)(24,213,194)(25,195,214)(26,215,196)(27,197,216)(28,217,198)(29,199,218)(30,219,200)(31,201,220)(32,221,202)(49,228,272)(50,257,229)(51,230,258)(52,259,231)(53,232,260)(54,261,233)(55,234,262)(56,263,235)(57,236,264)(58,265,237)(59,238,266)(60,267,239)(61,240,268)(62,269,225)(63,226,270)(64,271,227)(65,88,166)(66,167,89)(67,90,168)(68,169,91)(69,92,170)(70,171,93)(71,94,172)(72,173,95)(73,96,174)(74,175,81)(75,82,176)(76,161,83)(77,84,162)(78,163,85)(79,86,164)(80,165,87)(97,159,120)(98,121,160)(99,145,122)(100,123,146)(101,147,124)(102,125,148)(103,149,126)(104,127,150)(105,151,128)(106,113,152)(107,153,114)(108,115,154)(109,155,116)(110,117,156)(111,157,118)(112,119,158)(129,246,188)(130,189,247)(131,248,190)(132,191,249)(133,250,192)(134,177,251)(135,252,178)(136,179,253)(137,254,180)(138,181,255)(139,256,182)(140,183,241)(141,242,184)(142,185,243)(143,244,186)(144,187,245), (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), (1,121,9,113)(2,120,10,128)(3,119,11,127)(4,118,12,126)(5,117,13,125)(6,116,14,124)(7,115,15,123)(8,114,16,122)(17,237,25,229)(18,236,26,228)(19,235,27,227)(20,234,28,226)(21,233,29,225)(22,232,30,240)(23,231,31,239)(24,230,32,238)(33,102,41,110)(34,101,42,109)(35,100,43,108)(36,99,44,107)(37,98,45,106)(38,97,46,105)(39,112,47,104)(40,111,48,103)(49,204,57,196)(50,203,58,195)(51,202,59,194)(52,201,60,193)(53,200,61,208)(54,199,62,207)(55,198,63,206)(56,197,64,205)(65,181,73,189)(66,180,74,188)(67,179,75,187)(68,178,76,186)(69,177,77,185)(70,192,78,184)(71,191,79,183)(72,190,80,182)(81,246,89,254)(82,245,90,253)(83,244,91,252)(84,243,92,251)(85,242,93,250)(86,241,94,249)(87,256,95,248)(88,255,96,247)(129,167,137,175)(130,166,138,174)(131,165,139,173)(132,164,140,172)(133,163,141,171)(134,162,142,170)(135,161,143,169)(136,176,144,168)(145,283,153,275)(146,282,154,274)(147,281,155,273)(148,280,156,288)(149,279,157,287)(150,278,158,286)(151,277,159,285)(152,276,160,284)(209,262,217,270)(210,261,218,269)(211,260,219,268)(212,259,220,267)(213,258,221,266)(214,257,222,265)(215,272,223,264)(216,271,224,263)>;
G:=Group( (1,223,256)(2,241,224)(3,209,242)(4,243,210)(5,211,244)(6,245,212)(7,213,246)(8,247,214)(9,215,248)(10,249,216)(11,217,250)(12,251,218)(13,219,252)(14,253,220)(15,221,254)(16,255,222)(17,36,138)(18,139,37)(19,38,140)(20,141,39)(21,40,142)(22,143,41)(23,42,144)(24,129,43)(25,44,130)(26,131,45)(27,46,132)(28,133,47)(29,48,134)(30,135,33)(31,34,136)(32,137,35)(49,80,152)(50,153,65)(51,66,154)(52,155,67)(53,68,156)(54,157,69)(55,70,158)(56,159,71)(57,72,160)(58,145,73)(59,74,146)(60,147,75)(61,76,148)(62,149,77)(63,78,150)(64,151,79)(81,123,266)(82,267,124)(83,125,268)(84,269,126)(85,127,270)(86,271,128)(87,113,272)(88,257,114)(89,115,258)(90,259,116)(91,117,260)(92,261,118)(93,119,262)(94,263,120)(95,121,264)(96,265,122)(97,172,235)(98,236,173)(99,174,237)(100,238,175)(101,176,239)(102,240,161)(103,162,225)(104,226,163)(105,164,227)(106,228,165)(107,166,229)(108,230,167)(109,168,231)(110,232,169)(111,170,233)(112,234,171)(177,199,287)(178,288,200)(179,201,273)(180,274,202)(181,203,275)(182,276,204)(183,205,277)(184,278,206)(185,207,279)(186,280,208)(187,193,281)(188,282,194)(189,195,283)(190,284,196)(191,197,285)(192,286,198), (1,276,37)(2,38,277)(3,278,39)(4,40,279)(5,280,41)(6,42,281)(7,282,43)(8,44,283)(9,284,45)(10,46,285)(11,286,47)(12,48,287)(13,288,33)(14,34,273)(15,274,35)(16,36,275)(17,203,222)(18,223,204)(19,205,224)(20,209,206)(21,207,210)(22,211,208)(23,193,212)(24,213,194)(25,195,214)(26,215,196)(27,197,216)(28,217,198)(29,199,218)(30,219,200)(31,201,220)(32,221,202)(49,228,272)(50,257,229)(51,230,258)(52,259,231)(53,232,260)(54,261,233)(55,234,262)(56,263,235)(57,236,264)(58,265,237)(59,238,266)(60,267,239)(61,240,268)(62,269,225)(63,226,270)(64,271,227)(65,88,166)(66,167,89)(67,90,168)(68,169,91)(69,92,170)(70,171,93)(71,94,172)(72,173,95)(73,96,174)(74,175,81)(75,82,176)(76,161,83)(77,84,162)(78,163,85)(79,86,164)(80,165,87)(97,159,120)(98,121,160)(99,145,122)(100,123,146)(101,147,124)(102,125,148)(103,149,126)(104,127,150)(105,151,128)(106,113,152)(107,153,114)(108,115,154)(109,155,116)(110,117,156)(111,157,118)(112,119,158)(129,246,188)(130,189,247)(131,248,190)(132,191,249)(133,250,192)(134,177,251)(135,252,178)(136,179,253)(137,254,180)(138,181,255)(139,256,182)(140,183,241)(141,242,184)(142,185,243)(143,244,186)(144,187,245), (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), (1,121,9,113)(2,120,10,128)(3,119,11,127)(4,118,12,126)(5,117,13,125)(6,116,14,124)(7,115,15,123)(8,114,16,122)(17,237,25,229)(18,236,26,228)(19,235,27,227)(20,234,28,226)(21,233,29,225)(22,232,30,240)(23,231,31,239)(24,230,32,238)(33,102,41,110)(34,101,42,109)(35,100,43,108)(36,99,44,107)(37,98,45,106)(38,97,46,105)(39,112,47,104)(40,111,48,103)(49,204,57,196)(50,203,58,195)(51,202,59,194)(52,201,60,193)(53,200,61,208)(54,199,62,207)(55,198,63,206)(56,197,64,205)(65,181,73,189)(66,180,74,188)(67,179,75,187)(68,178,76,186)(69,177,77,185)(70,192,78,184)(71,191,79,183)(72,190,80,182)(81,246,89,254)(82,245,90,253)(83,244,91,252)(84,243,92,251)(85,242,93,250)(86,241,94,249)(87,256,95,248)(88,255,96,247)(129,167,137,175)(130,166,138,174)(131,165,139,173)(132,164,140,172)(133,163,141,171)(134,162,142,170)(135,161,143,169)(136,176,144,168)(145,283,153,275)(146,282,154,274)(147,281,155,273)(148,280,156,288)(149,279,157,287)(150,278,158,286)(151,277,159,285)(152,276,160,284)(209,262,217,270)(210,261,218,269)(211,260,219,268)(212,259,220,267)(213,258,221,266)(214,257,222,265)(215,272,223,264)(216,271,224,263) );
G=PermutationGroup([[(1,223,256),(2,241,224),(3,209,242),(4,243,210),(5,211,244),(6,245,212),(7,213,246),(8,247,214),(9,215,248),(10,249,216),(11,217,250),(12,251,218),(13,219,252),(14,253,220),(15,221,254),(16,255,222),(17,36,138),(18,139,37),(19,38,140),(20,141,39),(21,40,142),(22,143,41),(23,42,144),(24,129,43),(25,44,130),(26,131,45),(27,46,132),(28,133,47),(29,48,134),(30,135,33),(31,34,136),(32,137,35),(49,80,152),(50,153,65),(51,66,154),(52,155,67),(53,68,156),(54,157,69),(55,70,158),(56,159,71),(57,72,160),(58,145,73),(59,74,146),(60,147,75),(61,76,148),(62,149,77),(63,78,150),(64,151,79),(81,123,266),(82,267,124),(83,125,268),(84,269,126),(85,127,270),(86,271,128),(87,113,272),(88,257,114),(89,115,258),(90,259,116),(91,117,260),(92,261,118),(93,119,262),(94,263,120),(95,121,264),(96,265,122),(97,172,235),(98,236,173),(99,174,237),(100,238,175),(101,176,239),(102,240,161),(103,162,225),(104,226,163),(105,164,227),(106,228,165),(107,166,229),(108,230,167),(109,168,231),(110,232,169),(111,170,233),(112,234,171),(177,199,287),(178,288,200),(179,201,273),(180,274,202),(181,203,275),(182,276,204),(183,205,277),(184,278,206),(185,207,279),(186,280,208),(187,193,281),(188,282,194),(189,195,283),(190,284,196),(191,197,285),(192,286,198)], [(1,276,37),(2,38,277),(3,278,39),(4,40,279),(5,280,41),(6,42,281),(7,282,43),(8,44,283),(9,284,45),(10,46,285),(11,286,47),(12,48,287),(13,288,33),(14,34,273),(15,274,35),(16,36,275),(17,203,222),(18,223,204),(19,205,224),(20,209,206),(21,207,210),(22,211,208),(23,193,212),(24,213,194),(25,195,214),(26,215,196),(27,197,216),(28,217,198),(29,199,218),(30,219,200),(31,201,220),(32,221,202),(49,228,272),(50,257,229),(51,230,258),(52,259,231),(53,232,260),(54,261,233),(55,234,262),(56,263,235),(57,236,264),(58,265,237),(59,238,266),(60,267,239),(61,240,268),(62,269,225),(63,226,270),(64,271,227),(65,88,166),(66,167,89),(67,90,168),(68,169,91),(69,92,170),(70,171,93),(71,94,172),(72,173,95),(73,96,174),(74,175,81),(75,82,176),(76,161,83),(77,84,162),(78,163,85),(79,86,164),(80,165,87),(97,159,120),(98,121,160),(99,145,122),(100,123,146),(101,147,124),(102,125,148),(103,149,126),(104,127,150),(105,151,128),(106,113,152),(107,153,114),(108,115,154),(109,155,116),(110,117,156),(111,157,118),(112,119,158),(129,246,188),(130,189,247),(131,248,190),(132,191,249),(133,250,192),(134,177,251),(135,252,178),(136,179,253),(137,254,180),(138,181,255),(139,256,182),(140,183,241),(141,242,184),(142,185,243),(143,244,186),(144,187,245)], [(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)], [(1,121,9,113),(2,120,10,128),(3,119,11,127),(4,118,12,126),(5,117,13,125),(6,116,14,124),(7,115,15,123),(8,114,16,122),(17,237,25,229),(18,236,26,228),(19,235,27,227),(20,234,28,226),(21,233,29,225),(22,232,30,240),(23,231,31,239),(24,230,32,238),(33,102,41,110),(34,101,42,109),(35,100,43,108),(36,99,44,107),(37,98,45,106),(38,97,46,105),(39,112,47,104),(40,111,48,103),(49,204,57,196),(50,203,58,195),(51,202,59,194),(52,201,60,193),(53,200,61,208),(54,199,62,207),(55,198,63,206),(56,197,64,205),(65,181,73,189),(66,180,74,188),(67,179,75,187),(68,178,76,186),(69,177,77,185),(70,192,78,184),(71,191,79,183),(72,190,80,182),(81,246,89,254),(82,245,90,253),(83,244,91,252),(84,243,92,251),(85,242,93,250),(86,241,94,249),(87,256,95,248),(88,255,96,247),(129,167,137,175),(130,166,138,174),(131,165,139,173),(132,164,140,172),(133,163,141,171),(134,162,142,170),(135,161,143,169),(136,176,144,168),(145,283,153,275),(146,282,154,274),(147,281,155,273),(148,280,156,288),(149,279,157,287),(150,278,158,286),(151,277,159,285),(152,276,160,284),(209,262,217,270),(210,261,218,269),(211,260,219,268),(212,259,220,267),(213,258,221,266),(214,257,222,265),(215,272,223,264),(216,271,224,263)]])
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 16A | 16B | 16C | 16D | 24A | ··· | 24H |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | 16 | 16 | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 72 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D8 | C3⋊D4 | Q32 | D4⋊S3 | C3⋊Q32 |
| kernel | C32⋊7Q32 | C24.S3 | C32⋊5Q16 | C32×Q16 | C3×Q16 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 4 | 4 | 8 |
Matrix representation of C32⋊7Q32 ►in GL6(𝔽97)
| 96 | 1 | 0 | 0 | 0 | 0 |
| 96 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 96 |
| 0 | 0 | 0 | 0 | 1 | 96 |
| 96 | 1 | 0 | 0 | 0 | 0 |
| 96 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 1 |
| 0 | 0 | 0 | 0 | 96 | 0 |
| 39 | 12 | 0 | 0 | 0 | 0 |
| 51 | 58 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 26 | 0 | 0 |
| 0 | 0 | 71 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 89 |
| 0 | 0 | 0 | 0 | 53 | 36 |
| 41 | 15 | 0 | 0 | 0 | 0 |
| 82 | 56 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 62 | 0 | 0 |
| 0 | 0 | 62 | 61 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 15 |
| 0 | 0 | 0 | 0 | 82 | 56 |
G:=sub<GL(6,GF(97))| [96,96,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96],[96,96,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[39,51,0,0,0,0,12,58,0,0,0,0,0,0,2,71,0,0,0,0,26,2,0,0,0,0,0,0,61,53,0,0,0,0,89,36],[41,82,0,0,0,0,15,56,0,0,0,0,0,0,36,62,0,0,0,0,62,61,0,0,0,0,0,0,41,82,0,0,0,0,15,56] >;
C32⋊7Q32 in GAP, Magma, Sage, TeX
C_3^2\rtimes_7Q_{32} % in TeX
G:=Group("C3^2:7Q32"); // GroupNames label
G:=SmallGroup(288,304);
// by ID
G=gap.SmallGroup(288,304);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,120,254,135,142,675,346,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^16=1,d^2=c^8,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations