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G = C325Q32order 288 = 25·32

2nd semidirect product of C32 and Q32 acting via Q32/C16=C2

metabelian, supersoluble, monomial

Aliases: C48.3S3, C6.9D24, C325Q32, C31Dic24, C24.74D6, C12.45D12, C16.(C3⋊S3), (C3×C48).1C2, (C3×C6).25D8, (C3×C12).120D4, C4.3(C12⋊S3), C2.5(C325D8), C325Q16.1C2, (C3×C24).52C22, C8.15(C2×C3⋊S3), SmallGroup(288,276)

Series: Derived Chief Lower central Upper central

C1C3×C24 — C325Q32
C1C3C32C3×C6C3×C12C3×C24C325Q16 — C325Q32
C32C3×C6C3×C12C3×C24 — C325Q32
C1C2C4C8C16

Generators and relations for C325Q32
 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 [×4], C4, C4 [×2], C6 [×4], C8, Q8 [×2], C32, Dic3 [×8], C12 [×4], C16, Q16 [×2], C3×C6, C24 [×4], Dic6 [×8], Q32, C3⋊Dic3 [×2], C3×C12, C48 [×4], Dic12 [×8], C3×C24, C324Q8 [×2], Dic24 [×4], C3×C48, C325Q16 [×2], C325Q32
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, C3⋊S3, D12 [×4], Q32, C2×C3⋊S3, D24 [×4], C12⋊S3, Dic24 [×4], C325D8, C325Q32

Smallest permutation representation of C325Q32
Regular action on 288 points
Generators in S288
(1 90 147)(2 91 148)(3 92 149)(4 93 150)(5 94 151)(6 95 152)(7 96 153)(8 81 154)(9 82 155)(10 83 156)(11 84 157)(12 85 158)(13 86 159)(14 87 160)(15 88 145)(16 89 146)(17 68 167)(18 69 168)(19 70 169)(20 71 170)(21 72 171)(22 73 172)(23 74 173)(24 75 174)(25 76 175)(26 77 176)(27 78 161)(28 79 162)(29 80 163)(30 65 164)(31 66 165)(32 67 166)(33 252 202)(34 253 203)(35 254 204)(36 255 205)(37 256 206)(38 241 207)(39 242 208)(40 243 193)(41 244 194)(42 245 195)(43 246 196)(44 247 197)(45 248 198)(46 249 199)(47 250 200)(48 251 201)(49 143 225)(50 144 226)(51 129 227)(52 130 228)(53 131 229)(54 132 230)(55 133 231)(56 134 232)(57 135 233)(58 136 234)(59 137 235)(60 138 236)(61 139 237)(62 140 238)(63 141 239)(64 142 240)(97 116 177)(98 117 178)(99 118 179)(100 119 180)(101 120 181)(102 121 182)(103 122 183)(104 123 184)(105 124 185)(106 125 186)(107 126 187)(108 127 188)(109 128 189)(110 113 190)(111 114 191)(112 115 192)(209 260 282)(210 261 283)(211 262 284)(212 263 285)(213 264 286)(214 265 287)(215 266 288)(216 267 273)(217 268 274)(218 269 275)(219 270 276)(220 271 277)(221 272 278)(222 257 279)(223 258 280)(224 259 281)
(1 203 128)(2 204 113)(3 205 114)(4 206 115)(5 207 116)(6 208 117)(7 193 118)(8 194 119)(9 195 120)(10 196 121)(11 197 122)(12 198 123)(13 199 124)(14 200 125)(15 201 126)(16 202 127)(17 60 219)(18 61 220)(19 62 221)(20 63 222)(21 64 223)(22 49 224)(23 50 209)(24 51 210)(25 52 211)(26 53 212)(27 54 213)(28 55 214)(29 56 215)(30 57 216)(31 58 217)(32 59 218)(33 188 89)(34 189 90)(35 190 91)(36 191 92)(37 192 93)(38 177 94)(39 178 95)(40 179 96)(41 180 81)(42 181 82)(43 182 83)(44 183 84)(45 184 85)(46 185 86)(47 186 87)(48 187 88)(65 135 267)(66 136 268)(67 137 269)(68 138 270)(69 139 271)(70 140 272)(71 141 257)(72 142 258)(73 143 259)(74 144 260)(75 129 261)(76 130 262)(77 131 263)(78 132 264)(79 133 265)(80 134 266)(97 151 241)(98 152 242)(99 153 243)(100 154 244)(101 155 245)(102 156 246)(103 157 247)(104 158 248)(105 159 249)(106 160 250)(107 145 251)(108 146 252)(109 147 253)(110 148 254)(111 149 255)(112 150 256)(161 230 286)(162 231 287)(163 232 288)(164 233 273)(165 234 274)(166 235 275)(167 236 276)(168 237 277)(169 238 278)(170 239 279)(171 240 280)(172 225 281)(173 226 282)(174 227 283)(175 228 284)(176 229 285)
(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 27 9 19)(2 26 10 18)(3 25 11 17)(4 24 12 32)(5 23 13 31)(6 22 14 30)(7 21 15 29)(8 20 16 28)(33 287 41 279)(34 286 42 278)(35 285 43 277)(36 284 44 276)(37 283 45 275)(38 282 46 274)(39 281 47 273)(40 280 48 288)(49 125 57 117)(50 124 58 116)(51 123 59 115)(52 122 60 114)(53 121 61 113)(54 120 62 128)(55 119 63 127)(56 118 64 126)(65 152 73 160)(66 151 74 159)(67 150 75 158)(68 149 76 157)(69 148 77 156)(70 147 78 155)(71 146 79 154)(72 145 80 153)(81 170 89 162)(82 169 90 161)(83 168 91 176)(84 167 92 175)(85 166 93 174)(86 165 94 173)(87 164 95 172)(88 163 96 171)(97 144 105 136)(98 143 106 135)(99 142 107 134)(100 141 108 133)(101 140 109 132)(102 139 110 131)(103 138 111 130)(104 137 112 129)(177 226 185 234)(178 225 186 233)(179 240 187 232)(180 239 188 231)(181 238 189 230)(182 237 190 229)(183 236 191 228)(184 235 192 227)(193 223 201 215)(194 222 202 214)(195 221 203 213)(196 220 204 212)(197 219 205 211)(198 218 206 210)(199 217 207 209)(200 216 208 224)(241 260 249 268)(242 259 250 267)(243 258 251 266)(244 257 252 265)(245 272 253 264)(246 271 254 263)(247 270 255 262)(248 269 256 261)

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

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

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

75 conjugacy classes

class 1  2 3A3B3C3D4A4B4C6A6B6C6D8A8B12A···12H16A16B16C16D24A···24P48A···48AF
order12333344466668812···121616161624···2448···48
size112222272722222222···222222···22···2

75 irreducible representations

dim11122222222
type++++++++-+-
imageC1C2C2S3D4D6D8D12Q32D24Dic24
kernelC325Q32C3×C48C325Q16C48C3×C12C24C3×C6C12C32C6C3
# reps1124142841632

Matrix representation of C325Q32 in GL4(𝔽97) generated by

0100
969600
009696
0010
,
0100
969600
0001
009696
,
321900
781300
00218
007981
,
503300
804700
009336
00404
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] >;

C325Q32 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

׿
×
𝔽