Copied to
clipboard

G = C327Q32order 288 = 25·32

2nd semidirect product of C32 and Q32 acting via Q32/Q16=C2

metabelian, supersoluble, monomial

Aliases: C327Q32, C24.21D6, (C3×C6).40D8, Q16.(C3⋊S3), (C3×C12).55D4, C33(C3⋊Q32), (C3×Q16).5S3, C6.26(D4⋊S3), C12.37(C3⋊D4), C325Q16.3C2, C24.S3.2C2, (C3×C24).20C22, C4.4(C327D4), C2.7(C327D8), (C32×Q16).2C2, C8.7(C2×C3⋊S3), SmallGroup(288,304)

Series: Derived Chief Lower central Upper central

C1C3×C24 — C327Q32
C1C3C32C3×C6C3×C12C3×C24C325Q16 — C327Q32
C32C3×C6C3×C12C3×C24 — C327Q32
C1C2C4C8Q16

Generators and relations for C327Q32
 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 [×4], C4, C4 [×2], C6 [×4], C8, Q8 [×2], C32, Dic3 [×4], C12 [×4], C12 [×4], C16, Q16, Q16, C3×C6, C24 [×4], Dic6 [×4], C3×Q8 [×4], Q32, C3⋊Dic3, C3×C12, C3×C12, C3⋊C16 [×4], Dic12 [×4], C3×Q16 [×4], C3×C24, C324Q8, Q8×C32, C3⋊Q32 [×4], C24.S3, C325Q16, C32×Q16, C327Q32
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, C3⋊S3, C3⋊D4 [×4], Q32, C2×C3⋊S3, D4⋊S3 [×4], C327D4, C3⋊Q32 [×4], C327D8, C327Q32

Smallest permutation representation of C327Q32
Regular action on 288 points
Generators in S288
(1 212 253)(2 254 213)(3 214 255)(4 256 215)(5 216 241)(6 242 217)(7 218 243)(8 244 219)(9 220 245)(10 246 221)(11 222 247)(12 248 223)(13 224 249)(14 250 209)(15 210 251)(16 252 211)(17 279 150)(18 151 280)(19 281 152)(20 153 282)(21 283 154)(22 155 284)(23 285 156)(24 157 286)(25 287 158)(26 159 288)(27 273 160)(28 145 274)(29 275 146)(30 147 276)(31 277 148)(32 149 278)(33 52 237)(34 238 53)(35 54 239)(36 240 55)(37 56 225)(38 226 57)(39 58 227)(40 228 59)(41 60 229)(42 230 61)(43 62 231)(44 232 63)(45 64 233)(46 234 49)(47 50 235)(48 236 51)(65 95 112)(66 97 96)(67 81 98)(68 99 82)(69 83 100)(70 101 84)(71 85 102)(72 103 86)(73 87 104)(74 105 88)(75 89 106)(76 107 90)(77 91 108)(78 109 92)(79 93 110)(80 111 94)(113 175 177)(114 178 176)(115 161 179)(116 180 162)(117 163 181)(118 182 164)(119 165 183)(120 184 166)(121 167 185)(122 186 168)(123 169 187)(124 188 170)(125 171 189)(126 190 172)(127 173 191)(128 192 174)(129 261 197)(130 198 262)(131 263 199)(132 200 264)(133 265 201)(134 202 266)(135 267 203)(136 204 268)(137 269 205)(138 206 270)(139 271 207)(140 208 272)(141 257 193)(142 194 258)(143 259 195)(144 196 260)
(1 159 62)(2 63 160)(3 145 64)(4 49 146)(5 147 50)(6 51 148)(7 149 52)(8 53 150)(9 151 54)(10 55 152)(11 153 56)(12 57 154)(13 155 58)(14 59 156)(15 157 60)(16 61 158)(17 244 34)(18 35 245)(19 246 36)(20 37 247)(21 248 38)(22 39 249)(23 250 40)(24 41 251)(25 252 42)(26 43 253)(27 254 44)(28 45 255)(29 256 46)(30 47 241)(31 242 48)(32 33 243)(65 179 197)(66 198 180)(67 181 199)(68 200 182)(69 183 201)(70 202 184)(71 185 203)(72 204 186)(73 187 205)(74 206 188)(75 189 207)(76 208 190)(77 191 193)(78 194 192)(79 177 195)(80 196 178)(81 117 131)(82 132 118)(83 119 133)(84 134 120)(85 121 135)(86 136 122)(87 123 137)(88 138 124)(89 125 139)(90 140 126)(91 127 141)(92 142 128)(93 113 143)(94 144 114)(95 115 129)(96 130 116)(97 262 162)(98 163 263)(99 264 164)(100 165 265)(101 266 166)(102 167 267)(103 268 168)(104 169 269)(105 270 170)(106 171 271)(107 272 172)(108 173 257)(109 258 174)(110 175 259)(111 260 176)(112 161 261)(209 228 285)(210 286 229)(211 230 287)(212 288 231)(213 232 273)(214 274 233)(215 234 275)(216 276 235)(217 236 277)(218 278 237)(219 238 279)(220 280 239)(221 240 281)(222 282 225)(223 226 283)(224 284 227)
(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 69 9 77)(2 68 10 76)(3 67 11 75)(4 66 12 74)(5 65 13 73)(6 80 14 72)(7 79 15 71)(8 78 16 70)(17 174 25 166)(18 173 26 165)(19 172 27 164)(20 171 28 163)(21 170 29 162)(22 169 30 161)(23 168 31 176)(24 167 32 175)(33 259 41 267)(34 258 42 266)(35 257 43 265)(36 272 44 264)(37 271 45 263)(38 270 46 262)(39 269 47 261)(40 268 48 260)(49 198 57 206)(50 197 58 205)(51 196 59 204)(52 195 60 203)(53 194 61 202)(54 193 62 201)(55 208 63 200)(56 207 64 199)(81 222 89 214)(82 221 90 213)(83 220 91 212)(84 219 92 211)(85 218 93 210)(86 217 94 209)(87 216 95 224)(88 215 96 223)(97 248 105 256)(98 247 106 255)(99 246 107 254)(100 245 108 253)(101 244 109 252)(102 243 110 251)(103 242 111 250)(104 241 112 249)(113 286 121 278)(114 285 122 277)(115 284 123 276)(116 283 124 275)(117 282 125 274)(118 281 126 273)(119 280 127 288)(120 279 128 287)(129 227 137 235)(130 226 138 234)(131 225 139 233)(132 240 140 232)(133 239 141 231)(134 238 142 230)(135 237 143 229)(136 236 144 228)(145 181 153 189)(146 180 154 188)(147 179 155 187)(148 178 156 186)(149 177 157 185)(150 192 158 184)(151 191 159 183)(152 190 160 182)

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

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

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

39 conjugacy classes

class 1  2 3A3B3C3D4A4B4C6A6B6C6D8A8B12A12B12C12D12E···12L16A16B16C16D24A···24H
order1233334446666881212121212···121616161624···24
size112222287222222244448···8181818184···4

39 irreducible representations

dim111122222244
type++++++++-+-
imageC1C2C2C2S3D4D6D8C3⋊D4Q32D4⋊S3C3⋊Q32
kernelC327Q32C24.S3C325Q16C32×Q16C3×Q16C3×C12C24C3×C6C12C32C6C3
# reps111141428448

Matrix representation of C327Q32 in GL6(𝔽97)

9610000
9600000
001000
000100
0000096
0000196
,
9610000
9600000
001000
000100
0000961
0000960
,
39120000
51580000
0022600
0071200
00006189
00005336
,
41150000
82560000
00366200
00626100
00004115
00008256

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] >;

C327Q32 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

׿
×
𝔽