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## G = C32×Q32order 288 = 25·32

### Direct product of C32 and Q32

direct product, metacyclic, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C32×Q32
 Chief series C1 — C2 — C4 — C8 — C24 — C3×C24 — C32×Q16 — C32×Q32
 Lower central C1 — C2 — C4 — C8 — C32×Q32
 Upper central C1 — C3×C6 — C3×C12 — C3×C24 — C32×Q32

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 [×4], C4, C4 [×2], C6 [×4], C8, Q8 [×2], C32, C12 [×4], C12 [×8], C16, Q16 [×2], C3×C6, C24 [×4], C3×Q8 [×8], Q32, C3×C12, C3×C12 [×2], C48 [×4], C3×Q16 [×8], C3×C24, Q8×C32 [×2], C3×Q32 [×4], C3×C48, C32×Q16 [×2], C32×Q32
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], D4, C32, C2×C6 [×4], D8, C3×C6 [×3], C3×D4 [×4], Q32, C62, C3×D8 [×4], D4×C32, C3×Q32 [×4], C32×D8, C32×Q32

Smallest permutation representation of C32×Q32
Regular action on 288 points
Generators in S288
(1 280 60)(2 281 61)(3 282 62)(4 283 63)(5 284 64)(6 285 49)(7 286 50)(8 287 51)(9 288 52)(10 273 53)(11 274 54)(12 275 55)(13 276 56)(14 277 57)(15 278 58)(16 279 59)(17 190 41)(18 191 42)(19 192 43)(20 177 44)(21 178 45)(22 179 46)(23 180 47)(24 181 48)(25 182 33)(26 183 34)(27 184 35)(28 185 36)(29 186 37)(30 187 38)(31 188 39)(32 189 40)(65 127 105)(66 128 106)(67 113 107)(68 114 108)(69 115 109)(70 116 110)(71 117 111)(72 118 112)(73 119 97)(74 120 98)(75 121 99)(76 122 100)(77 123 101)(78 124 102)(79 125 103)(80 126 104)(81 233 206)(82 234 207)(83 235 208)(84 236 193)(85 237 194)(86 238 195)(87 239 196)(88 240 197)(89 225 198)(90 226 199)(91 227 200)(92 228 201)(93 229 202)(94 230 203)(95 231 204)(96 232 205)(129 242 220)(130 243 221)(131 244 222)(132 245 223)(133 246 224)(134 247 209)(135 248 210)(136 249 211)(137 250 212)(138 251 213)(139 252 214)(140 253 215)(141 254 216)(142 255 217)(143 256 218)(144 241 219)(145 169 267)(146 170 268)(147 171 269)(148 172 270)(149 173 271)(150 174 272)(151 175 257)(152 176 258)(153 161 259)(154 162 260)(155 163 261)(156 164 262)(157 165 263)(158 166 264)(159 167 265)(160 168 266)
(1 268 124)(2 269 125)(3 270 126)(4 271 127)(5 272 128)(6 257 113)(7 258 114)(8 259 115)(9 260 116)(10 261 117)(11 262 118)(12 263 119)(13 264 120)(14 265 121)(15 266 122)(16 267 123)(17 234 218)(18 235 219)(19 236 220)(20 237 221)(21 238 222)(22 239 223)(23 240 224)(24 225 209)(25 226 210)(26 227 211)(27 228 212)(28 229 213)(29 230 214)(30 231 215)(31 232 216)(32 233 217)(33 90 248)(34 91 249)(35 92 250)(36 93 251)(37 94 252)(38 95 253)(39 96 254)(40 81 255)(41 82 256)(42 83 241)(43 84 242)(44 85 243)(45 86 244)(46 87 245)(47 88 246)(48 89 247)(49 175 67)(50 176 68)(51 161 69)(52 162 70)(53 163 71)(54 164 72)(55 165 73)(56 166 74)(57 167 75)(58 168 76)(59 169 77)(60 170 78)(61 171 79)(62 172 80)(63 173 65)(64 174 66)(97 275 157)(98 276 158)(99 277 159)(100 278 160)(101 279 145)(102 280 146)(103 281 147)(104 282 148)(105 283 149)(106 284 150)(107 285 151)(108 286 152)(109 287 153)(110 288 154)(111 273 155)(112 274 156)(129 192 193)(130 177 194)(131 178 195)(132 179 196)(133 180 197)(134 181 198)(135 182 199)(136 183 200)(137 184 201)(138 185 202)(139 186 203)(140 187 204)(141 188 205)(142 189 206)(143 190 207)(144 191 208)
(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)
(1 181 9 189)(2 180 10 188)(3 179 11 187)(4 178 12 186)(5 177 13 185)(6 192 14 184)(7 191 15 183)(8 190 16 182)(17 59 25 51)(18 58 26 50)(19 57 27 49)(20 56 28 64)(21 55 29 63)(22 54 30 62)(23 53 31 61)(24 52 32 60)(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)(65 222 73 214)(66 221 74 213)(67 220 75 212)(68 219 76 211)(69 218 77 210)(70 217 78 209)(71 216 79 224)(72 215 80 223)(81 146 89 154)(82 145 90 153)(83 160 91 152)(84 159 92 151)(85 158 93 150)(86 157 94 149)(87 156 95 148)(88 155 96 147)(97 252 105 244)(98 251 106 243)(99 250 107 242)(100 249 108 241)(101 248 109 256)(102 247 110 255)(103 246 111 254)(104 245 112 253)(113 129 121 137)(114 144 122 136)(115 143 123 135)(116 142 124 134)(117 141 125 133)(118 140 126 132)(119 139 127 131)(120 138 128 130)(161 234 169 226)(162 233 170 225)(163 232 171 240)(164 231 172 239)(165 230 173 238)(166 229 174 237)(167 228 175 236)(168 227 176 235)(193 265 201 257)(194 264 202 272)(195 263 203 271)(196 262 204 270)(197 261 205 269)(198 260 206 268)(199 259 207 267)(200 258 208 266)

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

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

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

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

׿
×
𝔽