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## G = C2×C24.S3order 288 = 25·32

### Direct product of C2 and C24.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C24.S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C24.S3 — C2×C24.S3
 Lower central C32 — C2×C24.S3
 Upper central C1 — C2×C8

Generators and relations for C2×C24.S3
G = < a,b,c,d | a2=b24=c3=1, d2=b9, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b17, dcd-1=c-1 >

Subgroups: 132 in 84 conjugacy classes, 69 normal (19 characteristic)
C1, C2, C2, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C16, C2×C8, C3×C6, C3×C6, C24, C2×C12, C2×C16, C3×C12, C62, C3⋊C16, C2×C24, C3×C24, C6×C12, C2×C3⋊C16, C24.S3, C6×C24, C2×C24.S3
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C16, C2×C8, C3⋊S3, C3⋊C8, C2×Dic3, C2×C16, C3⋊Dic3, C2×C3⋊S3, C3⋊C16, C2×C3⋊C8, C324C8, C2×C3⋊Dic3, C2×C3⋊C16, C24.S3, C2×C324C8, C2×C24.S3

Smallest permutation representation of C2×C24.S3
Regular action on 288 points
Generators in S288
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 205)(26 206)(27 207)(28 208)(29 209)(30 210)(31 211)(32 212)(33 213)(34 214)(35 215)(36 216)(37 193)(38 194)(39 195)(40 196)(41 197)(42 198)(43 199)(44 200)(45 201)(46 202)(47 203)(48 204)(73 182)(74 183)(75 184)(76 185)(77 186)(78 187)(79 188)(80 189)(81 190)(82 191)(83 192)(84 169)(85 170)(86 171)(87 172)(88 173)(89 174)(90 175)(91 176)(92 177)(93 178)(94 179)(95 180)(96 181)(97 157)(98 158)(99 159)(100 160)(101 161)(102 162)(103 163)(104 164)(105 165)(106 166)(107 167)(108 168)(109 145)(110 146)(111 147)(112 148)(113 149)(114 150)(115 151)(116 152)(117 153)(118 154)(119 155)(120 156)(121 286)(122 287)(123 288)(124 265)(125 266)(126 267)(127 268)(128 269)(129 270)(130 271)(131 272)(132 273)(133 274)(134 275)(135 276)(136 277)(137 278)(138 279)(139 280)(140 281)(141 282)(142 283)(143 284)(144 285)(217 264)(218 241)(219 242)(220 243)(221 244)(222 245)(223 246)(224 247)(225 248)(226 249)(227 250)(228 251)(229 252)(230 253)(231 254)(232 255)(233 256)(234 257)(235 258)(236 259)(237 260)(238 261)(239 262)(240 263)
(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)
(1 151 171)(2 152 172)(3 153 173)(4 154 174)(5 155 175)(6 156 176)(7 157 177)(8 158 178)(9 159 179)(10 160 180)(11 161 181)(12 162 182)(13 163 183)(14 164 184)(15 165 185)(16 166 186)(17 167 187)(18 168 188)(19 145 189)(20 146 190)(21 147 191)(22 148 192)(23 149 169)(24 150 170)(25 288 243)(26 265 244)(27 266 245)(28 267 246)(29 268 247)(30 269 248)(31 270 249)(32 271 250)(33 272 251)(34 273 252)(35 274 253)(36 275 254)(37 276 255)(38 277 256)(39 278 257)(40 279 258)(41 280 259)(42 281 260)(43 282 261)(44 283 262)(45 284 263)(46 285 264)(47 286 241)(48 287 242)(49 101 96)(50 102 73)(51 103 74)(52 104 75)(53 105 76)(54 106 77)(55 107 78)(56 108 79)(57 109 80)(58 110 81)(59 111 82)(60 112 83)(61 113 84)(62 114 85)(63 115 86)(64 116 87)(65 117 88)(66 118 89)(67 119 90)(68 120 91)(69 97 92)(70 98 93)(71 99 94)(72 100 95)(121 218 203)(122 219 204)(123 220 205)(124 221 206)(125 222 207)(126 223 208)(127 224 209)(128 225 210)(129 226 211)(130 227 212)(131 228 213)(132 229 214)(133 230 215)(134 231 216)(135 232 193)(136 233 194)(137 234 195)(138 235 196)(139 236 197)(140 237 198)(141 238 199)(142 239 200)(143 240 201)(144 217 202)
(1 193 10 202 19 211 4 196 13 205 22 214 7 199 16 208)(2 210 11 195 20 204 5 213 14 198 23 207 8 216 17 201)(3 203 12 212 21 197 6 206 15 215 24 200 9 209 18 194)(25 60 34 69 43 54 28 63 37 72 46 57 31 66 40 51)(26 53 35 62 44 71 29 56 38 65 47 50 32 59 41 68)(27 70 36 55 45 64 30 49 39 58 48 67 33 52 42 61)(73 271 82 280 91 265 76 274 85 283 94 268 79 277 88 286)(74 288 83 273 92 282 77 267 86 276 95 285 80 270 89 279)(75 281 84 266 93 275 78 284 87 269 96 278 81 287 90 272)(97 261 106 246 115 255 100 264 109 249 118 258 103 243 112 252)(98 254 107 263 116 248 101 257 110 242 119 251 104 260 113 245)(99 247 108 256 117 241 102 250 111 259 120 244 105 253 114 262)(121 182 130 191 139 176 124 185 133 170 142 179 127 188 136 173)(122 175 131 184 140 169 125 178 134 187 143 172 128 181 137 190)(123 192 132 177 141 186 126 171 135 180 144 189 129 174 138 183)(145 226 154 235 163 220 148 229 157 238 166 223 151 232 160 217)(146 219 155 228 164 237 149 222 158 231 167 240 152 225 161 234)(147 236 156 221 165 230 150 239 159 224 168 233 153 218 162 227)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 8A ··· 8H 12A ··· 12P 16A ··· 16P 24A ··· 24AF order 1 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 16 ··· 16 24 ··· 24 size 1 1 1 1 2 2 2 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 C16 S3 Dic3 D6 Dic3 C3⋊C8 C3⋊C8 C3⋊C16 kernel C2×C24.S3 C24.S3 C6×C24 C3×C24 C6×C12 C3×C12 C62 C3×C6 C2×C24 C24 C24 C2×C12 C12 C2×C6 C6 # reps 1 2 1 2 2 4 4 16 4 4 4 4 8 8 32

Matrix representation of C2×C24.S3 in GL4(𝔽97) generated by

 96 0 0 0 0 96 0 0 0 0 96 0 0 0 0 96
,
 0 22 0 0 75 22 0 0 0 0 33 0 0 0 0 33
,
 1 0 0 0 0 1 0 0 0 0 0 96 0 0 1 96
,
 96 76 0 0 75 1 0 0 0 0 66 72 0 0 41 31
G:=sub<GL(4,GF(97))| [96,0,0,0,0,96,0,0,0,0,96,0,0,0,0,96],[0,75,0,0,22,22,0,0,0,0,33,0,0,0,0,33],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,96,96],[96,75,0,0,76,1,0,0,0,0,66,41,0,0,72,31] >;

C2×C24.S3 in GAP, Magma, Sage, TeX

C_2\times C_{24}.S_3
% in TeX

G:=Group("C2xC24.S3");
// GroupNames label

G:=SmallGroup(288,286);
// by ID

G=gap.SmallGroup(288,286);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,58,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^24=c^3=1,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^17,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽