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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 [×2], C3 [×4], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C16 [×2], C2×C8, C3×C6, C3×C6 [×2], C24 [×8], C2×C12 [×4], C2×C16, C3×C12 [×2], C62, C3⋊C16 [×8], C2×C24 [×4], C3×C24 [×2], C6×C12, C2×C3⋊C16 [×4], C24.S3 [×2], C6×C24, C2×C24.S3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, Dic3 [×8], D6 [×4], C16 [×2], C2×C8, C3⋊S3, C3⋊C8 [×8], C2×Dic3 [×4], C2×C16, C3⋊Dic3 [×2], C2×C3⋊S3, C3⋊C16 [×8], C2×C3⋊C8 [×4], C324C8 [×2], C2×C3⋊Dic3, C2×C3⋊C16 [×4], C24.S3 [×2], C2×C324C8, C2×C24.S3

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

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

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

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

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