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## G = C24⋊2Dic3order 288 = 25·32

### 2nd semidirect product of C24 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C24⋊2Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×C12 — C12⋊Dic3 — C24⋊2Dic3
 Lower central C32 — C3×C6 — C3×C12 — C24⋊2Dic3
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C242Dic3
G = < a,b,c | a24=b6=1, c2=b3, ab=ba, cac-1=a11, cbc-1=b-1 >

Subgroups: 396 in 108 conjugacy classes, 69 normal (15 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×8], C12 [×8], C2×C6 [×4], C4⋊C4 [×2], C2×C8, C3×C6, C3×C6 [×2], C24 [×8], C2×Dic3 [×8], C2×C12 [×4], C4.Q8, C3⋊Dic3 [×2], C3×C12 [×2], C62, C4⋊Dic3 [×8], C2×C24 [×4], C3×C24 [×2], C2×C3⋊Dic3 [×2], C6×C12, C8⋊Dic3 [×4], C12⋊Dic3 [×2], C6×C24, C242Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, Dic3 [×8], D6 [×4], C4⋊C4, SD16 [×2], C3⋊S3, Dic6 [×4], D12 [×4], C2×Dic3 [×4], C4.Q8, C3⋊Dic3 [×2], C2×C3⋊S3, C24⋊C2 [×8], C4⋊Dic3 [×4], C324Q8, C12⋊S3, C2×C3⋊Dic3, C8⋊Dic3 [×4], C242S3 [×2], C12⋊Dic3, C242Dic3

Smallest permutation representation of C242Dic3
Regular action on 288 points
Generators in S288
(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)
(1 197 131 90 244 116)(2 198 132 91 245 117)(3 199 133 92 246 118)(4 200 134 93 247 119)(5 201 135 94 248 120)(6 202 136 95 249 97)(7 203 137 96 250 98)(8 204 138 73 251 99)(9 205 139 74 252 100)(10 206 140 75 253 101)(11 207 141 76 254 102)(12 208 142 77 255 103)(13 209 143 78 256 104)(14 210 144 79 257 105)(15 211 121 80 258 106)(16 212 122 81 259 107)(17 213 123 82 260 108)(18 214 124 83 261 109)(19 215 125 84 262 110)(20 216 126 85 263 111)(21 193 127 86 264 112)(22 194 128 87 241 113)(23 195 129 88 242 114)(24 196 130 89 243 115)(25 231 274 155 173 59)(26 232 275 156 174 60)(27 233 276 157 175 61)(28 234 277 158 176 62)(29 235 278 159 177 63)(30 236 279 160 178 64)(31 237 280 161 179 65)(32 238 281 162 180 66)(33 239 282 163 181 67)(34 240 283 164 182 68)(35 217 284 165 183 69)(36 218 285 166 184 70)(37 219 286 167 185 71)(38 220 287 168 186 72)(39 221 288 145 187 49)(40 222 265 146 188 50)(41 223 266 147 189 51)(42 224 267 148 190 52)(43 225 268 149 191 53)(44 226 269 150 192 54)(45 227 270 151 169 55)(46 228 271 152 170 56)(47 229 272 153 171 57)(48 230 273 154 172 58)
(1 283 90 68)(2 270 91 55)(3 281 92 66)(4 268 93 53)(5 279 94 64)(6 266 95 51)(7 277 96 62)(8 288 73 49)(9 275 74 60)(10 286 75 71)(11 273 76 58)(12 284 77 69)(13 271 78 56)(14 282 79 67)(15 269 80 54)(16 280 81 65)(17 267 82 52)(18 278 83 63)(19 265 84 50)(20 276 85 61)(21 287 86 72)(22 274 87 59)(23 285 88 70)(24 272 89 57)(25 113 155 128)(26 100 156 139)(27 111 157 126)(28 98 158 137)(29 109 159 124)(30 120 160 135)(31 107 161 122)(32 118 162 133)(33 105 163 144)(34 116 164 131)(35 103 165 142)(36 114 166 129)(37 101 167 140)(38 112 168 127)(39 99 145 138)(40 110 146 125)(41 97 147 136)(42 108 148 123)(43 119 149 134)(44 106 150 121)(45 117 151 132)(46 104 152 143)(47 115 153 130)(48 102 154 141)(169 198 227 245)(170 209 228 256)(171 196 229 243)(172 207 230 254)(173 194 231 241)(174 205 232 252)(175 216 233 263)(176 203 234 250)(177 214 235 261)(178 201 236 248)(179 212 237 259)(180 199 238 246)(181 210 239 257)(182 197 240 244)(183 208 217 255)(184 195 218 242)(185 206 219 253)(186 193 220 264)(187 204 221 251)(188 215 222 262)(189 202 223 249)(190 213 224 260)(191 200 225 247)(192 211 226 258)

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A ··· 6L 8A 8B 8C 8D 12A ··· 12P 24A ··· 24AF order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 2 2 2 2 36 36 36 36 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + - + - + - + image C1 C2 C2 C4 S3 Q8 D4 Dic3 D6 SD16 Dic6 D12 C24⋊C2 kernel C24⋊2Dic3 C12⋊Dic3 C6×C24 C3×C24 C2×C24 C3×C12 C62 C24 C2×C12 C3×C6 C12 C2×C6 C6 # reps 1 2 1 4 4 1 1 8 4 4 8 8 32

Matrix representation of C242Dic3 in GL5(𝔽73)

 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 25 36 0 0 0 37 62
,
 72 0 0 0 0 0 72 1 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 72
,
 27 0 0 0 0 0 0 72 0 0 0 72 0 0 0 0 0 0 61 51 0 0 0 63 12

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,25,37,0,0,0,36,62],[72,0,0,0,0,0,72,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72],[27,0,0,0,0,0,0,72,0,0,0,72,0,0,0,0,0,0,61,63,0,0,0,51,12] >;

C242Dic3 in GAP, Magma, Sage, TeX

C_{24}\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("C24:2Dic3");
// GroupNames label

G:=SmallGroup(288,292);
// by ID

G=gap.SmallGroup(288,292);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,64,675,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=b^6=1,c^2=b^3,a*b=b*a,c*a*c^-1=a^11,c*b*c^-1=b^-1>;
// generators/relations

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