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## G = C24⋊1Dic3order 288 = 25·32

### 1st semidirect product of C24 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C24⋊1Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×C12 — C12⋊Dic3 — C24⋊1Dic3
 Lower central C32 — C3×C6 — C3×C12 — C24⋊1Dic3
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C241Dic3
G = < a,b,c | a24=b6=1, c2=b3, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 396 in 108 conjugacy classes, 69 normal (19 characteristic)
C1, C2 [×3], 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 [×3], C24 [×8], C2×Dic3 [×8], C2×C12 [×4], C2.D8, C3⋊Dic3 [×2], C3×C12 [×2], C62, C4⋊Dic3 [×8], C2×C24 [×4], C3×C24 [×2], C2×C3⋊Dic3 [×2], C6×C12, C241C4 [×4], C12⋊Dic3 [×2], C6×C24, C241Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, Dic3 [×8], D6 [×4], C4⋊C4, D8, Q16, C3⋊S3, Dic6 [×4], D12 [×4], C2×Dic3 [×4], C2.D8, C3⋊Dic3 [×2], C2×C3⋊S3, D24 [×4], Dic12 [×4], C4⋊Dic3 [×4], C324Q8, C12⋊S3, C2×C3⋊Dic3, C241C4 [×4], C325D8, C325Q16, C12⋊Dic3, C241Dic3

Smallest permutation representation of C241Dic3
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)
(1 212 71 32 89 101)(2 213 72 33 90 102)(3 214 49 34 91 103)(4 215 50 35 92 104)(5 216 51 36 93 105)(6 193 52 37 94 106)(7 194 53 38 95 107)(8 195 54 39 96 108)(9 196 55 40 73 109)(10 197 56 41 74 110)(11 198 57 42 75 111)(12 199 58 43 76 112)(13 200 59 44 77 113)(14 201 60 45 78 114)(15 202 61 46 79 115)(16 203 62 47 80 116)(17 204 63 48 81 117)(18 205 64 25 82 118)(19 206 65 26 83 119)(20 207 66 27 84 120)(21 208 67 28 85 97)(22 209 68 29 86 98)(23 210 69 30 87 99)(24 211 70 31 88 100)(121 188 246 154 284 231)(122 189 247 155 285 232)(123 190 248 156 286 233)(124 191 249 157 287 234)(125 192 250 158 288 235)(126 169 251 159 265 236)(127 170 252 160 266 237)(128 171 253 161 267 238)(129 172 254 162 268 239)(130 173 255 163 269 240)(131 174 256 164 270 217)(132 175 257 165 271 218)(133 176 258 166 272 219)(134 177 259 167 273 220)(135 178 260 168 274 221)(136 179 261 145 275 222)(137 180 262 146 276 223)(138 181 263 147 277 224)(139 182 264 148 278 225)(140 183 241 149 279 226)(141 184 242 150 280 227)(142 185 243 151 281 228)(143 186 244 152 282 229)(144 187 245 153 283 230)
(1 138 32 147)(2 137 33 146)(3 136 34 145)(4 135 35 168)(5 134 36 167)(6 133 37 166)(7 132 38 165)(8 131 39 164)(9 130 40 163)(10 129 41 162)(11 128 42 161)(12 127 43 160)(13 126 44 159)(14 125 45 158)(15 124 46 157)(16 123 47 156)(17 122 48 155)(18 121 25 154)(19 144 26 153)(20 143 27 152)(21 142 28 151)(22 141 29 150)(23 140 30 149)(24 139 31 148)(49 275 103 179)(50 274 104 178)(51 273 105 177)(52 272 106 176)(53 271 107 175)(54 270 108 174)(55 269 109 173)(56 268 110 172)(57 267 111 171)(58 266 112 170)(59 265 113 169)(60 288 114 192)(61 287 115 191)(62 286 116 190)(63 285 117 189)(64 284 118 188)(65 283 119 187)(66 282 120 186)(67 281 97 185)(68 280 98 184)(69 279 99 183)(70 278 100 182)(71 277 101 181)(72 276 102 180)(73 255 196 240)(74 254 197 239)(75 253 198 238)(76 252 199 237)(77 251 200 236)(78 250 201 235)(79 249 202 234)(80 248 203 233)(81 247 204 232)(82 246 205 231)(83 245 206 230)(84 244 207 229)(85 243 208 228)(86 242 209 227)(87 241 210 226)(88 264 211 225)(89 263 212 224)(90 262 213 223)(91 261 214 222)(92 260 215 221)(93 259 216 220)(94 258 193 219)(95 257 194 218)(96 256 195 217)

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

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

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

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 2 2 type + + + + - + - + + - - + + - image C1 C2 C2 C4 S3 Q8 D4 Dic3 D6 D8 Q16 Dic6 D12 D24 Dic12 kernel C24⋊1Dic3 C12⋊Dic3 C6×C24 C3×C24 C2×C24 C3×C12 C62 C24 C2×C12 C3×C6 C3×C6 C12 C2×C6 C6 C6 # reps 1 2 1 4 4 1 1 8 4 2 2 8 8 16 16

Matrix representation of C241Dic3 in GL5(𝔽73)

 1 0 0 0 0 0 68 18 0 0 0 55 50 0 0 0 0 0 18 23 0 0 0 50 68
,
 72 0 0 0 0 0 0 1 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 72 72
,
 27 0 0 0 0 0 3 31 0 0 0 28 70 0 0 0 0 0 3 31 0 0 0 28 70

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,68,55,0,0,0,18,50,0,0,0,0,0,18,50,0,0,0,23,68],[72,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,1,72],[27,0,0,0,0,0,3,28,0,0,0,31,70,0,0,0,0,0,3,28,0,0,0,31,70] >;

C241Dic3 in GAP, Magma, Sage, TeX

C_{24}\rtimes_1{\rm Dic}_3
% in TeX

G:=Group("C24:1Dic3");
// GroupNames label

G:=SmallGroup(288,293);
// by ID

G=gap.SmallGroup(288,293);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,176,675,80,2693,9414]);
// Polycyclic

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

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