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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, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C4⋊C4, C2×C8, C3×C6, C3×C6, C24, C2×Dic3, C2×C12, C4.Q8, C3⋊Dic3, C3×C12, C62, C4⋊Dic3, C2×C24, C3×C24, C2×C3⋊Dic3, C6×C12, C8⋊Dic3, C12⋊Dic3, C6×C24, C242Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, SD16, C3⋊S3, Dic6, D12, C2×Dic3, C4.Q8, C3⋊Dic3, C2×C3⋊S3, C24⋊C2, C4⋊Dic3, C324Q8, C12⋊S3, C2×C3⋊Dic3, C8⋊Dic3, C242S3, 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)
(1 125 66 192 147 217)(2 126 67 169 148 218)(3 127 68 170 149 219)(4 128 69 171 150 220)(5 129 70 172 151 221)(6 130 71 173 152 222)(7 131 72 174 153 223)(8 132 49 175 154 224)(9 133 50 176 155 225)(10 134 51 177 156 226)(11 135 52 178 157 227)(12 136 53 179 158 228)(13 137 54 180 159 229)(14 138 55 181 160 230)(15 139 56 182 161 231)(16 140 57 183 162 232)(17 141 58 184 163 233)(18 142 59 185 164 234)(19 143 60 186 165 235)(20 144 61 187 166 236)(21 121 62 188 167 237)(22 122 63 189 168 238)(23 123 64 190 145 239)(24 124 65 191 146 240)(25 95 275 100 196 248)(26 96 276 101 197 249)(27 73 277 102 198 250)(28 74 278 103 199 251)(29 75 279 104 200 252)(30 76 280 105 201 253)(31 77 281 106 202 254)(32 78 282 107 203 255)(33 79 283 108 204 256)(34 80 284 109 205 257)(35 81 285 110 206 258)(36 82 286 111 207 259)(37 83 287 112 208 260)(38 84 288 113 209 261)(39 85 265 114 210 262)(40 86 266 115 211 263)(41 87 267 116 212 264)(42 88 268 117 213 241)(43 89 269 118 214 242)(44 90 270 119 215 243)(45 91 271 120 216 244)(46 92 272 97 193 245)(47 93 273 98 194 246)(48 94 274 99 195 247)
(1 46 192 97)(2 33 169 108)(3 44 170 119)(4 31 171 106)(5 42 172 117)(6 29 173 104)(7 40 174 115)(8 27 175 102)(9 38 176 113)(10 25 177 100)(11 36 178 111)(12 47 179 98)(13 34 180 109)(14 45 181 120)(15 32 182 107)(16 43 183 118)(17 30 184 105)(18 41 185 116)(19 28 186 103)(20 39 187 114)(21 26 188 101)(22 37 189 112)(23 48 190 99)(24 35 191 110)(49 198 224 73)(50 209 225 84)(51 196 226 95)(52 207 227 82)(53 194 228 93)(54 205 229 80)(55 216 230 91)(56 203 231 78)(57 214 232 89)(58 201 233 76)(59 212 234 87)(60 199 235 74)(61 210 236 85)(62 197 237 96)(63 208 238 83)(64 195 239 94)(65 206 240 81)(66 193 217 92)(67 204 218 79)(68 215 219 90)(69 202 220 77)(70 213 221 88)(71 200 222 75)(72 211 223 86)(121 249 167 276)(122 260 168 287)(123 247 145 274)(124 258 146 285)(125 245 147 272)(126 256 148 283)(127 243 149 270)(128 254 150 281)(129 241 151 268)(130 252 152 279)(131 263 153 266)(132 250 154 277)(133 261 155 288)(134 248 156 275)(135 259 157 286)(136 246 158 273)(137 257 159 284)(138 244 160 271)(139 255 161 282)(140 242 162 269)(141 253 163 280)(142 264 164 267)(143 251 165 278)(144 262 166 265)

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

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

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

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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