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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, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C4⋊C4, C2×C8, C3×C6, C24, C2×Dic3, C2×C12, C2.D8, C3⋊Dic3, C3×C12, C62, C4⋊Dic3, C2×C24, C3×C24, C2×C3⋊Dic3, C6×C12, C241C4, C12⋊Dic3, C6×C24, C241Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, D8, Q16, C3⋊S3, Dic6, D12, C2×Dic3, C2.D8, C3⋊Dic3, C2×C3⋊S3, D24, Dic12, C4⋊Dic3, C324Q8, C12⋊S3, C2×C3⋊Dic3, C241C4, 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)
(1 158 242 201 75 185)(2 159 243 202 76 186)(3 160 244 203 77 187)(4 161 245 204 78 188)(5 162 246 205 79 189)(6 163 247 206 80 190)(7 164 248 207 81 191)(8 165 249 208 82 192)(9 166 250 209 83 169)(10 167 251 210 84 170)(11 168 252 211 85 171)(12 145 253 212 86 172)(13 146 254 213 87 173)(14 147 255 214 88 174)(15 148 256 215 89 175)(16 149 257 216 90 176)(17 150 258 193 91 177)(18 151 259 194 92 178)(19 152 260 195 93 179)(20 153 261 196 94 180)(21 154 262 197 95 181)(22 155 263 198 96 182)(23 156 264 199 73 183)(24 157 241 200 74 184)(25 271 120 132 62 231)(26 272 97 133 63 232)(27 273 98 134 64 233)(28 274 99 135 65 234)(29 275 100 136 66 235)(30 276 101 137 67 236)(31 277 102 138 68 237)(32 278 103 139 69 238)(33 279 104 140 70 239)(34 280 105 141 71 240)(35 281 106 142 72 217)(36 282 107 143 49 218)(37 283 108 144 50 219)(38 284 109 121 51 220)(39 285 110 122 52 221)(40 286 111 123 53 222)(41 287 112 124 54 223)(42 288 113 125 55 224)(43 265 114 126 56 225)(44 266 115 127 57 226)(45 267 116 128 58 227)(46 268 117 129 59 228)(47 269 118 130 60 229)(48 270 119 131 61 230)
(1 66 201 275)(2 65 202 274)(3 64 203 273)(4 63 204 272)(5 62 205 271)(6 61 206 270)(7 60 207 269)(8 59 208 268)(9 58 209 267)(10 57 210 266)(11 56 211 265)(12 55 212 288)(13 54 213 287)(14 53 214 286)(15 52 215 285)(16 51 216 284)(17 50 193 283)(18 49 194 282)(19 72 195 281)(20 71 196 280)(21 70 197 279)(22 69 198 278)(23 68 199 277)(24 67 200 276)(25 162 132 79)(26 161 133 78)(27 160 134 77)(28 159 135 76)(29 158 136 75)(30 157 137 74)(31 156 138 73)(32 155 139 96)(33 154 140 95)(34 153 141 94)(35 152 142 93)(36 151 143 92)(37 150 144 91)(38 149 121 90)(39 148 122 89)(40 147 123 88)(41 146 124 87)(42 145 125 86)(43 168 126 85)(44 167 127 84)(45 166 128 83)(46 165 129 82)(47 164 130 81)(48 163 131 80)(97 188 232 245)(98 187 233 244)(99 186 234 243)(100 185 235 242)(101 184 236 241)(102 183 237 264)(103 182 238 263)(104 181 239 262)(105 180 240 261)(106 179 217 260)(107 178 218 259)(108 177 219 258)(109 176 220 257)(110 175 221 256)(111 174 222 255)(112 173 223 254)(113 172 224 253)(114 171 225 252)(115 170 226 251)(116 169 227 250)(117 192 228 249)(118 191 229 248)(119 190 230 247)(120 189 231 246)

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

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

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

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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