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## G = C24⋊Dic3order 288 = 25·32

### 5th semidirect product of C24 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for C24⋊Dic3
G = < a,b,c | a24=b6=1, c2=b3, ab=ba, cac-1=a5, cbc-1=b-1 >

Subgroups: 308 in 120 conjugacy classes, 73 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C42, C2×C8, C2×C8, C3×C6, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C8⋊C4, C3⋊Dic3, C3×C12, C62, C2×C3⋊C8, C4×Dic3, C2×C24, C324C8, C3×C24, C2×C3⋊Dic3, C6×C12, C24⋊C4, C2×C324C8, C4×C3⋊Dic3, C6×C24, C24⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, M4(2), C3⋊S3, C4×S3, C2×Dic3, C8⋊C4, C3⋊Dic3, C2×C3⋊S3, C8⋊S3, C4×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C24⋊C4, C24⋊S3, C4×C3⋊Dic3, C24⋊Dic3

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6L 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12P 24A ··· 24AF order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 8 8 8 8 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 2 2 1 1 1 1 18 18 18 18 2 ··· 2 2 2 2 2 18 18 18 18 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C2 C4 C4 C4 S3 Dic3 D6 M4(2) C4×S3 C4×S3 C8⋊S3 kernel C24⋊Dic3 C2×C32⋊4C8 C4×C3⋊Dic3 C6×C24 C32⋊4C8 C3×C24 C2×C3⋊Dic3 C2×C24 C24 C2×C12 C3×C6 C12 C2×C6 C6 # reps 1 1 1 1 4 4 4 4 8 4 4 8 8 32

Matrix representation of C24⋊Dic3 in GL6(𝔽73)

 27 71 0 0 0 0 59 46 0 0 0 0 0 0 0 46 0 0 0 0 27 46 0 0 0 0 0 0 16 8 0 0 0 0 65 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 1 1
,
 70 6 0 0 0 0 23 3 0 0 0 0 0 0 63 32 0 0 0 0 22 10 0 0 0 0 0 0 61 10 0 0 0 0 22 12

G:=sub<GL(6,GF(73))| [27,59,0,0,0,0,71,46,0,0,0,0,0,0,0,27,0,0,0,0,46,46,0,0,0,0,0,0,16,65,0,0,0,0,8,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,1],[70,23,0,0,0,0,6,3,0,0,0,0,0,0,63,22,0,0,0,0,32,10,0,0,0,0,0,0,61,22,0,0,0,0,10,12] >;

C24⋊Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,290);
// by ID

G=gap.SmallGroup(288,290);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,64,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=b^6=1,c^2=b^3,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations

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