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G = C24⋊Dic3order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: C245Dic3, (C3×C24)⋊12C4, C12.80(C4×S3), (C6×C24).23C2, (C2×C24).33S3, C83(C3⋊Dic3), C324C88C4, C33(C24⋊C4), C6.9(C8⋊S3), C328(C8⋊C4), (C2×C12).419D6, (C3×C6).19C42, C62.73(C2×C4), C6.15(C4×Dic3), C2.2(C24⋊S3), C12.59(C2×Dic3), (C3×C6).16M4(2), (C6×C12).341C22, C4.21(C4×C3⋊S3), (C2×C8).8(C3⋊S3), (C2×C6).48(C4×S3), C2.4(C4×C3⋊Dic3), C22.10(C4×C3⋊S3), C4.12(C2×C3⋊Dic3), (C3×C12).111(C2×C4), (C2×C3⋊Dic3).11C4, (C4×C3⋊Dic3).13C2, (C2×C324C8).18C2, (C2×C4).92(C2×C3⋊S3), SmallGroup(288,290)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C24⋊Dic3
C1C3C32C3×C6C62C6×C12C4×C3⋊Dic3 — C24⋊Dic3
C32C3×C6 — C24⋊Dic3
C1C2×C4C2×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 [×2], C3 [×4], C4 [×2], C4 [×2], C22, C6 [×12], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×8], C12 [×8], C2×C6 [×4], C42, C2×C8, C2×C8, C3×C6, C3×C6 [×2], C3⋊C8 [×8], C24 [×8], C2×Dic3 [×8], C2×C12 [×4], C8⋊C4, C3⋊Dic3 [×2], C3×C12 [×2], C62, C2×C3⋊C8 [×4], C4×Dic3 [×4], C2×C24 [×4], C324C8 [×2], C3×C24 [×2], C2×C3⋊Dic3 [×2], C6×C12, C24⋊C4 [×4], C2×C324C8, C4×C3⋊Dic3, C6×C24, C24⋊Dic3
Quotients: C1, C2 [×3], C4 [×6], C22, S3 [×4], C2×C4 [×3], Dic3 [×8], D6 [×4], C42, M4(2) [×2], C3⋊S3, C4×S3 [×8], C2×Dic3 [×4], C8⋊C4, C3⋊Dic3 [×2], C2×C3⋊S3, C8⋊S3 [×8], C4×Dic3 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C24⋊C4 [×4], C24⋊S3 [×2], 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)
(1 113 87 35 60 269)(2 114 88 36 61 270)(3 115 89 37 62 271)(4 116 90 38 63 272)(5 117 91 39 64 273)(6 118 92 40 65 274)(7 119 93 41 66 275)(8 120 94 42 67 276)(9 97 95 43 68 277)(10 98 96 44 69 278)(11 99 73 45 70 279)(12 100 74 46 71 280)(13 101 75 47 72 281)(14 102 76 48 49 282)(15 103 77 25 50 283)(16 104 78 26 51 284)(17 105 79 27 52 285)(18 106 80 28 53 286)(19 107 81 29 54 287)(20 108 82 30 55 288)(21 109 83 31 56 265)(22 110 84 32 57 266)(23 111 85 33 58 267)(24 112 86 34 59 268)(121 258 187 217 202 163)(122 259 188 218 203 164)(123 260 189 219 204 165)(124 261 190 220 205 166)(125 262 191 221 206 167)(126 263 192 222 207 168)(127 264 169 223 208 145)(128 241 170 224 209 146)(129 242 171 225 210 147)(130 243 172 226 211 148)(131 244 173 227 212 149)(132 245 174 228 213 150)(133 246 175 229 214 151)(134 247 176 230 215 152)(135 248 177 231 216 153)(136 249 178 232 193 154)(137 250 179 233 194 155)(138 251 180 234 195 156)(139 252 181 235 196 157)(140 253 182 236 197 158)(141 254 183 237 198 159)(142 255 184 238 199 160)(143 256 185 239 200 161)(144 257 186 240 201 162)
(1 230 35 134)(2 235 36 139)(3 240 37 144)(4 221 38 125)(5 226 39 130)(6 231 40 135)(7 236 41 140)(8 217 42 121)(9 222 43 126)(10 227 44 131)(11 232 45 136)(12 237 46 141)(13 218 47 122)(14 223 48 127)(15 228 25 132)(16 233 26 137)(17 238 27 142)(18 219 28 123)(19 224 29 128)(20 229 30 133)(21 234 31 138)(22 239 32 143)(23 220 33 124)(24 225 34 129)(49 145 102 169)(50 150 103 174)(51 155 104 179)(52 160 105 184)(53 165 106 189)(54 146 107 170)(55 151 108 175)(56 156 109 180)(57 161 110 185)(58 166 111 190)(59 147 112 171)(60 152 113 176)(61 157 114 181)(62 162 115 186)(63 167 116 191)(64 148 117 172)(65 153 118 177)(66 158 119 182)(67 163 120 187)(68 168 97 192)(69 149 98 173)(70 154 99 178)(71 159 100 183)(72 164 101 188)(73 249 279 193)(74 254 280 198)(75 259 281 203)(76 264 282 208)(77 245 283 213)(78 250 284 194)(79 255 285 199)(80 260 286 204)(81 241 287 209)(82 246 288 214)(83 251 265 195)(84 256 266 200)(85 261 267 205)(86 242 268 210)(87 247 269 215)(88 252 270 196)(89 257 271 201)(90 262 272 206)(91 243 273 211)(92 248 274 216)(93 253 275 197)(94 258 276 202)(95 263 277 207)(96 244 278 212)

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

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

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

84 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F4G4H6A···6L8A8B8C8D8E8F8G8H12A···12P24A···24AF
order12223333444444446···68888888812···1224···24
size111122221111181818182···22222181818182···22···2

84 irreducible representations

dim11111112222222
type+++++-+
imageC1C2C2C2C4C4C4S3Dic3D6M4(2)C4×S3C4×S3C8⋊S3
kernelC24⋊Dic3C2×C324C8C4×C3⋊Dic3C6×C24C324C8C3×C24C2×C3⋊Dic3C2×C24C24C2×C12C3×C6C12C2×C6C6
# reps111144448448832

Matrix representation of C24⋊Dic3 in GL6(𝔽73)

27710000
59460000
0004600
00274600
0000168
0000658
,
100000
010000
0072100
0072000
0000072
000011
,
7060000
2330000
00633200
00221000
00006110
00002212

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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