metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C72⋊2C4, C8⋊2Dic9, C36.4Q8, C4.4Dic18, C24.4Dic3, C18.2SD16, C22.8D36, C12.19Dic6, (C2×C8).6D9, C9⋊2(C4.Q8), (C2×C72).8C2, C18.6(C4⋊C4), C3.(C8⋊Dic3), C36.36(C2×C4), (C2×C24).17S3, (C2×C4).74D18, (C2×C18).13D4, (C2×C6).21D12, C4⋊Dic9.2C2, C4.7(C2×Dic9), C6.2(C24⋊C2), (C2×C12).363D6, C2.2(C72⋊C2), C2.4(C4⋊Dic9), C6.8(C4⋊Dic3), (C2×C36).82C22, C12.42(C2×Dic3), SmallGroup(288,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊Dic9
G = < a,b,c | a8=b18=1, c2=b9, ab=ba, cac-1=a3, cbc-1=b-1 >
Smallest permutation representation of C8⋊Dic9
►Regular action on 288 points
Generators in S288
(1 85 32 145 256 245 71 48)(2 86 33 146 257 246 72 49)(3 87 34 147 258 247 55 50)(4 88 35 148 259 248 56 51)(5 89 36 149 260 249 57 52)(6 90 19 150 261 250 58 53)(7 73 20 151 262 251 59 54)(8 74 21 152 263 252 60 37)(9 75 22 153 264 235 61 38)(10 76 23 154 265 236 62 39)(11 77 24 155 266 237 63 40)(12 78 25 156 267 238 64 41)(13 79 26 157 268 239 65 42)(14 80 27 158 269 240 66 43)(15 81 28 159 270 241 67 44)(16 82 29 160 253 242 68 45)(17 83 30 161 254 243 69 46)(18 84 31 162 255 244 70 47)(91 142 274 203 164 119 191 228)(92 143 275 204 165 120 192 229)(93 144 276 205 166 121 193 230)(94 127 277 206 167 122 194 231)(95 128 278 207 168 123 195 232)(96 129 279 208 169 124 196 233)(97 130 280 209 170 125 197 234)(98 131 281 210 171 126 198 217)(99 132 282 211 172 109 181 218)(100 133 283 212 173 110 182 219)(101 134 284 213 174 111 183 220)(102 135 285 214 175 112 184 221)(103 136 286 215 176 113 185 222)(104 137 287 216 177 114 186 223)(105 138 288 199 178 115 187 224)(106 139 271 200 179 116 188 225)(107 140 272 201 180 117 189 226)(108 141 273 202 163 118 190 227)
(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)
(1 278 10 287)(2 277 11 286)(3 276 12 285)(4 275 13 284)(5 274 14 283)(6 273 15 282)(7 272 16 281)(8 271 17 280)(9 288 18 279)(19 108 28 99)(20 107 29 98)(21 106 30 97)(22 105 31 96)(23 104 32 95)(24 103 33 94)(25 102 34 93)(26 101 35 92)(27 100 36 91)(37 225 46 234)(38 224 47 233)(39 223 48 232)(40 222 49 231)(41 221 50 230)(42 220 51 229)(43 219 52 228)(44 218 53 227)(45 217 54 226)(55 166 64 175)(56 165 65 174)(57 164 66 173)(58 163 67 172)(59 180 68 171)(60 179 69 170)(61 178 70 169)(62 177 71 168)(63 176 72 167)(73 117 82 126)(74 116 83 125)(75 115 84 124)(76 114 85 123)(77 113 86 122)(78 112 87 121)(79 111 88 120)(80 110 89 119)(81 109 90 118)(127 237 136 246)(128 236 137 245)(129 235 138 244)(130 252 139 243)(131 251 140 242)(132 250 141 241)(133 249 142 240)(134 248 143 239)(135 247 144 238)(145 207 154 216)(146 206 155 215)(147 205 156 214)(148 204 157 213)(149 203 158 212)(150 202 159 211)(151 201 160 210)(152 200 161 209)(153 199 162 208)(181 261 190 270)(182 260 191 269)(183 259 192 268)(184 258 193 267)(185 257 194 266)(186 256 195 265)(187 255 196 264)(188 254 197 263)(189 253 198 262)
G:=sub<Sym(288)| (1,85,32,145,256,245,71,48)(2,86,33,146,257,246,72,49)(3,87,34,147,258,247,55,50)(4,88,35,148,259,248,56,51)(5,89,36,149,260,249,57,52)(6,90,19,150,261,250,58,53)(7,73,20,151,262,251,59,54)(8,74,21,152,263,252,60,37)(9,75,22,153,264,235,61,38)(10,76,23,154,265,236,62,39)(11,77,24,155,266,237,63,40)(12,78,25,156,267,238,64,41)(13,79,26,157,268,239,65,42)(14,80,27,158,269,240,66,43)(15,81,28,159,270,241,67,44)(16,82,29,160,253,242,68,45)(17,83,30,161,254,243,69,46)(18,84,31,162,255,244,70,47)(91,142,274,203,164,119,191,228)(92,143,275,204,165,120,192,229)(93,144,276,205,166,121,193,230)(94,127,277,206,167,122,194,231)(95,128,278,207,168,123,195,232)(96,129,279,208,169,124,196,233)(97,130,280,209,170,125,197,234)(98,131,281,210,171,126,198,217)(99,132,282,211,172,109,181,218)(100,133,283,212,173,110,182,219)(101,134,284,213,174,111,183,220)(102,135,285,214,175,112,184,221)(103,136,286,215,176,113,185,222)(104,137,287,216,177,114,186,223)(105,138,288,199,178,115,187,224)(106,139,271,200,179,116,188,225)(107,140,272,201,180,117,189,226)(108,141,273,202,163,118,190,227), (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), (1,278,10,287)(2,277,11,286)(3,276,12,285)(4,275,13,284)(5,274,14,283)(6,273,15,282)(7,272,16,281)(8,271,17,280)(9,288,18,279)(19,108,28,99)(20,107,29,98)(21,106,30,97)(22,105,31,96)(23,104,32,95)(24,103,33,94)(25,102,34,93)(26,101,35,92)(27,100,36,91)(37,225,46,234)(38,224,47,233)(39,223,48,232)(40,222,49,231)(41,221,50,230)(42,220,51,229)(43,219,52,228)(44,218,53,227)(45,217,54,226)(55,166,64,175)(56,165,65,174)(57,164,66,173)(58,163,67,172)(59,180,68,171)(60,179,69,170)(61,178,70,169)(62,177,71,168)(63,176,72,167)(73,117,82,126)(74,116,83,125)(75,115,84,124)(76,114,85,123)(77,113,86,122)(78,112,87,121)(79,111,88,120)(80,110,89,119)(81,109,90,118)(127,237,136,246)(128,236,137,245)(129,235,138,244)(130,252,139,243)(131,251,140,242)(132,250,141,241)(133,249,142,240)(134,248,143,239)(135,247,144,238)(145,207,154,216)(146,206,155,215)(147,205,156,214)(148,204,157,213)(149,203,158,212)(150,202,159,211)(151,201,160,210)(152,200,161,209)(153,199,162,208)(181,261,190,270)(182,260,191,269)(183,259,192,268)(184,258,193,267)(185,257,194,266)(186,256,195,265)(187,255,196,264)(188,254,197,263)(189,253,198,262)>;
G:=Group( (1,85,32,145,256,245,71,48)(2,86,33,146,257,246,72,49)(3,87,34,147,258,247,55,50)(4,88,35,148,259,248,56,51)(5,89,36,149,260,249,57,52)(6,90,19,150,261,250,58,53)(7,73,20,151,262,251,59,54)(8,74,21,152,263,252,60,37)(9,75,22,153,264,235,61,38)(10,76,23,154,265,236,62,39)(11,77,24,155,266,237,63,40)(12,78,25,156,267,238,64,41)(13,79,26,157,268,239,65,42)(14,80,27,158,269,240,66,43)(15,81,28,159,270,241,67,44)(16,82,29,160,253,242,68,45)(17,83,30,161,254,243,69,46)(18,84,31,162,255,244,70,47)(91,142,274,203,164,119,191,228)(92,143,275,204,165,120,192,229)(93,144,276,205,166,121,193,230)(94,127,277,206,167,122,194,231)(95,128,278,207,168,123,195,232)(96,129,279,208,169,124,196,233)(97,130,280,209,170,125,197,234)(98,131,281,210,171,126,198,217)(99,132,282,211,172,109,181,218)(100,133,283,212,173,110,182,219)(101,134,284,213,174,111,183,220)(102,135,285,214,175,112,184,221)(103,136,286,215,176,113,185,222)(104,137,287,216,177,114,186,223)(105,138,288,199,178,115,187,224)(106,139,271,200,179,116,188,225)(107,140,272,201,180,117,189,226)(108,141,273,202,163,118,190,227), (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), (1,278,10,287)(2,277,11,286)(3,276,12,285)(4,275,13,284)(5,274,14,283)(6,273,15,282)(7,272,16,281)(8,271,17,280)(9,288,18,279)(19,108,28,99)(20,107,29,98)(21,106,30,97)(22,105,31,96)(23,104,32,95)(24,103,33,94)(25,102,34,93)(26,101,35,92)(27,100,36,91)(37,225,46,234)(38,224,47,233)(39,223,48,232)(40,222,49,231)(41,221,50,230)(42,220,51,229)(43,219,52,228)(44,218,53,227)(45,217,54,226)(55,166,64,175)(56,165,65,174)(57,164,66,173)(58,163,67,172)(59,180,68,171)(60,179,69,170)(61,178,70,169)(62,177,71,168)(63,176,72,167)(73,117,82,126)(74,116,83,125)(75,115,84,124)(76,114,85,123)(77,113,86,122)(78,112,87,121)(79,111,88,120)(80,110,89,119)(81,109,90,118)(127,237,136,246)(128,236,137,245)(129,235,138,244)(130,252,139,243)(131,251,140,242)(132,250,141,241)(133,249,142,240)(134,248,143,239)(135,247,144,238)(145,207,154,216)(146,206,155,215)(147,205,156,214)(148,204,157,213)(149,203,158,212)(150,202,159,211)(151,201,160,210)(152,200,161,209)(153,199,162,208)(181,261,190,270)(182,260,191,269)(183,259,192,268)(184,258,193,267)(185,257,194,266)(186,256,195,265)(187,255,196,264)(188,254,197,263)(189,253,198,262) );
G=PermutationGroup([[(1,85,32,145,256,245,71,48),(2,86,33,146,257,246,72,49),(3,87,34,147,258,247,55,50),(4,88,35,148,259,248,56,51),(5,89,36,149,260,249,57,52),(6,90,19,150,261,250,58,53),(7,73,20,151,262,251,59,54),(8,74,21,152,263,252,60,37),(9,75,22,153,264,235,61,38),(10,76,23,154,265,236,62,39),(11,77,24,155,266,237,63,40),(12,78,25,156,267,238,64,41),(13,79,26,157,268,239,65,42),(14,80,27,158,269,240,66,43),(15,81,28,159,270,241,67,44),(16,82,29,160,253,242,68,45),(17,83,30,161,254,243,69,46),(18,84,31,162,255,244,70,47),(91,142,274,203,164,119,191,228),(92,143,275,204,165,120,192,229),(93,144,276,205,166,121,193,230),(94,127,277,206,167,122,194,231),(95,128,278,207,168,123,195,232),(96,129,279,208,169,124,196,233),(97,130,280,209,170,125,197,234),(98,131,281,210,171,126,198,217),(99,132,282,211,172,109,181,218),(100,133,283,212,173,110,182,219),(101,134,284,213,174,111,183,220),(102,135,285,214,175,112,184,221),(103,136,286,215,176,113,185,222),(104,137,287,216,177,114,186,223),(105,138,288,199,178,115,187,224),(106,139,271,200,179,116,188,225),(107,140,272,201,180,117,189,226),(108,141,273,202,163,118,190,227)], [(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)], [(1,278,10,287),(2,277,11,286),(3,276,12,285),(4,275,13,284),(5,274,14,283),(6,273,15,282),(7,272,16,281),(8,271,17,280),(9,288,18,279),(19,108,28,99),(20,107,29,98),(21,106,30,97),(22,105,31,96),(23,104,32,95),(24,103,33,94),(25,102,34,93),(26,101,35,92),(27,100,36,91),(37,225,46,234),(38,224,47,233),(39,223,48,232),(40,222,49,231),(41,221,50,230),(42,220,51,229),(43,219,52,228),(44,218,53,227),(45,217,54,226),(55,166,64,175),(56,165,65,174),(57,164,66,173),(58,163,67,172),(59,180,68,171),(60,179,69,170),(61,178,70,169),(62,177,71,168),(63,176,72,167),(73,117,82,126),(74,116,83,125),(75,115,84,124),(76,114,85,123),(77,113,86,122),(78,112,87,121),(79,111,88,120),(80,110,89,119),(81,109,90,118),(127,237,136,246),(128,236,137,245),(129,235,138,244),(130,252,139,243),(131,251,140,242),(132,250,141,241),(133,249,142,240),(134,248,143,239),(135,247,144,238),(145,207,154,216),(146,206,155,215),(147,205,156,214),(148,204,157,213),(149,203,158,212),(150,202,159,211),(151,201,160,210),(152,200,161,209),(153,199,162,208),(181,261,190,270),(182,260,191,269),(183,259,192,268),(184,258,193,267),(185,257,194,266),(186,256,195,265),(187,255,196,264),(188,254,197,263),(189,253,198,262)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 36 | 36 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | + | - | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | S3 | Q8 | D4 | Dic3 | D6 | SD16 | D9 | Dic6 | D12 | Dic9 | D18 | C24⋊C2 | Dic18 | D36 | C72⋊C2 |
| kernel | C8⋊Dic9 | C4⋊Dic9 | C2×C72 | C72 | C2×C24 | C36 | C2×C18 | C24 | C2×C12 | C18 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 4 | 3 | 2 | 2 | 6 | 3 | 8 | 6 | 6 | 24 |
Matrix representation of C8⋊Dic9 ►in GL4(𝔽73) generated by
| 48 | 11 | 0 | 0 |
| 62 | 37 | 0 | 0 |
| 0 | 0 | 48 | 62 |
| 0 | 0 | 11 | 37 |
| 0 | 72 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 70 | 45 |
| 0 | 0 | 28 | 42 |
| 51 | 63 | 0 | 0 |
| 12 | 22 | 0 | 0 |
| 0 | 0 | 23 | 68 |
| 0 | 0 | 18 | 50 |
G:=sub<GL(4,GF(73))| [48,62,0,0,11,37,0,0,0,0,48,11,0,0,62,37],[0,1,0,0,72,1,0,0,0,0,70,28,0,0,45,42],[51,12,0,0,63,22,0,0,0,0,23,18,0,0,68,50] >;
C8⋊Dic9 in GAP, Magma, Sage, TeX
C_8\rtimes {\rm Dic}_9 % in TeX
G:=Group("C8:Dic9"); // GroupNames label
G:=SmallGroup(288,25);
// by ID
G=gap.SmallGroup(288,25);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,64,675,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^8=b^18=1,c^2=b^9,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations
Export