metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic9⋊C8, C36.8Q8, C36.52D4, C4.8Dic18, C12.27Dic6, C18.1M4(2), C9⋊2(C4⋊C8), C6.9(S3×C8), (C2×C8).1D9, C2.4(C8×D9), (C2×C24).1S3, C18.4(C2×C8), (C2×C72).1C2, C18.5(C4⋊C4), (C2×C4).91D18, C3.(Dic3⋊C8), C22.9(C4×D9), C6.4(C8⋊S3), (C2×C12).405D6, C2.1(C8⋊D9), C4.26(C9⋊D4), (C4×Dic9).5C2, (C2×Dic9).2C4, C2.1(Dic9⋊C4), C12.121(C3⋊D4), C6.12(Dic3⋊C4), (C2×C36).103C22, (C2×C9⋊C8).9C2, (C2×C6).35(C4×S3), (C2×C18).10(C2×C4), SmallGroup(288,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic9⋊C8
G = < a,b,c | a18=c8=1, b2=a9, bab-1=a-1, ac=ca, cbc-1=a9b >
Smallest permutation representation of Dic9⋊C8
►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)
(1 273 10 282)(2 272 11 281)(3 271 12 280)(4 288 13 279)(5 287 14 278)(6 286 15 277)(7 285 16 276)(8 284 17 275)(9 283 18 274)(19 175 28 166)(20 174 29 165)(21 173 30 164)(22 172 31 163)(23 171 32 180)(24 170 33 179)(25 169 34 178)(26 168 35 177)(27 167 36 176)(37 71 46 62)(38 70 47 61)(39 69 48 60)(40 68 49 59)(41 67 50 58)(42 66 51 57)(43 65 52 56)(44 64 53 55)(45 63 54 72)(73 206 82 215)(74 205 83 214)(75 204 84 213)(76 203 85 212)(77 202 86 211)(78 201 87 210)(79 200 88 209)(80 199 89 208)(81 216 90 207)(91 217 100 226)(92 234 101 225)(93 233 102 224)(94 232 103 223)(95 231 104 222)(96 230 105 221)(97 229 106 220)(98 228 107 219)(99 227 108 218)(109 242 118 251)(110 241 119 250)(111 240 120 249)(112 239 121 248)(113 238 122 247)(114 237 123 246)(115 236 124 245)(116 235 125 244)(117 252 126 243)(127 195 136 186)(128 194 137 185)(129 193 138 184)(130 192 139 183)(131 191 140 182)(132 190 141 181)(133 189 142 198)(134 188 143 197)(135 187 144 196)(145 261 154 270)(146 260 155 269)(147 259 156 268)(148 258 157 267)(149 257 158 266)(150 256 159 265)(151 255 160 264)(152 254 161 263)(153 253 162 262)
(1 249 226 90 181 35 158 62)(2 250 227 73 182 36 159 63)(3 251 228 74 183 19 160 64)(4 252 229 75 184 20 161 65)(5 235 230 76 185 21 162 66)(6 236 231 77 186 22 145 67)(7 237 232 78 187 23 146 68)(8 238 233 79 188 24 147 69)(9 239 234 80 189 25 148 70)(10 240 217 81 190 26 149 71)(11 241 218 82 191 27 150 72)(12 242 219 83 192 28 151 55)(13 243 220 84 193 29 152 56)(14 244 221 85 194 30 153 57)(15 245 222 86 195 31 154 58)(16 246 223 87 196 32 155 59)(17 247 224 88 197 33 156 60)(18 248 225 89 198 34 157 61)(37 282 111 100 207 141 177 257)(38 283 112 101 208 142 178 258)(39 284 113 102 209 143 179 259)(40 285 114 103 210 144 180 260)(41 286 115 104 211 127 163 261)(42 287 116 105 212 128 164 262)(43 288 117 106 213 129 165 263)(44 271 118 107 214 130 166 264)(45 272 119 108 215 131 167 265)(46 273 120 91 216 132 168 266)(47 274 121 92 199 133 169 267)(48 275 122 93 200 134 170 268)(49 276 123 94 201 135 171 269)(50 277 124 95 202 136 172 270)(51 278 125 96 203 137 173 253)(52 279 126 97 204 138 174 254)(53 280 109 98 205 139 175 255)(54 281 110 99 206 140 176 256)
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), (1,273,10,282)(2,272,11,281)(3,271,12,280)(4,288,13,279)(5,287,14,278)(6,286,15,277)(7,285,16,276)(8,284,17,275)(9,283,18,274)(19,175,28,166)(20,174,29,165)(21,173,30,164)(22,172,31,163)(23,171,32,180)(24,170,33,179)(25,169,34,178)(26,168,35,177)(27,167,36,176)(37,71,46,62)(38,70,47,61)(39,69,48,60)(40,68,49,59)(41,67,50,58)(42,66,51,57)(43,65,52,56)(44,64,53,55)(45,63,54,72)(73,206,82,215)(74,205,83,214)(75,204,84,213)(76,203,85,212)(77,202,86,211)(78,201,87,210)(79,200,88,209)(80,199,89,208)(81,216,90,207)(91,217,100,226)(92,234,101,225)(93,233,102,224)(94,232,103,223)(95,231,104,222)(96,230,105,221)(97,229,106,220)(98,228,107,219)(99,227,108,218)(109,242,118,251)(110,241,119,250)(111,240,120,249)(112,239,121,248)(113,238,122,247)(114,237,123,246)(115,236,124,245)(116,235,125,244)(117,252,126,243)(127,195,136,186)(128,194,137,185)(129,193,138,184)(130,192,139,183)(131,191,140,182)(132,190,141,181)(133,189,142,198)(134,188,143,197)(135,187,144,196)(145,261,154,270)(146,260,155,269)(147,259,156,268)(148,258,157,267)(149,257,158,266)(150,256,159,265)(151,255,160,264)(152,254,161,263)(153,253,162,262), (1,249,226,90,181,35,158,62)(2,250,227,73,182,36,159,63)(3,251,228,74,183,19,160,64)(4,252,229,75,184,20,161,65)(5,235,230,76,185,21,162,66)(6,236,231,77,186,22,145,67)(7,237,232,78,187,23,146,68)(8,238,233,79,188,24,147,69)(9,239,234,80,189,25,148,70)(10,240,217,81,190,26,149,71)(11,241,218,82,191,27,150,72)(12,242,219,83,192,28,151,55)(13,243,220,84,193,29,152,56)(14,244,221,85,194,30,153,57)(15,245,222,86,195,31,154,58)(16,246,223,87,196,32,155,59)(17,247,224,88,197,33,156,60)(18,248,225,89,198,34,157,61)(37,282,111,100,207,141,177,257)(38,283,112,101,208,142,178,258)(39,284,113,102,209,143,179,259)(40,285,114,103,210,144,180,260)(41,286,115,104,211,127,163,261)(42,287,116,105,212,128,164,262)(43,288,117,106,213,129,165,263)(44,271,118,107,214,130,166,264)(45,272,119,108,215,131,167,265)(46,273,120,91,216,132,168,266)(47,274,121,92,199,133,169,267)(48,275,122,93,200,134,170,268)(49,276,123,94,201,135,171,269)(50,277,124,95,202,136,172,270)(51,278,125,96,203,137,173,253)(52,279,126,97,204,138,174,254)(53,280,109,98,205,139,175,255)(54,281,110,99,206,140,176,256)>;
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), (1,273,10,282)(2,272,11,281)(3,271,12,280)(4,288,13,279)(5,287,14,278)(6,286,15,277)(7,285,16,276)(8,284,17,275)(9,283,18,274)(19,175,28,166)(20,174,29,165)(21,173,30,164)(22,172,31,163)(23,171,32,180)(24,170,33,179)(25,169,34,178)(26,168,35,177)(27,167,36,176)(37,71,46,62)(38,70,47,61)(39,69,48,60)(40,68,49,59)(41,67,50,58)(42,66,51,57)(43,65,52,56)(44,64,53,55)(45,63,54,72)(73,206,82,215)(74,205,83,214)(75,204,84,213)(76,203,85,212)(77,202,86,211)(78,201,87,210)(79,200,88,209)(80,199,89,208)(81,216,90,207)(91,217,100,226)(92,234,101,225)(93,233,102,224)(94,232,103,223)(95,231,104,222)(96,230,105,221)(97,229,106,220)(98,228,107,219)(99,227,108,218)(109,242,118,251)(110,241,119,250)(111,240,120,249)(112,239,121,248)(113,238,122,247)(114,237,123,246)(115,236,124,245)(116,235,125,244)(117,252,126,243)(127,195,136,186)(128,194,137,185)(129,193,138,184)(130,192,139,183)(131,191,140,182)(132,190,141,181)(133,189,142,198)(134,188,143,197)(135,187,144,196)(145,261,154,270)(146,260,155,269)(147,259,156,268)(148,258,157,267)(149,257,158,266)(150,256,159,265)(151,255,160,264)(152,254,161,263)(153,253,162,262), (1,249,226,90,181,35,158,62)(2,250,227,73,182,36,159,63)(3,251,228,74,183,19,160,64)(4,252,229,75,184,20,161,65)(5,235,230,76,185,21,162,66)(6,236,231,77,186,22,145,67)(7,237,232,78,187,23,146,68)(8,238,233,79,188,24,147,69)(9,239,234,80,189,25,148,70)(10,240,217,81,190,26,149,71)(11,241,218,82,191,27,150,72)(12,242,219,83,192,28,151,55)(13,243,220,84,193,29,152,56)(14,244,221,85,194,30,153,57)(15,245,222,86,195,31,154,58)(16,246,223,87,196,32,155,59)(17,247,224,88,197,33,156,60)(18,248,225,89,198,34,157,61)(37,282,111,100,207,141,177,257)(38,283,112,101,208,142,178,258)(39,284,113,102,209,143,179,259)(40,285,114,103,210,144,180,260)(41,286,115,104,211,127,163,261)(42,287,116,105,212,128,164,262)(43,288,117,106,213,129,165,263)(44,271,118,107,214,130,166,264)(45,272,119,108,215,131,167,265)(46,273,120,91,216,132,168,266)(47,274,121,92,199,133,169,267)(48,275,122,93,200,134,170,268)(49,276,123,94,201,135,171,269)(50,277,124,95,202,136,172,270)(51,278,125,96,203,137,173,253)(52,279,126,97,204,138,174,254)(53,280,109,98,205,139,175,255)(54,281,110,99,206,140,176,256) );
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)], [(1,273,10,282),(2,272,11,281),(3,271,12,280),(4,288,13,279),(5,287,14,278),(6,286,15,277),(7,285,16,276),(8,284,17,275),(9,283,18,274),(19,175,28,166),(20,174,29,165),(21,173,30,164),(22,172,31,163),(23,171,32,180),(24,170,33,179),(25,169,34,178),(26,168,35,177),(27,167,36,176),(37,71,46,62),(38,70,47,61),(39,69,48,60),(40,68,49,59),(41,67,50,58),(42,66,51,57),(43,65,52,56),(44,64,53,55),(45,63,54,72),(73,206,82,215),(74,205,83,214),(75,204,84,213),(76,203,85,212),(77,202,86,211),(78,201,87,210),(79,200,88,209),(80,199,89,208),(81,216,90,207),(91,217,100,226),(92,234,101,225),(93,233,102,224),(94,232,103,223),(95,231,104,222),(96,230,105,221),(97,229,106,220),(98,228,107,219),(99,227,108,218),(109,242,118,251),(110,241,119,250),(111,240,120,249),(112,239,121,248),(113,238,122,247),(114,237,123,246),(115,236,124,245),(116,235,125,244),(117,252,126,243),(127,195,136,186),(128,194,137,185),(129,193,138,184),(130,192,139,183),(131,191,140,182),(132,190,141,181),(133,189,142,198),(134,188,143,197),(135,187,144,196),(145,261,154,270),(146,260,155,269),(147,259,156,268),(148,258,157,267),(149,257,158,266),(150,256,159,265),(151,255,160,264),(152,254,161,263),(153,253,162,262)], [(1,249,226,90,181,35,158,62),(2,250,227,73,182,36,159,63),(3,251,228,74,183,19,160,64),(4,252,229,75,184,20,161,65),(5,235,230,76,185,21,162,66),(6,236,231,77,186,22,145,67),(7,237,232,78,187,23,146,68),(8,238,233,79,188,24,147,69),(9,239,234,80,189,25,148,70),(10,240,217,81,190,26,149,71),(11,241,218,82,191,27,150,72),(12,242,219,83,192,28,151,55),(13,243,220,84,193,29,152,56),(14,244,221,85,194,30,153,57),(15,245,222,86,195,31,154,58),(16,246,223,87,196,32,155,59),(17,247,224,88,197,33,156,60),(18,248,225,89,198,34,157,61),(37,282,111,100,207,141,177,257),(38,283,112,101,208,142,178,258),(39,284,113,102,209,143,179,259),(40,285,114,103,210,144,180,260),(41,286,115,104,211,127,163,261),(42,287,116,105,212,128,164,262),(43,288,117,106,213,129,165,263),(44,271,118,107,214,130,166,264),(45,272,119,108,215,131,167,265),(46,273,120,91,216,132,168,266),(47,274,121,92,199,133,169,267),(48,275,122,93,200,134,170,268),(49,276,123,94,201,135,171,269),(50,277,124,95,202,136,172,270),(51,278,125,96,203,137,173,253),(52,279,126,97,204,138,174,254),(53,280,109,98,205,139,175,255),(54,281,110,99,206,140,176,256)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 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 | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C8 | S3 | D4 | Q8 | D6 | M4(2) | D9 | Dic6 | C3⋊D4 | C4×S3 | D18 | S3×C8 | C8⋊S3 | Dic18 | C9⋊D4 | C4×D9 | C8×D9 | C8⋊D9 |
| kernel | Dic9⋊C8 | C2×C9⋊C8 | C4×Dic9 | C2×C72 | C2×Dic9 | Dic9 | C2×C24 | C36 | C36 | C2×C12 | C18 | C2×C8 | C12 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 12 | 12 |
Matrix representation of Dic9⋊C8 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 31 | 28 |
| 0 | 45 | 3 |
| 72 | 0 | 0 |
| 0 | 2 | 55 |
| 0 | 53 | 71 |
| 51 | 0 | 0 |
| 0 | 70 | 67 |
| 0 | 6 | 3 |
G:=sub<GL(3,GF(73))| [1,0,0,0,31,45,0,28,3],[72,0,0,0,2,53,0,55,71],[51,0,0,0,70,6,0,67,3] >;
Dic9⋊C8 in GAP, Magma, Sage, TeX
{\rm Dic}_9\rtimes C_8 % in TeX
G:=Group("Dic9:C8"); // GroupNames label
G:=SmallGroup(288,22);
// by ID
G=gap.SmallGroup(288,22);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^18=c^8=1,b^2=a^9,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^9*b>;
// generators/relations
Export