Aliases: Q8⋊Dic9, C6.1GL2(𝔽3), C6.1CSU2(𝔽3), Q8⋊C9⋊1C4, (C2×C6).6S4, C2.(Q8⋊D9), (C2×Q8).1D9, (C6×Q8).1S3, C6.2(A4⋊C4), C2.(Q8.D9), C3.(Q8⋊Dic3), (C3×Q8).1Dic3, C2.2(C6.S4), C22.3(C3.S4), (C2×Q8⋊C9).1C2, SmallGroup(288,69)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8⋊C9 — Q8⋊Dic9 |
Generators and relations for Q8⋊Dic9
G = < a,b,c,d | a4=c18=1, b2=a2, d2=c9, bab-1=dbd-1=a-1, cac-1=b, dad-1=a2b, cbc-1=ab, dcd-1=c-1 >
Character table of Q8⋊Dic9
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 18A | 18B | 18C | 18D | 18E | 18F | 18G | 18H | 18I | |
| size | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 36 | 36 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 8 | 8 | 8 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | -1 | 1 | -1 | i | -i | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | -1 | 1 | -1 | -i | i | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ8 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ9 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -√2 | -√2 | √2 | √2 | -1 | -1 | -1 | 0 | 0 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | 1 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ95-ζ94 | ζ97+ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | symplectic lifted from Dic9, Schur index 2 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | √2 | √2 | -√2 | -√2 | -1 | -1 | -1 | 0 | 0 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ13 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | 1 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ97-ζ92 | ζ98+ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | symplectic lifted from Dic9, Schur index 2 |
| ρ14 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | 1 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ98-ζ9 | ζ95+ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | symplectic lifted from Dic9, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | √-2 | -√-2 | √-2 | -√-2 | -1 | -1 | -1 | 0 | 0 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | complex lifted from GL2(𝔽3) |
| ρ16 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -√-2 | √-2 | -√-2 | √-2 | -1 | -1 | -1 | 0 | 0 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | complex lifted from GL2(𝔽3) |
| ρ17 | 3 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ18 | 3 | 3 | 3 | 3 | 3 | -1 | -1 | 1 | 1 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ19 | 3 | 3 | -3 | -3 | 3 | -1 | 1 | -i | i | -3 | 3 | -3 | i | -i | -i | i | 0 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ20 | 3 | 3 | -3 | -3 | 3 | -1 | 1 | i | -i | -3 | 3 | -3 | -i | i | i | -i | 0 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ21 | 4 | -4 | -4 | 4 | 4 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | orthogonal lifted from GL2(𝔽3) |
| ρ22 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | 0 | 0 | ζ97+ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ98-ζ9 | ζ95+ζ94 | ζ98+ζ9 | -ζ97-ζ92 | orthogonal lifted from Q8⋊D9 |
| ρ23 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | 0 | 0 | ζ98+ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ95-ζ94 | ζ97+ζ92 | ζ95+ζ94 | -ζ98-ζ9 | orthogonal lifted from Q8⋊D9 |
| ρ24 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | 0 | 0 | ζ95+ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ97-ζ92 | ζ98+ζ9 | ζ97+ζ92 | -ζ95-ζ94 | orthogonal lifted from Q8⋊D9 |
| ρ25 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | 0 | 0 | -ζ98-ζ9 | ζ97+ζ92 | -ζ97-ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | -ζ95-ζ94 | ζ98+ζ9 | symplectic lifted from Q8.D9, Schur index 2 |
| ρ26 | 4 | -4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ27 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | 0 | 0 | -ζ97-ζ92 | ζ95+ζ94 | -ζ95-ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | -ζ98-ζ9 | ζ97+ζ92 | symplectic lifted from Q8.D9, Schur index 2 |
| ρ28 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | 0 | 0 | -ζ95-ζ94 | ζ98+ζ9 | -ζ98-ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | -ζ97-ζ92 | ζ95+ζ94 | symplectic lifted from Q8.D9, Schur index 2 |
| ρ29 | 6 | 6 | 6 | 6 | -3 | -2 | -2 | 0 | 0 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
| ρ30 | 6 | 6 | -6 | -6 | -3 | -2 | 2 | 0 | 0 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C6.S4, Schur index 2 |
►Regular action on 288 points
Generators in S288
(1 181 173 80)(2 96 174 59)(3 272 175 215)(4 184 176 83)(5 99 177 62)(6 275 178 200)(7 187 179 86)(8 102 180 65)(9 278 163 203)(10 190 164 89)(11 105 165 68)(12 281 166 206)(13 193 167 74)(14 108 168 71)(15 284 169 209)(16 196 170 77)(17 93 171 56)(18 287 172 212)(19 254 137 120)(20 243 138 50)(21 161 139 231)(22 257 140 123)(23 246 141 53)(24 146 142 234)(25 260 143 126)(26 249 144 38)(27 149 127 219)(28 263 128 111)(29 252 129 41)(30 152 130 222)(31 266 131 114)(32 237 132 44)(33 155 133 225)(34 269 134 117)(35 240 135 47)(36 158 136 228)(37 147 248 217)(39 110 250 262)(40 150 251 220)(42 113 235 265)(43 153 236 223)(45 116 238 268)(46 156 239 226)(48 119 241 253)(49 159 242 229)(51 122 244 256)(52 162 245 232)(54 125 247 259)(55 285 92 210)(57 79 94 198)(58 288 95 213)(60 82 97 183)(61 273 98 216)(63 85 100 186)(64 276 101 201)(66 88 103 189)(67 279 104 204)(69 73 106 192)(70 282 107 207)(72 76 91 195)(75 208 194 283)(78 211 197 286)(81 214 182 271)(84 199 185 274)(87 202 188 277)(90 205 191 280)(109 218 261 148)(112 221 264 151)(115 224 267 154)(118 227 270 157)(121 230 255 160)(124 233 258 145)
(1 95 173 58)(2 271 174 214)(3 183 175 82)(4 98 176 61)(5 274 177 199)(6 186 178 85)(7 101 179 64)(8 277 180 202)(9 189 163 88)(10 104 164 67)(11 280 165 205)(12 192 166 73)(13 107 167 70)(14 283 168 208)(15 195 169 76)(16 92 170 55)(17 286 171 211)(18 198 172 79)(19 242 137 49)(20 160 138 230)(21 256 139 122)(22 245 140 52)(23 145 141 233)(24 259 142 125)(25 248 143 37)(26 148 144 218)(27 262 127 110)(28 251 128 40)(29 151 129 221)(30 265 130 113)(31 236 131 43)(32 154 132 224)(33 268 133 116)(34 239 134 46)(35 157 135 227)(36 253 136 119)(38 109 249 261)(39 149 250 219)(41 112 252 264)(42 152 235 222)(44 115 237 267)(45 155 238 225)(47 118 240 270)(48 158 241 228)(50 121 243 255)(51 161 244 231)(53 124 246 258)(54 146 247 234)(56 78 93 197)(57 287 94 212)(59 81 96 182)(60 272 97 215)(62 84 99 185)(63 275 100 200)(65 87 102 188)(66 278 103 203)(68 90 105 191)(69 281 106 206)(71 75 108 194)(72 284 91 209)(74 207 193 282)(77 210 196 285)(80 213 181 288)(83 216 184 273)(86 201 187 276)(89 204 190 279)(111 220 263 150)(114 223 266 153)(117 226 269 156)(120 229 254 159)(123 232 257 162)(126 217 260 147)
(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)
(1 238 10 247)(2 237 11 246)(3 236 12 245)(4 235 13 244)(5 252 14 243)(6 251 15 242)(7 250 16 241)(8 249 17 240)(9 248 18 239)(19 275 28 284)(20 274 29 283)(21 273 30 282)(22 272 31 281)(23 271 32 280)(24 288 33 279)(25 287 34 278)(26 286 35 277)(27 285 36 276)(37 172 46 163)(38 171 47 180)(39 170 48 179)(40 169 49 178)(41 168 50 177)(42 167 51 176)(43 166 52 175)(44 165 53 174)(45 164 54 173)(55 253 64 262)(56 270 65 261)(57 269 66 260)(58 268 67 259)(59 267 68 258)(60 266 69 257)(61 265 70 256)(62 264 71 255)(63 263 72 254)(73 232 82 223)(74 231 83 222)(75 230 84 221)(76 229 85 220)(77 228 86 219)(78 227 87 218)(79 226 88 217)(80 225 89 234)(81 224 90 233)(91 120 100 111)(92 119 101 110)(93 118 102 109)(94 117 103 126)(95 116 104 125)(96 115 105 124)(97 114 106 123)(98 113 107 122)(99 112 108 121)(127 210 136 201)(128 209 137 200)(129 208 138 199)(130 207 139 216)(131 206 140 215)(132 205 141 214)(133 204 142 213)(134 203 143 212)(135 202 144 211)(145 182 154 191)(146 181 155 190)(147 198 156 189)(148 197 157 188)(149 196 158 187)(150 195 159 186)(151 194 160 185)(152 193 161 184)(153 192 162 183)
G:=sub<Sym(288)| (1,181,173,80)(2,96,174,59)(3,272,175,215)(4,184,176,83)(5,99,177,62)(6,275,178,200)(7,187,179,86)(8,102,180,65)(9,278,163,203)(10,190,164,89)(11,105,165,68)(12,281,166,206)(13,193,167,74)(14,108,168,71)(15,284,169,209)(16,196,170,77)(17,93,171,56)(18,287,172,212)(19,254,137,120)(20,243,138,50)(21,161,139,231)(22,257,140,123)(23,246,141,53)(24,146,142,234)(25,260,143,126)(26,249,144,38)(27,149,127,219)(28,263,128,111)(29,252,129,41)(30,152,130,222)(31,266,131,114)(32,237,132,44)(33,155,133,225)(34,269,134,117)(35,240,135,47)(36,158,136,228)(37,147,248,217)(39,110,250,262)(40,150,251,220)(42,113,235,265)(43,153,236,223)(45,116,238,268)(46,156,239,226)(48,119,241,253)(49,159,242,229)(51,122,244,256)(52,162,245,232)(54,125,247,259)(55,285,92,210)(57,79,94,198)(58,288,95,213)(60,82,97,183)(61,273,98,216)(63,85,100,186)(64,276,101,201)(66,88,103,189)(67,279,104,204)(69,73,106,192)(70,282,107,207)(72,76,91,195)(75,208,194,283)(78,211,197,286)(81,214,182,271)(84,199,185,274)(87,202,188,277)(90,205,191,280)(109,218,261,148)(112,221,264,151)(115,224,267,154)(118,227,270,157)(121,230,255,160)(124,233,258,145), (1,95,173,58)(2,271,174,214)(3,183,175,82)(4,98,176,61)(5,274,177,199)(6,186,178,85)(7,101,179,64)(8,277,180,202)(9,189,163,88)(10,104,164,67)(11,280,165,205)(12,192,166,73)(13,107,167,70)(14,283,168,208)(15,195,169,76)(16,92,170,55)(17,286,171,211)(18,198,172,79)(19,242,137,49)(20,160,138,230)(21,256,139,122)(22,245,140,52)(23,145,141,233)(24,259,142,125)(25,248,143,37)(26,148,144,218)(27,262,127,110)(28,251,128,40)(29,151,129,221)(30,265,130,113)(31,236,131,43)(32,154,132,224)(33,268,133,116)(34,239,134,46)(35,157,135,227)(36,253,136,119)(38,109,249,261)(39,149,250,219)(41,112,252,264)(42,152,235,222)(44,115,237,267)(45,155,238,225)(47,118,240,270)(48,158,241,228)(50,121,243,255)(51,161,244,231)(53,124,246,258)(54,146,247,234)(56,78,93,197)(57,287,94,212)(59,81,96,182)(60,272,97,215)(62,84,99,185)(63,275,100,200)(65,87,102,188)(66,278,103,203)(68,90,105,191)(69,281,106,206)(71,75,108,194)(72,284,91,209)(74,207,193,282)(77,210,196,285)(80,213,181,288)(83,216,184,273)(86,201,187,276)(89,204,190,279)(111,220,263,150)(114,223,266,153)(117,226,269,156)(120,229,254,159)(123,232,257,162)(126,217,260,147), (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), (1,238,10,247)(2,237,11,246)(3,236,12,245)(4,235,13,244)(5,252,14,243)(6,251,15,242)(7,250,16,241)(8,249,17,240)(9,248,18,239)(19,275,28,284)(20,274,29,283)(21,273,30,282)(22,272,31,281)(23,271,32,280)(24,288,33,279)(25,287,34,278)(26,286,35,277)(27,285,36,276)(37,172,46,163)(38,171,47,180)(39,170,48,179)(40,169,49,178)(41,168,50,177)(42,167,51,176)(43,166,52,175)(44,165,53,174)(45,164,54,173)(55,253,64,262)(56,270,65,261)(57,269,66,260)(58,268,67,259)(59,267,68,258)(60,266,69,257)(61,265,70,256)(62,264,71,255)(63,263,72,254)(73,232,82,223)(74,231,83,222)(75,230,84,221)(76,229,85,220)(77,228,86,219)(78,227,87,218)(79,226,88,217)(80,225,89,234)(81,224,90,233)(91,120,100,111)(92,119,101,110)(93,118,102,109)(94,117,103,126)(95,116,104,125)(96,115,105,124)(97,114,106,123)(98,113,107,122)(99,112,108,121)(127,210,136,201)(128,209,137,200)(129,208,138,199)(130,207,139,216)(131,206,140,215)(132,205,141,214)(133,204,142,213)(134,203,143,212)(135,202,144,211)(145,182,154,191)(146,181,155,190)(147,198,156,189)(148,197,157,188)(149,196,158,187)(150,195,159,186)(151,194,160,185)(152,193,161,184)(153,192,162,183)>;
G:=Group( (1,181,173,80)(2,96,174,59)(3,272,175,215)(4,184,176,83)(5,99,177,62)(6,275,178,200)(7,187,179,86)(8,102,180,65)(9,278,163,203)(10,190,164,89)(11,105,165,68)(12,281,166,206)(13,193,167,74)(14,108,168,71)(15,284,169,209)(16,196,170,77)(17,93,171,56)(18,287,172,212)(19,254,137,120)(20,243,138,50)(21,161,139,231)(22,257,140,123)(23,246,141,53)(24,146,142,234)(25,260,143,126)(26,249,144,38)(27,149,127,219)(28,263,128,111)(29,252,129,41)(30,152,130,222)(31,266,131,114)(32,237,132,44)(33,155,133,225)(34,269,134,117)(35,240,135,47)(36,158,136,228)(37,147,248,217)(39,110,250,262)(40,150,251,220)(42,113,235,265)(43,153,236,223)(45,116,238,268)(46,156,239,226)(48,119,241,253)(49,159,242,229)(51,122,244,256)(52,162,245,232)(54,125,247,259)(55,285,92,210)(57,79,94,198)(58,288,95,213)(60,82,97,183)(61,273,98,216)(63,85,100,186)(64,276,101,201)(66,88,103,189)(67,279,104,204)(69,73,106,192)(70,282,107,207)(72,76,91,195)(75,208,194,283)(78,211,197,286)(81,214,182,271)(84,199,185,274)(87,202,188,277)(90,205,191,280)(109,218,261,148)(112,221,264,151)(115,224,267,154)(118,227,270,157)(121,230,255,160)(124,233,258,145), (1,95,173,58)(2,271,174,214)(3,183,175,82)(4,98,176,61)(5,274,177,199)(6,186,178,85)(7,101,179,64)(8,277,180,202)(9,189,163,88)(10,104,164,67)(11,280,165,205)(12,192,166,73)(13,107,167,70)(14,283,168,208)(15,195,169,76)(16,92,170,55)(17,286,171,211)(18,198,172,79)(19,242,137,49)(20,160,138,230)(21,256,139,122)(22,245,140,52)(23,145,141,233)(24,259,142,125)(25,248,143,37)(26,148,144,218)(27,262,127,110)(28,251,128,40)(29,151,129,221)(30,265,130,113)(31,236,131,43)(32,154,132,224)(33,268,133,116)(34,239,134,46)(35,157,135,227)(36,253,136,119)(38,109,249,261)(39,149,250,219)(41,112,252,264)(42,152,235,222)(44,115,237,267)(45,155,238,225)(47,118,240,270)(48,158,241,228)(50,121,243,255)(51,161,244,231)(53,124,246,258)(54,146,247,234)(56,78,93,197)(57,287,94,212)(59,81,96,182)(60,272,97,215)(62,84,99,185)(63,275,100,200)(65,87,102,188)(66,278,103,203)(68,90,105,191)(69,281,106,206)(71,75,108,194)(72,284,91,209)(74,207,193,282)(77,210,196,285)(80,213,181,288)(83,216,184,273)(86,201,187,276)(89,204,190,279)(111,220,263,150)(114,223,266,153)(117,226,269,156)(120,229,254,159)(123,232,257,162)(126,217,260,147), (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), (1,238,10,247)(2,237,11,246)(3,236,12,245)(4,235,13,244)(5,252,14,243)(6,251,15,242)(7,250,16,241)(8,249,17,240)(9,248,18,239)(19,275,28,284)(20,274,29,283)(21,273,30,282)(22,272,31,281)(23,271,32,280)(24,288,33,279)(25,287,34,278)(26,286,35,277)(27,285,36,276)(37,172,46,163)(38,171,47,180)(39,170,48,179)(40,169,49,178)(41,168,50,177)(42,167,51,176)(43,166,52,175)(44,165,53,174)(45,164,54,173)(55,253,64,262)(56,270,65,261)(57,269,66,260)(58,268,67,259)(59,267,68,258)(60,266,69,257)(61,265,70,256)(62,264,71,255)(63,263,72,254)(73,232,82,223)(74,231,83,222)(75,230,84,221)(76,229,85,220)(77,228,86,219)(78,227,87,218)(79,226,88,217)(80,225,89,234)(81,224,90,233)(91,120,100,111)(92,119,101,110)(93,118,102,109)(94,117,103,126)(95,116,104,125)(96,115,105,124)(97,114,106,123)(98,113,107,122)(99,112,108,121)(127,210,136,201)(128,209,137,200)(129,208,138,199)(130,207,139,216)(131,206,140,215)(132,205,141,214)(133,204,142,213)(134,203,143,212)(135,202,144,211)(145,182,154,191)(146,181,155,190)(147,198,156,189)(148,197,157,188)(149,196,158,187)(150,195,159,186)(151,194,160,185)(152,193,161,184)(153,192,162,183) );
G=PermutationGroup([[(1,181,173,80),(2,96,174,59),(3,272,175,215),(4,184,176,83),(5,99,177,62),(6,275,178,200),(7,187,179,86),(8,102,180,65),(9,278,163,203),(10,190,164,89),(11,105,165,68),(12,281,166,206),(13,193,167,74),(14,108,168,71),(15,284,169,209),(16,196,170,77),(17,93,171,56),(18,287,172,212),(19,254,137,120),(20,243,138,50),(21,161,139,231),(22,257,140,123),(23,246,141,53),(24,146,142,234),(25,260,143,126),(26,249,144,38),(27,149,127,219),(28,263,128,111),(29,252,129,41),(30,152,130,222),(31,266,131,114),(32,237,132,44),(33,155,133,225),(34,269,134,117),(35,240,135,47),(36,158,136,228),(37,147,248,217),(39,110,250,262),(40,150,251,220),(42,113,235,265),(43,153,236,223),(45,116,238,268),(46,156,239,226),(48,119,241,253),(49,159,242,229),(51,122,244,256),(52,162,245,232),(54,125,247,259),(55,285,92,210),(57,79,94,198),(58,288,95,213),(60,82,97,183),(61,273,98,216),(63,85,100,186),(64,276,101,201),(66,88,103,189),(67,279,104,204),(69,73,106,192),(70,282,107,207),(72,76,91,195),(75,208,194,283),(78,211,197,286),(81,214,182,271),(84,199,185,274),(87,202,188,277),(90,205,191,280),(109,218,261,148),(112,221,264,151),(115,224,267,154),(118,227,270,157),(121,230,255,160),(124,233,258,145)], [(1,95,173,58),(2,271,174,214),(3,183,175,82),(4,98,176,61),(5,274,177,199),(6,186,178,85),(7,101,179,64),(8,277,180,202),(9,189,163,88),(10,104,164,67),(11,280,165,205),(12,192,166,73),(13,107,167,70),(14,283,168,208),(15,195,169,76),(16,92,170,55),(17,286,171,211),(18,198,172,79),(19,242,137,49),(20,160,138,230),(21,256,139,122),(22,245,140,52),(23,145,141,233),(24,259,142,125),(25,248,143,37),(26,148,144,218),(27,262,127,110),(28,251,128,40),(29,151,129,221),(30,265,130,113),(31,236,131,43),(32,154,132,224),(33,268,133,116),(34,239,134,46),(35,157,135,227),(36,253,136,119),(38,109,249,261),(39,149,250,219),(41,112,252,264),(42,152,235,222),(44,115,237,267),(45,155,238,225),(47,118,240,270),(48,158,241,228),(50,121,243,255),(51,161,244,231),(53,124,246,258),(54,146,247,234),(56,78,93,197),(57,287,94,212),(59,81,96,182),(60,272,97,215),(62,84,99,185),(63,275,100,200),(65,87,102,188),(66,278,103,203),(68,90,105,191),(69,281,106,206),(71,75,108,194),(72,284,91,209),(74,207,193,282),(77,210,196,285),(80,213,181,288),(83,216,184,273),(86,201,187,276),(89,204,190,279),(111,220,263,150),(114,223,266,153),(117,226,269,156),(120,229,254,159),(123,232,257,162),(126,217,260,147)], [(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)], [(1,238,10,247),(2,237,11,246),(3,236,12,245),(4,235,13,244),(5,252,14,243),(6,251,15,242),(7,250,16,241),(8,249,17,240),(9,248,18,239),(19,275,28,284),(20,274,29,283),(21,273,30,282),(22,272,31,281),(23,271,32,280),(24,288,33,279),(25,287,34,278),(26,286,35,277),(27,285,36,276),(37,172,46,163),(38,171,47,180),(39,170,48,179),(40,169,49,178),(41,168,50,177),(42,167,51,176),(43,166,52,175),(44,165,53,174),(45,164,54,173),(55,253,64,262),(56,270,65,261),(57,269,66,260),(58,268,67,259),(59,267,68,258),(60,266,69,257),(61,265,70,256),(62,264,71,255),(63,263,72,254),(73,232,82,223),(74,231,83,222),(75,230,84,221),(76,229,85,220),(77,228,86,219),(78,227,87,218),(79,226,88,217),(80,225,89,234),(81,224,90,233),(91,120,100,111),(92,119,101,110),(93,118,102,109),(94,117,103,126),(95,116,104,125),(96,115,105,124),(97,114,106,123),(98,113,107,122),(99,112,108,121),(127,210,136,201),(128,209,137,200),(129,208,138,199),(130,207,139,216),(131,206,140,215),(132,205,141,214),(133,204,142,213),(134,203,143,212),(135,202,144,211),(145,182,154,191),(146,181,155,190),(147,198,156,189),(148,197,157,188),(149,196,158,187),(150,195,159,186),(151,194,160,185),(152,193,161,184),(153,192,162,183)]])
Matrix representation of Q8⋊Dic9 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 17 | 55 |
| 0 | 0 | 8 | 56 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 71 |
| 0 | 0 | 1 | 72 |
| 45 | 3 | 0 | 0 |
| 70 | 42 | 0 | 0 |
| 0 | 0 | 10 | 71 |
| 0 | 0 | 9 | 64 |
| 61 | 51 | 0 | 0 |
| 63 | 12 | 0 | 0 |
| 0 | 0 | 36 | 37 |
| 0 | 0 | 34 | 37 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,17,8,0,0,55,56],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,71,72],[45,70,0,0,3,42,0,0,0,0,10,9,0,0,71,64],[61,63,0,0,51,12,0,0,0,0,36,34,0,0,37,37] >;
Q8⋊Dic9 in GAP, Magma, Sage, TeX
Q_8\rtimes {\rm Dic}_9 % in TeX
G:=Group("Q8:Dic9"); // GroupNames label
G:=SmallGroup(288,69);
// by ID
G=gap.SmallGroup(288,69);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,14,422,142,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^18=1,b^2=a^2,d^2=c^9,b*a*b^-1=d*b*d^-1=a^-1,c*a*c^-1=b,d*a*d^-1=a^2*b,c*b*c^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of Q8⋊Dic9 in TeX
Character table of Q8⋊Dic9 in TeX