direct product, non-abelian, soluble
Aliases: C2×Q8.D9, Q8.1D18, C6.2CSU2(𝔽3), (C2×C6).7S4, C6.19(C2×S4), (C6×Q8).2S3, (C3×Q8).8D6, (C2×Q8).2D9, Q8⋊C9.1C22, C3.(C2×CSU2(𝔽3)), C22.4(C3.S4), C2.5(C2×C3.S4), (C2×Q8⋊C9).2C2, SmallGroup(288,335)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8⋊C9 — C2×Q8.D9 |
Generators and relations for C2×Q8.D9
G = < a,b,c,d,e | a2=b4=d9=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=c, ebe-1=b-1c, dcd-1=bc, ece-1=b2c, ede-1=d-1 >
Subgroups: 303 in 65 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, Q8, Q8, C9, Dic3, C12, C2×C6, C2×C8, Q16, C2×Q8, C2×Q8, C18, C3⋊C8, Dic6, C2×Dic3, C2×C12, C3×Q8, C3×Q8, C2×Q16, Dic9, C2×C18, C2×C3⋊C8, C3⋊Q16, C2×Dic6, C6×Q8, Q8⋊C9, C2×Dic9, C2×C3⋊Q16, Q8.D9, C2×Q8⋊C9, C2×Q8.D9
Quotients: C1, C2, C22, S3, D6, D9, S4, D18, CSU2(𝔽3), C2×S4, C3.S4, C2×CSU2(𝔽3), Q8.D9, C2×C3.S4, C2×Q8.D9
Character table of C2×Q8.D9
| 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 | -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 |
| ρ4 | 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 |
| ρ5 | 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 D6 |
| 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 | 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 D18 |
| ρ8 | 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 D18 |
| ρ9 | 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 |
| ρ10 | 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 |
| ρ11 | 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 D18 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| 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 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| 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 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| 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 C2×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 C2×S4 |
| ρ19 | 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 |
| ρ20 | 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 |
| 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 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ22 | 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 |
| 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 | symplectic lifted from Q8.D9, Schur index 2 |
| ρ24 | 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 |
| ρ25 | 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 |
| ρ26 | 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 |
| 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 C2×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 | orthogonal lifted from C3.S4 |
►Regular action on 288 points
Generators in S288
(1 138)(2 139)(3 140)(4 141)(5 142)(6 143)(7 144)(8 136)(9 137)(10 274)(11 275)(12 276)(13 277)(14 278)(15 279)(16 271)(17 272)(18 273)(19 267)(20 268)(21 269)(22 270)(23 262)(24 263)(25 264)(26 265)(27 266)(28 260)(29 261)(30 253)(31 254)(32 255)(33 256)(34 257)(35 258)(36 259)(37 244)(38 245)(39 246)(40 247)(41 248)(42 249)(43 250)(44 251)(45 252)(46 237)(47 238)(48 239)(49 240)(50 241)(51 242)(52 243)(53 235)(54 236)(55 230)(56 231)(57 232)(58 233)(59 234)(60 226)(61 227)(62 228)(63 229)(64 223)(65 224)(66 225)(67 217)(68 218)(69 219)(70 220)(71 221)(72 222)(73 178)(74 179)(75 180)(76 172)(77 173)(78 174)(79 175)(80 176)(81 177)(82 195)(83 196)(84 197)(85 198)(86 190)(87 191)(88 192)(89 193)(90 194)(91 188)(92 189)(93 181)(94 182)(95 183)(96 184)(97 185)(98 186)(99 187)(100 148)(101 149)(102 150)(103 151)(104 152)(105 153)(106 145)(107 146)(108 147)(109 165)(110 166)(111 167)(112 168)(113 169)(114 170)(115 171)(116 163)(117 164)(118 158)(119 159)(120 160)(121 161)(122 162)(123 154)(124 155)(125 156)(126 157)(127 199)(128 200)(129 201)(130 202)(131 203)(132 204)(133 205)(134 206)(135 207)(208 280)(209 281)(210 282)(211 283)(212 284)(213 285)(214 286)(215 287)(216 288)
(1 105 126 73)(2 85 118 128)(3 109 119 96)(4 108 120 76)(5 88 121 131)(6 112 122 99)(7 102 123 79)(8 82 124 134)(9 115 125 93)(10 36 21 46)(11 66 22 41)(12 63 23 288)(13 30 24 49)(14 69 25 44)(15 57 26 282)(16 33 27 52)(17 72 19 38)(18 60 20 285)(28 62 47 287)(29 42 48 67)(31 56 50 281)(32 45 51 70)(34 59 53 284)(35 39 54 64)(37 58 71 283)(40 61 65 286)(43 55 68 280)(74 95 106 117)(75 86 107 129)(77 98 100 111)(78 89 101 132)(80 92 103 114)(81 83 104 135)(84 94 127 116)(87 97 130 110)(90 91 133 113)(136 195 155 206)(137 171 156 181)(138 153 157 178)(139 198 158 200)(140 165 159 184)(141 147 160 172)(142 192 161 203)(143 168 162 187)(144 150 154 175)(145 164 179 183)(146 201 180 190)(148 167 173 186)(149 204 174 193)(151 170 176 189)(152 207 177 196)(163 197 182 199)(166 191 185 202)(169 194 188 205)(208 250 230 218)(209 254 231 241)(210 279 232 265)(211 244 233 221)(212 257 234 235)(213 273 226 268)(214 247 227 224)(215 260 228 238)(216 276 229 262)(217 261 249 239)(219 264 251 278)(220 255 252 242)(222 267 245 272)(223 258 246 236)(225 270 248 275)(237 274 259 269)(240 277 253 263)(243 271 256 266)
(1 84 126 127)(2 117 118 95)(3 107 119 75)(4 87 120 130)(5 111 121 98)(6 101 122 78)(7 90 123 133)(8 114 124 92)(9 104 125 81)(10 65 21 40)(11 62 22 287)(12 29 23 48)(13 68 24 43)(14 56 25 281)(15 32 26 51)(16 71 27 37)(17 59 19 284)(18 35 20 54)(28 41 47 66)(30 55 49 280)(31 44 50 69)(33 58 52 283)(34 38 53 72)(36 61 46 286)(39 60 64 285)(42 63 67 288)(45 57 70 282)(73 94 105 116)(74 85 106 128)(76 97 108 110)(77 88 100 131)(79 91 102 113)(80 82 103 134)(83 93 135 115)(86 96 129 109)(89 99 132 112)(136 170 155 189)(137 152 156 177)(138 197 157 199)(139 164 158 183)(140 146 159 180)(141 191 160 202)(142 167 161 186)(143 149 162 174)(144 194 154 205)(145 200 179 198)(147 166 172 185)(148 203 173 192)(150 169 175 188)(151 206 176 195)(153 163 178 182)(165 190 184 201)(168 193 187 204)(171 196 181 207)(208 253 230 240)(209 278 231 264)(210 252 232 220)(211 256 233 243)(212 272 234 267)(213 246 226 223)(214 259 227 237)(215 275 228 270)(216 249 229 217)(218 263 250 277)(219 254 251 241)(221 266 244 271)(222 257 245 235)(224 269 247 274)(225 260 248 238)(236 273 258 268)(239 276 261 262)(242 279 255 265)
(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 59 126 284)(2 58 118 283)(3 57 119 282)(4 56 120 281)(5 55 121 280)(6 63 122 288)(7 62 123 287)(8 61 124 286)(9 60 125 285)(10 134 21 82)(11 133 22 90)(12 132 23 89)(13 131 24 88)(14 130 25 87)(15 129 26 86)(16 128 27 85)(17 127 19 84)(18 135 20 83)(28 91 47 113)(29 99 48 112)(30 98 49 111)(31 97 50 110)(32 96 51 109)(33 95 52 117)(34 94 53 116)(35 93 54 115)(36 92 46 114)(37 74 71 106)(38 73 72 105)(39 81 64 104)(40 80 65 103)(41 79 66 102)(42 78 67 101)(43 77 68 100)(44 76 69 108)(45 75 70 107)(136 227 155 214)(137 226 156 213)(138 234 157 212)(139 233 158 211)(140 232 159 210)(141 231 160 209)(142 230 161 208)(143 229 162 216)(144 228 154 215)(145 244 179 221)(146 252 180 220)(147 251 172 219)(148 250 173 218)(149 249 174 217)(150 248 175 225)(151 247 176 224)(152 246 177 223)(153 245 178 222)(163 257 182 235)(164 256 183 243)(165 255 184 242)(166 254 185 241)(167 253 186 240)(168 261 187 239)(169 260 188 238)(170 259 189 237)(171 258 181 236)(190 279 201 265)(191 278 202 264)(192 277 203 263)(193 276 204 262)(194 275 205 270)(195 274 206 269)(196 273 207 268)(197 272 199 267)(198 271 200 266)
G:=sub<Sym(288)| (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,136)(9,137)(10,274)(11,275)(12,276)(13,277)(14,278)(15,279)(16,271)(17,272)(18,273)(19,267)(20,268)(21,269)(22,270)(23,262)(24,263)(25,264)(26,265)(27,266)(28,260)(29,261)(30,253)(31,254)(32,255)(33,256)(34,257)(35,258)(36,259)(37,244)(38,245)(39,246)(40,247)(41,248)(42,249)(43,250)(44,251)(45,252)(46,237)(47,238)(48,239)(49,240)(50,241)(51,242)(52,243)(53,235)(54,236)(55,230)(56,231)(57,232)(58,233)(59,234)(60,226)(61,227)(62,228)(63,229)(64,223)(65,224)(66,225)(67,217)(68,218)(69,219)(70,220)(71,221)(72,222)(73,178)(74,179)(75,180)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,195)(83,196)(84,197)(85,198)(86,190)(87,191)(88,192)(89,193)(90,194)(91,188)(92,189)(93,181)(94,182)(95,183)(96,184)(97,185)(98,186)(99,187)(100,148)(101,149)(102,150)(103,151)(104,152)(105,153)(106,145)(107,146)(108,147)(109,165)(110,166)(111,167)(112,168)(113,169)(114,170)(115,171)(116,163)(117,164)(118,158)(119,159)(120,160)(121,161)(122,162)(123,154)(124,155)(125,156)(126,157)(127,199)(128,200)(129,201)(130,202)(131,203)(132,204)(133,205)(134,206)(135,207)(208,280)(209,281)(210,282)(211,283)(212,284)(213,285)(214,286)(215,287)(216,288), (1,105,126,73)(2,85,118,128)(3,109,119,96)(4,108,120,76)(5,88,121,131)(6,112,122,99)(7,102,123,79)(8,82,124,134)(9,115,125,93)(10,36,21,46)(11,66,22,41)(12,63,23,288)(13,30,24,49)(14,69,25,44)(15,57,26,282)(16,33,27,52)(17,72,19,38)(18,60,20,285)(28,62,47,287)(29,42,48,67)(31,56,50,281)(32,45,51,70)(34,59,53,284)(35,39,54,64)(37,58,71,283)(40,61,65,286)(43,55,68,280)(74,95,106,117)(75,86,107,129)(77,98,100,111)(78,89,101,132)(80,92,103,114)(81,83,104,135)(84,94,127,116)(87,97,130,110)(90,91,133,113)(136,195,155,206)(137,171,156,181)(138,153,157,178)(139,198,158,200)(140,165,159,184)(141,147,160,172)(142,192,161,203)(143,168,162,187)(144,150,154,175)(145,164,179,183)(146,201,180,190)(148,167,173,186)(149,204,174,193)(151,170,176,189)(152,207,177,196)(163,197,182,199)(166,191,185,202)(169,194,188,205)(208,250,230,218)(209,254,231,241)(210,279,232,265)(211,244,233,221)(212,257,234,235)(213,273,226,268)(214,247,227,224)(215,260,228,238)(216,276,229,262)(217,261,249,239)(219,264,251,278)(220,255,252,242)(222,267,245,272)(223,258,246,236)(225,270,248,275)(237,274,259,269)(240,277,253,263)(243,271,256,266), (1,84,126,127)(2,117,118,95)(3,107,119,75)(4,87,120,130)(5,111,121,98)(6,101,122,78)(7,90,123,133)(8,114,124,92)(9,104,125,81)(10,65,21,40)(11,62,22,287)(12,29,23,48)(13,68,24,43)(14,56,25,281)(15,32,26,51)(16,71,27,37)(17,59,19,284)(18,35,20,54)(28,41,47,66)(30,55,49,280)(31,44,50,69)(33,58,52,283)(34,38,53,72)(36,61,46,286)(39,60,64,285)(42,63,67,288)(45,57,70,282)(73,94,105,116)(74,85,106,128)(76,97,108,110)(77,88,100,131)(79,91,102,113)(80,82,103,134)(83,93,135,115)(86,96,129,109)(89,99,132,112)(136,170,155,189)(137,152,156,177)(138,197,157,199)(139,164,158,183)(140,146,159,180)(141,191,160,202)(142,167,161,186)(143,149,162,174)(144,194,154,205)(145,200,179,198)(147,166,172,185)(148,203,173,192)(150,169,175,188)(151,206,176,195)(153,163,178,182)(165,190,184,201)(168,193,187,204)(171,196,181,207)(208,253,230,240)(209,278,231,264)(210,252,232,220)(211,256,233,243)(212,272,234,267)(213,246,226,223)(214,259,227,237)(215,275,228,270)(216,249,229,217)(218,263,250,277)(219,254,251,241)(221,266,244,271)(222,257,245,235)(224,269,247,274)(225,260,248,238)(236,273,258,268)(239,276,261,262)(242,279,255,265), (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), (1,59,126,284)(2,58,118,283)(3,57,119,282)(4,56,120,281)(5,55,121,280)(6,63,122,288)(7,62,123,287)(8,61,124,286)(9,60,125,285)(10,134,21,82)(11,133,22,90)(12,132,23,89)(13,131,24,88)(14,130,25,87)(15,129,26,86)(16,128,27,85)(17,127,19,84)(18,135,20,83)(28,91,47,113)(29,99,48,112)(30,98,49,111)(31,97,50,110)(32,96,51,109)(33,95,52,117)(34,94,53,116)(35,93,54,115)(36,92,46,114)(37,74,71,106)(38,73,72,105)(39,81,64,104)(40,80,65,103)(41,79,66,102)(42,78,67,101)(43,77,68,100)(44,76,69,108)(45,75,70,107)(136,227,155,214)(137,226,156,213)(138,234,157,212)(139,233,158,211)(140,232,159,210)(141,231,160,209)(142,230,161,208)(143,229,162,216)(144,228,154,215)(145,244,179,221)(146,252,180,220)(147,251,172,219)(148,250,173,218)(149,249,174,217)(150,248,175,225)(151,247,176,224)(152,246,177,223)(153,245,178,222)(163,257,182,235)(164,256,183,243)(165,255,184,242)(166,254,185,241)(167,253,186,240)(168,261,187,239)(169,260,188,238)(170,259,189,237)(171,258,181,236)(190,279,201,265)(191,278,202,264)(192,277,203,263)(193,276,204,262)(194,275,205,270)(195,274,206,269)(196,273,207,268)(197,272,199,267)(198,271,200,266)>;
G:=Group( (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,136)(9,137)(10,274)(11,275)(12,276)(13,277)(14,278)(15,279)(16,271)(17,272)(18,273)(19,267)(20,268)(21,269)(22,270)(23,262)(24,263)(25,264)(26,265)(27,266)(28,260)(29,261)(30,253)(31,254)(32,255)(33,256)(34,257)(35,258)(36,259)(37,244)(38,245)(39,246)(40,247)(41,248)(42,249)(43,250)(44,251)(45,252)(46,237)(47,238)(48,239)(49,240)(50,241)(51,242)(52,243)(53,235)(54,236)(55,230)(56,231)(57,232)(58,233)(59,234)(60,226)(61,227)(62,228)(63,229)(64,223)(65,224)(66,225)(67,217)(68,218)(69,219)(70,220)(71,221)(72,222)(73,178)(74,179)(75,180)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,195)(83,196)(84,197)(85,198)(86,190)(87,191)(88,192)(89,193)(90,194)(91,188)(92,189)(93,181)(94,182)(95,183)(96,184)(97,185)(98,186)(99,187)(100,148)(101,149)(102,150)(103,151)(104,152)(105,153)(106,145)(107,146)(108,147)(109,165)(110,166)(111,167)(112,168)(113,169)(114,170)(115,171)(116,163)(117,164)(118,158)(119,159)(120,160)(121,161)(122,162)(123,154)(124,155)(125,156)(126,157)(127,199)(128,200)(129,201)(130,202)(131,203)(132,204)(133,205)(134,206)(135,207)(208,280)(209,281)(210,282)(211,283)(212,284)(213,285)(214,286)(215,287)(216,288), (1,105,126,73)(2,85,118,128)(3,109,119,96)(4,108,120,76)(5,88,121,131)(6,112,122,99)(7,102,123,79)(8,82,124,134)(9,115,125,93)(10,36,21,46)(11,66,22,41)(12,63,23,288)(13,30,24,49)(14,69,25,44)(15,57,26,282)(16,33,27,52)(17,72,19,38)(18,60,20,285)(28,62,47,287)(29,42,48,67)(31,56,50,281)(32,45,51,70)(34,59,53,284)(35,39,54,64)(37,58,71,283)(40,61,65,286)(43,55,68,280)(74,95,106,117)(75,86,107,129)(77,98,100,111)(78,89,101,132)(80,92,103,114)(81,83,104,135)(84,94,127,116)(87,97,130,110)(90,91,133,113)(136,195,155,206)(137,171,156,181)(138,153,157,178)(139,198,158,200)(140,165,159,184)(141,147,160,172)(142,192,161,203)(143,168,162,187)(144,150,154,175)(145,164,179,183)(146,201,180,190)(148,167,173,186)(149,204,174,193)(151,170,176,189)(152,207,177,196)(163,197,182,199)(166,191,185,202)(169,194,188,205)(208,250,230,218)(209,254,231,241)(210,279,232,265)(211,244,233,221)(212,257,234,235)(213,273,226,268)(214,247,227,224)(215,260,228,238)(216,276,229,262)(217,261,249,239)(219,264,251,278)(220,255,252,242)(222,267,245,272)(223,258,246,236)(225,270,248,275)(237,274,259,269)(240,277,253,263)(243,271,256,266), (1,84,126,127)(2,117,118,95)(3,107,119,75)(4,87,120,130)(5,111,121,98)(6,101,122,78)(7,90,123,133)(8,114,124,92)(9,104,125,81)(10,65,21,40)(11,62,22,287)(12,29,23,48)(13,68,24,43)(14,56,25,281)(15,32,26,51)(16,71,27,37)(17,59,19,284)(18,35,20,54)(28,41,47,66)(30,55,49,280)(31,44,50,69)(33,58,52,283)(34,38,53,72)(36,61,46,286)(39,60,64,285)(42,63,67,288)(45,57,70,282)(73,94,105,116)(74,85,106,128)(76,97,108,110)(77,88,100,131)(79,91,102,113)(80,82,103,134)(83,93,135,115)(86,96,129,109)(89,99,132,112)(136,170,155,189)(137,152,156,177)(138,197,157,199)(139,164,158,183)(140,146,159,180)(141,191,160,202)(142,167,161,186)(143,149,162,174)(144,194,154,205)(145,200,179,198)(147,166,172,185)(148,203,173,192)(150,169,175,188)(151,206,176,195)(153,163,178,182)(165,190,184,201)(168,193,187,204)(171,196,181,207)(208,253,230,240)(209,278,231,264)(210,252,232,220)(211,256,233,243)(212,272,234,267)(213,246,226,223)(214,259,227,237)(215,275,228,270)(216,249,229,217)(218,263,250,277)(219,254,251,241)(221,266,244,271)(222,257,245,235)(224,269,247,274)(225,260,248,238)(236,273,258,268)(239,276,261,262)(242,279,255,265), (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), (1,59,126,284)(2,58,118,283)(3,57,119,282)(4,56,120,281)(5,55,121,280)(6,63,122,288)(7,62,123,287)(8,61,124,286)(9,60,125,285)(10,134,21,82)(11,133,22,90)(12,132,23,89)(13,131,24,88)(14,130,25,87)(15,129,26,86)(16,128,27,85)(17,127,19,84)(18,135,20,83)(28,91,47,113)(29,99,48,112)(30,98,49,111)(31,97,50,110)(32,96,51,109)(33,95,52,117)(34,94,53,116)(35,93,54,115)(36,92,46,114)(37,74,71,106)(38,73,72,105)(39,81,64,104)(40,80,65,103)(41,79,66,102)(42,78,67,101)(43,77,68,100)(44,76,69,108)(45,75,70,107)(136,227,155,214)(137,226,156,213)(138,234,157,212)(139,233,158,211)(140,232,159,210)(141,231,160,209)(142,230,161,208)(143,229,162,216)(144,228,154,215)(145,244,179,221)(146,252,180,220)(147,251,172,219)(148,250,173,218)(149,249,174,217)(150,248,175,225)(151,247,176,224)(152,246,177,223)(153,245,178,222)(163,257,182,235)(164,256,183,243)(165,255,184,242)(166,254,185,241)(167,253,186,240)(168,261,187,239)(169,260,188,238)(170,259,189,237)(171,258,181,236)(190,279,201,265)(191,278,202,264)(192,277,203,263)(193,276,204,262)(194,275,205,270)(195,274,206,269)(196,273,207,268)(197,272,199,267)(198,271,200,266) );
G=PermutationGroup([[(1,138),(2,139),(3,140),(4,141),(5,142),(6,143),(7,144),(8,136),(9,137),(10,274),(11,275),(12,276),(13,277),(14,278),(15,279),(16,271),(17,272),(18,273),(19,267),(20,268),(21,269),(22,270),(23,262),(24,263),(25,264),(26,265),(27,266),(28,260),(29,261),(30,253),(31,254),(32,255),(33,256),(34,257),(35,258),(36,259),(37,244),(38,245),(39,246),(40,247),(41,248),(42,249),(43,250),(44,251),(45,252),(46,237),(47,238),(48,239),(49,240),(50,241),(51,242),(52,243),(53,235),(54,236),(55,230),(56,231),(57,232),(58,233),(59,234),(60,226),(61,227),(62,228),(63,229),(64,223),(65,224),(66,225),(67,217),(68,218),(69,219),(70,220),(71,221),(72,222),(73,178),(74,179),(75,180),(76,172),(77,173),(78,174),(79,175),(80,176),(81,177),(82,195),(83,196),(84,197),(85,198),(86,190),(87,191),(88,192),(89,193),(90,194),(91,188),(92,189),(93,181),(94,182),(95,183),(96,184),(97,185),(98,186),(99,187),(100,148),(101,149),(102,150),(103,151),(104,152),(105,153),(106,145),(107,146),(108,147),(109,165),(110,166),(111,167),(112,168),(113,169),(114,170),(115,171),(116,163),(117,164),(118,158),(119,159),(120,160),(121,161),(122,162),(123,154),(124,155),(125,156),(126,157),(127,199),(128,200),(129,201),(130,202),(131,203),(132,204),(133,205),(134,206),(135,207),(208,280),(209,281),(210,282),(211,283),(212,284),(213,285),(214,286),(215,287),(216,288)], [(1,105,126,73),(2,85,118,128),(3,109,119,96),(4,108,120,76),(5,88,121,131),(6,112,122,99),(7,102,123,79),(8,82,124,134),(9,115,125,93),(10,36,21,46),(11,66,22,41),(12,63,23,288),(13,30,24,49),(14,69,25,44),(15,57,26,282),(16,33,27,52),(17,72,19,38),(18,60,20,285),(28,62,47,287),(29,42,48,67),(31,56,50,281),(32,45,51,70),(34,59,53,284),(35,39,54,64),(37,58,71,283),(40,61,65,286),(43,55,68,280),(74,95,106,117),(75,86,107,129),(77,98,100,111),(78,89,101,132),(80,92,103,114),(81,83,104,135),(84,94,127,116),(87,97,130,110),(90,91,133,113),(136,195,155,206),(137,171,156,181),(138,153,157,178),(139,198,158,200),(140,165,159,184),(141,147,160,172),(142,192,161,203),(143,168,162,187),(144,150,154,175),(145,164,179,183),(146,201,180,190),(148,167,173,186),(149,204,174,193),(151,170,176,189),(152,207,177,196),(163,197,182,199),(166,191,185,202),(169,194,188,205),(208,250,230,218),(209,254,231,241),(210,279,232,265),(211,244,233,221),(212,257,234,235),(213,273,226,268),(214,247,227,224),(215,260,228,238),(216,276,229,262),(217,261,249,239),(219,264,251,278),(220,255,252,242),(222,267,245,272),(223,258,246,236),(225,270,248,275),(237,274,259,269),(240,277,253,263),(243,271,256,266)], [(1,84,126,127),(2,117,118,95),(3,107,119,75),(4,87,120,130),(5,111,121,98),(6,101,122,78),(7,90,123,133),(8,114,124,92),(9,104,125,81),(10,65,21,40),(11,62,22,287),(12,29,23,48),(13,68,24,43),(14,56,25,281),(15,32,26,51),(16,71,27,37),(17,59,19,284),(18,35,20,54),(28,41,47,66),(30,55,49,280),(31,44,50,69),(33,58,52,283),(34,38,53,72),(36,61,46,286),(39,60,64,285),(42,63,67,288),(45,57,70,282),(73,94,105,116),(74,85,106,128),(76,97,108,110),(77,88,100,131),(79,91,102,113),(80,82,103,134),(83,93,135,115),(86,96,129,109),(89,99,132,112),(136,170,155,189),(137,152,156,177),(138,197,157,199),(139,164,158,183),(140,146,159,180),(141,191,160,202),(142,167,161,186),(143,149,162,174),(144,194,154,205),(145,200,179,198),(147,166,172,185),(148,203,173,192),(150,169,175,188),(151,206,176,195),(153,163,178,182),(165,190,184,201),(168,193,187,204),(171,196,181,207),(208,253,230,240),(209,278,231,264),(210,252,232,220),(211,256,233,243),(212,272,234,267),(213,246,226,223),(214,259,227,237),(215,275,228,270),(216,249,229,217),(218,263,250,277),(219,254,251,241),(221,266,244,271),(222,257,245,235),(224,269,247,274),(225,260,248,238),(236,273,258,268),(239,276,261,262),(242,279,255,265)], [(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)], [(1,59,126,284),(2,58,118,283),(3,57,119,282),(4,56,120,281),(5,55,121,280),(6,63,122,288),(7,62,123,287),(8,61,124,286),(9,60,125,285),(10,134,21,82),(11,133,22,90),(12,132,23,89),(13,131,24,88),(14,130,25,87),(15,129,26,86),(16,128,27,85),(17,127,19,84),(18,135,20,83),(28,91,47,113),(29,99,48,112),(30,98,49,111),(31,97,50,110),(32,96,51,109),(33,95,52,117),(34,94,53,116),(35,93,54,115),(36,92,46,114),(37,74,71,106),(38,73,72,105),(39,81,64,104),(40,80,65,103),(41,79,66,102),(42,78,67,101),(43,77,68,100),(44,76,69,108),(45,75,70,107),(136,227,155,214),(137,226,156,213),(138,234,157,212),(139,233,158,211),(140,232,159,210),(141,231,160,209),(142,230,161,208),(143,229,162,216),(144,228,154,215),(145,244,179,221),(146,252,180,220),(147,251,172,219),(148,250,173,218),(149,249,174,217),(150,248,175,225),(151,247,176,224),(152,246,177,223),(153,245,178,222),(163,257,182,235),(164,256,183,243),(165,255,184,242),(166,254,185,241),(167,253,186,240),(168,261,187,239),(169,260,188,238),(170,259,189,237),(171,258,181,236),(190,279,201,265),(191,278,202,264),(192,277,203,263),(193,276,204,262),(194,275,205,270),(195,274,206,269),(196,273,207,268),(197,272,199,267),(198,271,200,266)]])
Matrix representation of C2×Q8.D9 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 32 | 56 | 0 | 0 |
| 56 | 41 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 48 | 66 | 0 | 0 |
| 65 | 24 | 0 | 0 |
| 0 | 0 | 32 | 24 |
| 0 | 0 | 39 | 27 |
| 52 | 54 | 0 | 0 |
| 54 | 21 | 0 | 0 |
| 0 | 0 | 57 | 68 |
| 0 | 0 | 51 | 16 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[32,56,0,0,56,41,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[48,65,0,0,66,24,0,0,0,0,32,39,0,0,24,27],[52,54,0,0,54,21,0,0,0,0,57,51,0,0,68,16] >;
C2×Q8.D9 in GAP, Magma, Sage, TeX
C_2\times Q_8.D_9
% in TeX
G:=Group("C2xQ8.D9"); // GroupNames label
G:=SmallGroup(288,335);
// by ID
G=gap.SmallGroup(288,335);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,422,142,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^9=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=c,e*b*e^-1=b^-1*c,d*c*d^-1=b*c,e*c*e^-1=b^2*c,e*d*e^-1=d^-1>;
// generators/relations
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