metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2Dic9, C36.10D4, C18.5Q16, C18.8SD16, (Q8×C9)⋊1C4, C36.8(C2×C4), (C2×Q8).4D9, C9⋊3(Q8⋊C4), (C2×C18).35D4, (C2×C4).42D18, (C2×C12).48D6, C4⋊Dic9.9C2, (Q8×C18).2C2, (C6×Q8).10S3, C4.2(C2×Dic9), C4.15(C9⋊D4), C12.8(C3⋊D4), C3.(Q8⋊2Dic3), (C3×Q8).5Dic3, C12.2(C2×Dic3), C6.8(C3⋊Q16), C2.3(C9⋊Q16), (C2×C36).26C22, C2.3(Q8⋊2D9), C6.9(Q8⋊2S3), C18.17(C22⋊C4), C22.18(C9⋊D4), C6.18(C6.D4), C2.7(C18.D4), (C2×C9⋊C8).5C2, (C2×C6).73(C3⋊D4), SmallGroup(288,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2Dic9
G = < a,b,c,d | a4=c18=1, b2=a2, d2=c9, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >
Subgroups: 200 in 63 conjugacy classes, 36 normal (30 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, C9, Dic3, C12, C12, C2×C6, C4⋊C4, C2×C8, C2×Q8, C18, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, Dic9, C36, C36, C2×C18, C2×C3⋊C8, C4⋊Dic3, C6×Q8, C9⋊C8, C2×Dic9, C2×C36, C2×C36, Q8×C9, Q8×C9, Q8⋊2Dic3, C2×C9⋊C8, C4⋊Dic9, Q8×C18, Q8⋊2Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, SD16, Q16, D9, C2×Dic3, C3⋊D4, Q8⋊C4, Dic9, D18, Q8⋊2S3, C3⋊Q16, C6.D4, C2×Dic9, C9⋊D4, Q8⋊2Dic3, C9⋊Q16, Q8⋊2D9, C18.D4, Q8⋊2Dic9
►Regular action on 288 points
Generators in S288
(1 75 51 216)(2 76 52 199)(3 77 53 200)(4 78 54 201)(5 79 37 202)(6 80 38 203)(7 81 39 204)(8 82 40 205)(9 83 41 206)(10 84 42 207)(11 85 43 208)(12 86 44 209)(13 87 45 210)(14 88 46 211)(15 89 47 212)(16 90 48 213)(17 73 49 214)(18 74 50 215)(19 183 135 61)(20 184 136 62)(21 185 137 63)(22 186 138 64)(23 187 139 65)(24 188 140 66)(25 189 141 67)(26 190 142 68)(27 191 143 69)(28 192 144 70)(29 193 127 71)(30 194 128 72)(31 195 129 55)(32 196 130 56)(33 197 131 57)(34 198 132 58)(35 181 133 59)(36 182 134 60)(91 219 112 244)(92 220 113 245)(93 221 114 246)(94 222 115 247)(95 223 116 248)(96 224 117 249)(97 225 118 250)(98 226 119 251)(99 227 120 252)(100 228 121 235)(101 229 122 236)(102 230 123 237)(103 231 124 238)(104 232 125 239)(105 233 126 240)(106 234 109 241)(107 217 110 242)(108 218 111 243)(145 180 260 283)(146 163 261 284)(147 164 262 285)(148 165 263 286)(149 166 264 287)(150 167 265 288)(151 168 266 271)(152 169 267 272)(153 170 268 273)(154 171 269 274)(155 172 270 275)(156 173 253 276)(157 174 254 277)(158 175 255 278)(159 176 256 279)(160 177 257 280)(161 178 258 281)(162 179 259 282)
(1 276 51 173)(2 277 52 174)(3 278 53 175)(4 279 54 176)(5 280 37 177)(6 281 38 178)(7 282 39 179)(8 283 40 180)(9 284 41 163)(10 285 42 164)(11 286 43 165)(12 287 44 166)(13 288 45 167)(14 271 46 168)(15 272 47 169)(16 273 48 170)(17 274 49 171)(18 275 50 172)(19 241 135 234)(20 242 136 217)(21 243 137 218)(22 244 138 219)(23 245 139 220)(24 246 140 221)(25 247 141 222)(26 248 142 223)(27 249 143 224)(28 250 144 225)(29 251 127 226)(30 252 128 227)(31 235 129 228)(32 236 130 229)(33 237 131 230)(34 238 132 231)(35 239 133 232)(36 240 134 233)(55 100 195 121)(56 101 196 122)(57 102 197 123)(58 103 198 124)(59 104 181 125)(60 105 182 126)(61 106 183 109)(62 107 184 110)(63 108 185 111)(64 91 186 112)(65 92 187 113)(66 93 188 114)(67 94 189 115)(68 95 190 116)(69 96 191 117)(70 97 192 118)(71 98 193 119)(72 99 194 120)(73 269 214 154)(74 270 215 155)(75 253 216 156)(76 254 199 157)(77 255 200 158)(78 256 201 159)(79 257 202 160)(80 258 203 161)(81 259 204 162)(82 260 205 145)(83 261 206 146)(84 262 207 147)(85 263 208 148)(86 264 209 149)(87 265 210 150)(88 266 211 151)(89 267 212 152)(90 268 213 153)
(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 66 10 57)(2 65 11 56)(3 64 12 55)(4 63 13 72)(5 62 14 71)(6 61 15 70)(7 60 16 69)(8 59 17 68)(9 58 18 67)(19 212 28 203)(20 211 29 202)(21 210 30 201)(22 209 31 200)(23 208 32 199)(24 207 33 216)(25 206 34 215)(26 205 35 214)(27 204 36 213)(37 184 46 193)(38 183 47 192)(39 182 48 191)(40 181 49 190)(41 198 50 189)(42 197 51 188)(43 196 52 187)(44 195 53 186)(45 194 54 185)(73 142 82 133)(74 141 83 132)(75 140 84 131)(76 139 85 130)(77 138 86 129)(78 137 87 128)(79 136 88 127)(80 135 89 144)(81 134 90 143)(91 149 100 158)(92 148 101 157)(93 147 102 156)(94 146 103 155)(95 145 104 154)(96 162 105 153)(97 161 106 152)(98 160 107 151)(99 159 108 150)(109 267 118 258)(110 266 119 257)(111 265 120 256)(112 264 121 255)(113 263 122 254)(114 262 123 253)(115 261 124 270)(116 260 125 269)(117 259 126 268)(163 238 172 247)(164 237 173 246)(165 236 174 245)(166 235 175 244)(167 252 176 243)(168 251 177 242)(169 250 178 241)(170 249 179 240)(171 248 180 239)(217 271 226 280)(218 288 227 279)(219 287 228 278)(220 286 229 277)(221 285 230 276)(222 284 231 275)(223 283 232 274)(224 282 233 273)(225 281 234 272)
G:=sub<Sym(288)| (1,75,51,216)(2,76,52,199)(3,77,53,200)(4,78,54,201)(5,79,37,202)(6,80,38,203)(7,81,39,204)(8,82,40,205)(9,83,41,206)(10,84,42,207)(11,85,43,208)(12,86,44,209)(13,87,45,210)(14,88,46,211)(15,89,47,212)(16,90,48,213)(17,73,49,214)(18,74,50,215)(19,183,135,61)(20,184,136,62)(21,185,137,63)(22,186,138,64)(23,187,139,65)(24,188,140,66)(25,189,141,67)(26,190,142,68)(27,191,143,69)(28,192,144,70)(29,193,127,71)(30,194,128,72)(31,195,129,55)(32,196,130,56)(33,197,131,57)(34,198,132,58)(35,181,133,59)(36,182,134,60)(91,219,112,244)(92,220,113,245)(93,221,114,246)(94,222,115,247)(95,223,116,248)(96,224,117,249)(97,225,118,250)(98,226,119,251)(99,227,120,252)(100,228,121,235)(101,229,122,236)(102,230,123,237)(103,231,124,238)(104,232,125,239)(105,233,126,240)(106,234,109,241)(107,217,110,242)(108,218,111,243)(145,180,260,283)(146,163,261,284)(147,164,262,285)(148,165,263,286)(149,166,264,287)(150,167,265,288)(151,168,266,271)(152,169,267,272)(153,170,268,273)(154,171,269,274)(155,172,270,275)(156,173,253,276)(157,174,254,277)(158,175,255,278)(159,176,256,279)(160,177,257,280)(161,178,258,281)(162,179,259,282), (1,276,51,173)(2,277,52,174)(3,278,53,175)(4,279,54,176)(5,280,37,177)(6,281,38,178)(7,282,39,179)(8,283,40,180)(9,284,41,163)(10,285,42,164)(11,286,43,165)(12,287,44,166)(13,288,45,167)(14,271,46,168)(15,272,47,169)(16,273,48,170)(17,274,49,171)(18,275,50,172)(19,241,135,234)(20,242,136,217)(21,243,137,218)(22,244,138,219)(23,245,139,220)(24,246,140,221)(25,247,141,222)(26,248,142,223)(27,249,143,224)(28,250,144,225)(29,251,127,226)(30,252,128,227)(31,235,129,228)(32,236,130,229)(33,237,131,230)(34,238,132,231)(35,239,133,232)(36,240,134,233)(55,100,195,121)(56,101,196,122)(57,102,197,123)(58,103,198,124)(59,104,181,125)(60,105,182,126)(61,106,183,109)(62,107,184,110)(63,108,185,111)(64,91,186,112)(65,92,187,113)(66,93,188,114)(67,94,189,115)(68,95,190,116)(69,96,191,117)(70,97,192,118)(71,98,193,119)(72,99,194,120)(73,269,214,154)(74,270,215,155)(75,253,216,156)(76,254,199,157)(77,255,200,158)(78,256,201,159)(79,257,202,160)(80,258,203,161)(81,259,204,162)(82,260,205,145)(83,261,206,146)(84,262,207,147)(85,263,208,148)(86,264,209,149)(87,265,210,150)(88,266,211,151)(89,267,212,152)(90,268,213,153), (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), (1,66,10,57)(2,65,11,56)(3,64,12,55)(4,63,13,72)(5,62,14,71)(6,61,15,70)(7,60,16,69)(8,59,17,68)(9,58,18,67)(19,212,28,203)(20,211,29,202)(21,210,30,201)(22,209,31,200)(23,208,32,199)(24,207,33,216)(25,206,34,215)(26,205,35,214)(27,204,36,213)(37,184,46,193)(38,183,47,192)(39,182,48,191)(40,181,49,190)(41,198,50,189)(42,197,51,188)(43,196,52,187)(44,195,53,186)(45,194,54,185)(73,142,82,133)(74,141,83,132)(75,140,84,131)(76,139,85,130)(77,138,86,129)(78,137,87,128)(79,136,88,127)(80,135,89,144)(81,134,90,143)(91,149,100,158)(92,148,101,157)(93,147,102,156)(94,146,103,155)(95,145,104,154)(96,162,105,153)(97,161,106,152)(98,160,107,151)(99,159,108,150)(109,267,118,258)(110,266,119,257)(111,265,120,256)(112,264,121,255)(113,263,122,254)(114,262,123,253)(115,261,124,270)(116,260,125,269)(117,259,126,268)(163,238,172,247)(164,237,173,246)(165,236,174,245)(166,235,175,244)(167,252,176,243)(168,251,177,242)(169,250,178,241)(170,249,179,240)(171,248,180,239)(217,271,226,280)(218,288,227,279)(219,287,228,278)(220,286,229,277)(221,285,230,276)(222,284,231,275)(223,283,232,274)(224,282,233,273)(225,281,234,272)>;
G:=Group( (1,75,51,216)(2,76,52,199)(3,77,53,200)(4,78,54,201)(5,79,37,202)(6,80,38,203)(7,81,39,204)(8,82,40,205)(9,83,41,206)(10,84,42,207)(11,85,43,208)(12,86,44,209)(13,87,45,210)(14,88,46,211)(15,89,47,212)(16,90,48,213)(17,73,49,214)(18,74,50,215)(19,183,135,61)(20,184,136,62)(21,185,137,63)(22,186,138,64)(23,187,139,65)(24,188,140,66)(25,189,141,67)(26,190,142,68)(27,191,143,69)(28,192,144,70)(29,193,127,71)(30,194,128,72)(31,195,129,55)(32,196,130,56)(33,197,131,57)(34,198,132,58)(35,181,133,59)(36,182,134,60)(91,219,112,244)(92,220,113,245)(93,221,114,246)(94,222,115,247)(95,223,116,248)(96,224,117,249)(97,225,118,250)(98,226,119,251)(99,227,120,252)(100,228,121,235)(101,229,122,236)(102,230,123,237)(103,231,124,238)(104,232,125,239)(105,233,126,240)(106,234,109,241)(107,217,110,242)(108,218,111,243)(145,180,260,283)(146,163,261,284)(147,164,262,285)(148,165,263,286)(149,166,264,287)(150,167,265,288)(151,168,266,271)(152,169,267,272)(153,170,268,273)(154,171,269,274)(155,172,270,275)(156,173,253,276)(157,174,254,277)(158,175,255,278)(159,176,256,279)(160,177,257,280)(161,178,258,281)(162,179,259,282), (1,276,51,173)(2,277,52,174)(3,278,53,175)(4,279,54,176)(5,280,37,177)(6,281,38,178)(7,282,39,179)(8,283,40,180)(9,284,41,163)(10,285,42,164)(11,286,43,165)(12,287,44,166)(13,288,45,167)(14,271,46,168)(15,272,47,169)(16,273,48,170)(17,274,49,171)(18,275,50,172)(19,241,135,234)(20,242,136,217)(21,243,137,218)(22,244,138,219)(23,245,139,220)(24,246,140,221)(25,247,141,222)(26,248,142,223)(27,249,143,224)(28,250,144,225)(29,251,127,226)(30,252,128,227)(31,235,129,228)(32,236,130,229)(33,237,131,230)(34,238,132,231)(35,239,133,232)(36,240,134,233)(55,100,195,121)(56,101,196,122)(57,102,197,123)(58,103,198,124)(59,104,181,125)(60,105,182,126)(61,106,183,109)(62,107,184,110)(63,108,185,111)(64,91,186,112)(65,92,187,113)(66,93,188,114)(67,94,189,115)(68,95,190,116)(69,96,191,117)(70,97,192,118)(71,98,193,119)(72,99,194,120)(73,269,214,154)(74,270,215,155)(75,253,216,156)(76,254,199,157)(77,255,200,158)(78,256,201,159)(79,257,202,160)(80,258,203,161)(81,259,204,162)(82,260,205,145)(83,261,206,146)(84,262,207,147)(85,263,208,148)(86,264,209,149)(87,265,210,150)(88,266,211,151)(89,267,212,152)(90,268,213,153), (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), (1,66,10,57)(2,65,11,56)(3,64,12,55)(4,63,13,72)(5,62,14,71)(6,61,15,70)(7,60,16,69)(8,59,17,68)(9,58,18,67)(19,212,28,203)(20,211,29,202)(21,210,30,201)(22,209,31,200)(23,208,32,199)(24,207,33,216)(25,206,34,215)(26,205,35,214)(27,204,36,213)(37,184,46,193)(38,183,47,192)(39,182,48,191)(40,181,49,190)(41,198,50,189)(42,197,51,188)(43,196,52,187)(44,195,53,186)(45,194,54,185)(73,142,82,133)(74,141,83,132)(75,140,84,131)(76,139,85,130)(77,138,86,129)(78,137,87,128)(79,136,88,127)(80,135,89,144)(81,134,90,143)(91,149,100,158)(92,148,101,157)(93,147,102,156)(94,146,103,155)(95,145,104,154)(96,162,105,153)(97,161,106,152)(98,160,107,151)(99,159,108,150)(109,267,118,258)(110,266,119,257)(111,265,120,256)(112,264,121,255)(113,263,122,254)(114,262,123,253)(115,261,124,270)(116,260,125,269)(117,259,126,268)(163,238,172,247)(164,237,173,246)(165,236,174,245)(166,235,175,244)(167,252,176,243)(168,251,177,242)(169,250,178,241)(170,249,179,240)(171,248,180,239)(217,271,226,280)(218,288,227,279)(219,287,228,278)(220,286,229,277)(221,285,230,276)(222,284,231,275)(223,283,232,274)(224,282,233,273)(225,281,234,272) );
G=PermutationGroup([[(1,75,51,216),(2,76,52,199),(3,77,53,200),(4,78,54,201),(5,79,37,202),(6,80,38,203),(7,81,39,204),(8,82,40,205),(9,83,41,206),(10,84,42,207),(11,85,43,208),(12,86,44,209),(13,87,45,210),(14,88,46,211),(15,89,47,212),(16,90,48,213),(17,73,49,214),(18,74,50,215),(19,183,135,61),(20,184,136,62),(21,185,137,63),(22,186,138,64),(23,187,139,65),(24,188,140,66),(25,189,141,67),(26,190,142,68),(27,191,143,69),(28,192,144,70),(29,193,127,71),(30,194,128,72),(31,195,129,55),(32,196,130,56),(33,197,131,57),(34,198,132,58),(35,181,133,59),(36,182,134,60),(91,219,112,244),(92,220,113,245),(93,221,114,246),(94,222,115,247),(95,223,116,248),(96,224,117,249),(97,225,118,250),(98,226,119,251),(99,227,120,252),(100,228,121,235),(101,229,122,236),(102,230,123,237),(103,231,124,238),(104,232,125,239),(105,233,126,240),(106,234,109,241),(107,217,110,242),(108,218,111,243),(145,180,260,283),(146,163,261,284),(147,164,262,285),(148,165,263,286),(149,166,264,287),(150,167,265,288),(151,168,266,271),(152,169,267,272),(153,170,268,273),(154,171,269,274),(155,172,270,275),(156,173,253,276),(157,174,254,277),(158,175,255,278),(159,176,256,279),(160,177,257,280),(161,178,258,281),(162,179,259,282)], [(1,276,51,173),(2,277,52,174),(3,278,53,175),(4,279,54,176),(5,280,37,177),(6,281,38,178),(7,282,39,179),(8,283,40,180),(9,284,41,163),(10,285,42,164),(11,286,43,165),(12,287,44,166),(13,288,45,167),(14,271,46,168),(15,272,47,169),(16,273,48,170),(17,274,49,171),(18,275,50,172),(19,241,135,234),(20,242,136,217),(21,243,137,218),(22,244,138,219),(23,245,139,220),(24,246,140,221),(25,247,141,222),(26,248,142,223),(27,249,143,224),(28,250,144,225),(29,251,127,226),(30,252,128,227),(31,235,129,228),(32,236,130,229),(33,237,131,230),(34,238,132,231),(35,239,133,232),(36,240,134,233),(55,100,195,121),(56,101,196,122),(57,102,197,123),(58,103,198,124),(59,104,181,125),(60,105,182,126),(61,106,183,109),(62,107,184,110),(63,108,185,111),(64,91,186,112),(65,92,187,113),(66,93,188,114),(67,94,189,115),(68,95,190,116),(69,96,191,117),(70,97,192,118),(71,98,193,119),(72,99,194,120),(73,269,214,154),(74,270,215,155),(75,253,216,156),(76,254,199,157),(77,255,200,158),(78,256,201,159),(79,257,202,160),(80,258,203,161),(81,259,204,162),(82,260,205,145),(83,261,206,146),(84,262,207,147),(85,263,208,148),(86,264,209,149),(87,265,210,150),(88,266,211,151),(89,267,212,152),(90,268,213,153)], [(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)], [(1,66,10,57),(2,65,11,56),(3,64,12,55),(4,63,13,72),(5,62,14,71),(6,61,15,70),(7,60,16,69),(8,59,17,68),(9,58,18,67),(19,212,28,203),(20,211,29,202),(21,210,30,201),(22,209,31,200),(23,208,32,199),(24,207,33,216),(25,206,34,215),(26,205,35,214),(27,204,36,213),(37,184,46,193),(38,183,47,192),(39,182,48,191),(40,181,49,190),(41,198,50,189),(42,197,51,188),(43,196,52,187),(44,195,53,186),(45,194,54,185),(73,142,82,133),(74,141,83,132),(75,140,84,131),(76,139,85,130),(77,138,86,129),(78,137,87,128),(79,136,88,127),(80,135,89,144),(81,134,90,143),(91,149,100,158),(92,148,101,157),(93,147,102,156),(94,146,103,155),(95,145,104,154),(96,162,105,153),(97,161,106,152),(98,160,107,151),(99,159,108,150),(109,267,118,258),(110,266,119,257),(111,265,120,256),(112,264,121,255),(113,263,122,254),(114,262,123,253),(115,261,124,270),(116,260,125,269),(117,259,126,268),(163,238,172,247),(164,237,173,246),(165,236,174,245),(166,235,175,244),(167,252,176,243),(168,251,177,242),(169,250,178,241),(170,249,179,240),(171,248,180,239),(217,271,226,280),(218,288,227,279),(219,287,228,278),(220,286,229,277),(221,285,230,276),(222,284,231,275),(223,283,232,274),(224,282,233,273),(225,281,234,272)]])
54 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 | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | - | + | - | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | Dic3 | SD16 | Q16 | D9 | C3⋊D4 | C3⋊D4 | D18 | Dic9 | C9⋊D4 | C9⋊D4 | Q8⋊2S3 | C3⋊Q16 | C9⋊Q16 | Q8⋊2D9 |
| kernel | Q8⋊2Dic9 | C2×C9⋊C8 | C4⋊Dic9 | Q8×C18 | Q8×C9 | C6×Q8 | C36 | C2×C18 | C2×C12 | C3×Q8 | C18 | C18 | C2×Q8 | C12 | C2×C6 | C2×C4 | Q8 | C4 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 1 | 1 | 3 | 3 |
Matrix representation of Q8⋊2Dic9 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 52 | 19 |
| 0 | 0 | 19 | 21 |
| 70 | 42 | 0 | 0 |
| 31 | 28 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 25 | 62 | 0 | 0 |
| 37 | 48 | 0 | 0 |
| 0 | 0 | 32 | 17 |
| 0 | 0 | 17 | 41 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[72,0,0,0,0,72,0,0,0,0,52,19,0,0,19,21],[70,31,0,0,42,28,0,0,0,0,72,0,0,0,0,72],[25,37,0,0,62,48,0,0,0,0,32,17,0,0,17,41] >;
Q8⋊2Dic9 in GAP, Magma, Sage, TeX
Q_8\rtimes_2{\rm Dic}_9 % in TeX
G:=Group("Q8:2Dic9"); // GroupNames label
G:=SmallGroup(288,43);
// by ID
G=gap.SmallGroup(288,43);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,120,675,346,80,6725,292,9414]);
// 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*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations