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G = Q82Dic9order 288 = 25·32

1st semidirect product of Q8 and Dic9 acting via Dic9/C18=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q82Dic9, C36.10D4, C18.5Q16, C18.8SD16, (Q8×C9)⋊1C4, C36.8(C2×C4), (C2×Q8).4D9, C93(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.(Q82Dic3), (C3×Q8).5Dic3, C12.2(C2×Dic3), C6.8(C3⋊Q16), C2.3(C9⋊Q16), (C2×C36).26C22, C2.3(Q82D9), C6.9(Q82S3), 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

C1C36 — Q82Dic9
C1C3C9C18C2×C18C2×C36C4⋊Dic9 — Q82Dic9
C9C18C36 — Q82Dic9
C1C22C2×C4C2×Q8

Generators and relations for Q82Dic9
 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 [×3], C3, C4 [×2], C4 [×3], C22, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×2], Q8, C9, Dic3, C12 [×2], C12 [×2], C2×C6, C4⋊C4, C2×C8, C2×Q8, C18 [×3], C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, Q8⋊C4, Dic9, C36 [×2], C36 [×2], C2×C18, C2×C3⋊C8, C4⋊Dic3, C6×Q8, C9⋊C8, C2×Dic9, C2×C36, C2×C36, Q8×C9 [×2], Q8×C9, Q82Dic3, C2×C9⋊C8, C4⋊Dic9, Q8×C18, Q82Dic9
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, SD16, Q16, D9, C2×Dic3, C3⋊D4 [×2], Q8⋊C4, Dic9 [×2], D18, Q82S3, C3⋊Q16, C6.D4, C2×Dic9, C9⋊D4 [×2], Q82Dic3, C9⋊Q16, Q82D9, C18.D4, Q82Dic9

Smallest permutation representation of Q82Dic9
Regular action on 288 points
Generators in S288
(1 26 156 117)(2 27 157 118)(3 28 158 119)(4 29 159 120)(5 30 160 121)(6 31 161 122)(7 32 162 123)(8 33 145 124)(9 34 146 125)(10 35 147 126)(11 36 148 109)(12 19 149 110)(13 20 150 111)(14 21 151 112)(15 22 152 113)(16 23 153 114)(17 24 154 115)(18 25 155 116)(37 287 206 86)(38 288 207 87)(39 271 208 88)(40 272 209 89)(41 273 210 90)(42 274 211 73)(43 275 212 74)(44 276 213 75)(45 277 214 76)(46 278 215 77)(47 279 216 78)(48 280 199 79)(49 281 200 80)(50 282 201 81)(51 283 202 82)(52 284 203 83)(53 285 204 84)(54 286 205 85)(55 270 239 163)(56 253 240 164)(57 254 241 165)(58 255 242 166)(59 256 243 167)(60 257 244 168)(61 258 245 169)(62 259 246 170)(63 260 247 171)(64 261 248 172)(65 262 249 173)(66 263 250 174)(67 264 251 175)(68 265 252 176)(69 266 235 177)(70 267 236 178)(71 268 237 179)(72 269 238 180)(91 233 128 190)(92 234 129 191)(93 217 130 192)(94 218 131 193)(95 219 132 194)(96 220 133 195)(97 221 134 196)(98 222 135 197)(99 223 136 198)(100 224 137 181)(101 225 138 182)(102 226 139 183)(103 227 140 184)(104 228 141 185)(105 229 142 186)(106 230 143 187)(107 231 144 188)(108 232 127 189)
(1 38 156 207)(2 39 157 208)(3 40 158 209)(4 41 159 210)(5 42 160 211)(6 43 161 212)(7 44 162 213)(8 45 145 214)(9 46 146 215)(10 47 147 216)(11 48 148 199)(12 49 149 200)(13 50 150 201)(14 51 151 202)(15 52 152 203)(16 53 153 204)(17 54 154 205)(18 37 155 206)(19 80 110 281)(20 81 111 282)(21 82 112 283)(22 83 113 284)(23 84 114 285)(24 85 115 286)(25 86 116 287)(26 87 117 288)(27 88 118 271)(28 89 119 272)(29 90 120 273)(30 73 121 274)(31 74 122 275)(32 75 123 276)(33 76 124 277)(34 77 125 278)(35 78 126 279)(36 79 109 280)(55 139 239 102)(56 140 240 103)(57 141 241 104)(58 142 242 105)(59 143 243 106)(60 144 244 107)(61 127 245 108)(62 128 246 91)(63 129 247 92)(64 130 248 93)(65 131 249 94)(66 132 250 95)(67 133 251 96)(68 134 252 97)(69 135 235 98)(70 136 236 99)(71 137 237 100)(72 138 238 101)(163 183 270 226)(164 184 253 227)(165 185 254 228)(166 186 255 229)(167 187 256 230)(168 188 257 231)(169 189 258 232)(170 190 259 233)(171 191 260 234)(172 192 261 217)(173 193 262 218)(174 194 263 219)(175 195 264 220)(176 196 265 221)(177 197 266 222)(178 198 267 223)(179 181 268 224)(180 182 269 225)
(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)
(1 227 10 218)(2 226 11 217)(3 225 12 234)(4 224 13 233)(5 223 14 232)(6 222 15 231)(7 221 16 230)(8 220 17 229)(9 219 18 228)(19 92 28 101)(20 91 29 100)(21 108 30 99)(22 107 31 98)(23 106 32 97)(24 105 33 96)(25 104 34 95)(26 103 35 94)(27 102 36 93)(37 57 46 66)(38 56 47 65)(39 55 48 64)(40 72 49 63)(41 71 50 62)(42 70 51 61)(43 69 52 60)(44 68 53 59)(45 67 54 58)(73 267 82 258)(74 266 83 257)(75 265 84 256)(76 264 85 255)(77 263 86 254)(78 262 87 253)(79 261 88 270)(80 260 89 269)(81 259 90 268)(109 130 118 139)(110 129 119 138)(111 128 120 137)(112 127 121 136)(113 144 122 135)(114 143 123 134)(115 142 124 133)(116 141 125 132)(117 140 126 131)(145 195 154 186)(146 194 155 185)(147 193 156 184)(148 192 157 183)(149 191 158 182)(150 190 159 181)(151 189 160 198)(152 188 161 197)(153 187 162 196)(163 280 172 271)(164 279 173 288)(165 278 174 287)(166 277 175 286)(167 276 176 285)(168 275 177 284)(169 274 178 283)(170 273 179 282)(171 272 180 281)(199 248 208 239)(200 247 209 238)(201 246 210 237)(202 245 211 236)(203 244 212 235)(204 243 213 252)(205 242 214 251)(206 241 215 250)(207 240 216 249)

G:=sub<Sym(288)| (1,26,156,117)(2,27,157,118)(3,28,158,119)(4,29,159,120)(5,30,160,121)(6,31,161,122)(7,32,162,123)(8,33,145,124)(9,34,146,125)(10,35,147,126)(11,36,148,109)(12,19,149,110)(13,20,150,111)(14,21,151,112)(15,22,152,113)(16,23,153,114)(17,24,154,115)(18,25,155,116)(37,287,206,86)(38,288,207,87)(39,271,208,88)(40,272,209,89)(41,273,210,90)(42,274,211,73)(43,275,212,74)(44,276,213,75)(45,277,214,76)(46,278,215,77)(47,279,216,78)(48,280,199,79)(49,281,200,80)(50,282,201,81)(51,283,202,82)(52,284,203,83)(53,285,204,84)(54,286,205,85)(55,270,239,163)(56,253,240,164)(57,254,241,165)(58,255,242,166)(59,256,243,167)(60,257,244,168)(61,258,245,169)(62,259,246,170)(63,260,247,171)(64,261,248,172)(65,262,249,173)(66,263,250,174)(67,264,251,175)(68,265,252,176)(69,266,235,177)(70,267,236,178)(71,268,237,179)(72,269,238,180)(91,233,128,190)(92,234,129,191)(93,217,130,192)(94,218,131,193)(95,219,132,194)(96,220,133,195)(97,221,134,196)(98,222,135,197)(99,223,136,198)(100,224,137,181)(101,225,138,182)(102,226,139,183)(103,227,140,184)(104,228,141,185)(105,229,142,186)(106,230,143,187)(107,231,144,188)(108,232,127,189), (1,38,156,207)(2,39,157,208)(3,40,158,209)(4,41,159,210)(5,42,160,211)(6,43,161,212)(7,44,162,213)(8,45,145,214)(9,46,146,215)(10,47,147,216)(11,48,148,199)(12,49,149,200)(13,50,150,201)(14,51,151,202)(15,52,152,203)(16,53,153,204)(17,54,154,205)(18,37,155,206)(19,80,110,281)(20,81,111,282)(21,82,112,283)(22,83,113,284)(23,84,114,285)(24,85,115,286)(25,86,116,287)(26,87,117,288)(27,88,118,271)(28,89,119,272)(29,90,120,273)(30,73,121,274)(31,74,122,275)(32,75,123,276)(33,76,124,277)(34,77,125,278)(35,78,126,279)(36,79,109,280)(55,139,239,102)(56,140,240,103)(57,141,241,104)(58,142,242,105)(59,143,243,106)(60,144,244,107)(61,127,245,108)(62,128,246,91)(63,129,247,92)(64,130,248,93)(65,131,249,94)(66,132,250,95)(67,133,251,96)(68,134,252,97)(69,135,235,98)(70,136,236,99)(71,137,237,100)(72,138,238,101)(163,183,270,226)(164,184,253,227)(165,185,254,228)(166,186,255,229)(167,187,256,230)(168,188,257,231)(169,189,258,232)(170,190,259,233)(171,191,260,234)(172,192,261,217)(173,193,262,218)(174,194,263,219)(175,195,264,220)(176,196,265,221)(177,197,266,222)(178,198,267,223)(179,181,268,224)(180,182,269,225), (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), (1,227,10,218)(2,226,11,217)(3,225,12,234)(4,224,13,233)(5,223,14,232)(6,222,15,231)(7,221,16,230)(8,220,17,229)(9,219,18,228)(19,92,28,101)(20,91,29,100)(21,108,30,99)(22,107,31,98)(23,106,32,97)(24,105,33,96)(25,104,34,95)(26,103,35,94)(27,102,36,93)(37,57,46,66)(38,56,47,65)(39,55,48,64)(40,72,49,63)(41,71,50,62)(42,70,51,61)(43,69,52,60)(44,68,53,59)(45,67,54,58)(73,267,82,258)(74,266,83,257)(75,265,84,256)(76,264,85,255)(77,263,86,254)(78,262,87,253)(79,261,88,270)(80,260,89,269)(81,259,90,268)(109,130,118,139)(110,129,119,138)(111,128,120,137)(112,127,121,136)(113,144,122,135)(114,143,123,134)(115,142,124,133)(116,141,125,132)(117,140,126,131)(145,195,154,186)(146,194,155,185)(147,193,156,184)(148,192,157,183)(149,191,158,182)(150,190,159,181)(151,189,160,198)(152,188,161,197)(153,187,162,196)(163,280,172,271)(164,279,173,288)(165,278,174,287)(166,277,175,286)(167,276,176,285)(168,275,177,284)(169,274,178,283)(170,273,179,282)(171,272,180,281)(199,248,208,239)(200,247,209,238)(201,246,210,237)(202,245,211,236)(203,244,212,235)(204,243,213,252)(205,242,214,251)(206,241,215,250)(207,240,216,249)>;

G:=Group( (1,26,156,117)(2,27,157,118)(3,28,158,119)(4,29,159,120)(5,30,160,121)(6,31,161,122)(7,32,162,123)(8,33,145,124)(9,34,146,125)(10,35,147,126)(11,36,148,109)(12,19,149,110)(13,20,150,111)(14,21,151,112)(15,22,152,113)(16,23,153,114)(17,24,154,115)(18,25,155,116)(37,287,206,86)(38,288,207,87)(39,271,208,88)(40,272,209,89)(41,273,210,90)(42,274,211,73)(43,275,212,74)(44,276,213,75)(45,277,214,76)(46,278,215,77)(47,279,216,78)(48,280,199,79)(49,281,200,80)(50,282,201,81)(51,283,202,82)(52,284,203,83)(53,285,204,84)(54,286,205,85)(55,270,239,163)(56,253,240,164)(57,254,241,165)(58,255,242,166)(59,256,243,167)(60,257,244,168)(61,258,245,169)(62,259,246,170)(63,260,247,171)(64,261,248,172)(65,262,249,173)(66,263,250,174)(67,264,251,175)(68,265,252,176)(69,266,235,177)(70,267,236,178)(71,268,237,179)(72,269,238,180)(91,233,128,190)(92,234,129,191)(93,217,130,192)(94,218,131,193)(95,219,132,194)(96,220,133,195)(97,221,134,196)(98,222,135,197)(99,223,136,198)(100,224,137,181)(101,225,138,182)(102,226,139,183)(103,227,140,184)(104,228,141,185)(105,229,142,186)(106,230,143,187)(107,231,144,188)(108,232,127,189), (1,38,156,207)(2,39,157,208)(3,40,158,209)(4,41,159,210)(5,42,160,211)(6,43,161,212)(7,44,162,213)(8,45,145,214)(9,46,146,215)(10,47,147,216)(11,48,148,199)(12,49,149,200)(13,50,150,201)(14,51,151,202)(15,52,152,203)(16,53,153,204)(17,54,154,205)(18,37,155,206)(19,80,110,281)(20,81,111,282)(21,82,112,283)(22,83,113,284)(23,84,114,285)(24,85,115,286)(25,86,116,287)(26,87,117,288)(27,88,118,271)(28,89,119,272)(29,90,120,273)(30,73,121,274)(31,74,122,275)(32,75,123,276)(33,76,124,277)(34,77,125,278)(35,78,126,279)(36,79,109,280)(55,139,239,102)(56,140,240,103)(57,141,241,104)(58,142,242,105)(59,143,243,106)(60,144,244,107)(61,127,245,108)(62,128,246,91)(63,129,247,92)(64,130,248,93)(65,131,249,94)(66,132,250,95)(67,133,251,96)(68,134,252,97)(69,135,235,98)(70,136,236,99)(71,137,237,100)(72,138,238,101)(163,183,270,226)(164,184,253,227)(165,185,254,228)(166,186,255,229)(167,187,256,230)(168,188,257,231)(169,189,258,232)(170,190,259,233)(171,191,260,234)(172,192,261,217)(173,193,262,218)(174,194,263,219)(175,195,264,220)(176,196,265,221)(177,197,266,222)(178,198,267,223)(179,181,268,224)(180,182,269,225), (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), (1,227,10,218)(2,226,11,217)(3,225,12,234)(4,224,13,233)(5,223,14,232)(6,222,15,231)(7,221,16,230)(8,220,17,229)(9,219,18,228)(19,92,28,101)(20,91,29,100)(21,108,30,99)(22,107,31,98)(23,106,32,97)(24,105,33,96)(25,104,34,95)(26,103,35,94)(27,102,36,93)(37,57,46,66)(38,56,47,65)(39,55,48,64)(40,72,49,63)(41,71,50,62)(42,70,51,61)(43,69,52,60)(44,68,53,59)(45,67,54,58)(73,267,82,258)(74,266,83,257)(75,265,84,256)(76,264,85,255)(77,263,86,254)(78,262,87,253)(79,261,88,270)(80,260,89,269)(81,259,90,268)(109,130,118,139)(110,129,119,138)(111,128,120,137)(112,127,121,136)(113,144,122,135)(114,143,123,134)(115,142,124,133)(116,141,125,132)(117,140,126,131)(145,195,154,186)(146,194,155,185)(147,193,156,184)(148,192,157,183)(149,191,158,182)(150,190,159,181)(151,189,160,198)(152,188,161,197)(153,187,162,196)(163,280,172,271)(164,279,173,288)(165,278,174,287)(166,277,175,286)(167,276,176,285)(168,275,177,284)(169,274,178,283)(170,273,179,282)(171,272,180,281)(199,248,208,239)(200,247,209,238)(201,246,210,237)(202,245,211,236)(203,244,212,235)(204,243,213,252)(205,242,214,251)(206,241,215,250)(207,240,216,249) );

G=PermutationGroup([(1,26,156,117),(2,27,157,118),(3,28,158,119),(4,29,159,120),(5,30,160,121),(6,31,161,122),(7,32,162,123),(8,33,145,124),(9,34,146,125),(10,35,147,126),(11,36,148,109),(12,19,149,110),(13,20,150,111),(14,21,151,112),(15,22,152,113),(16,23,153,114),(17,24,154,115),(18,25,155,116),(37,287,206,86),(38,288,207,87),(39,271,208,88),(40,272,209,89),(41,273,210,90),(42,274,211,73),(43,275,212,74),(44,276,213,75),(45,277,214,76),(46,278,215,77),(47,279,216,78),(48,280,199,79),(49,281,200,80),(50,282,201,81),(51,283,202,82),(52,284,203,83),(53,285,204,84),(54,286,205,85),(55,270,239,163),(56,253,240,164),(57,254,241,165),(58,255,242,166),(59,256,243,167),(60,257,244,168),(61,258,245,169),(62,259,246,170),(63,260,247,171),(64,261,248,172),(65,262,249,173),(66,263,250,174),(67,264,251,175),(68,265,252,176),(69,266,235,177),(70,267,236,178),(71,268,237,179),(72,269,238,180),(91,233,128,190),(92,234,129,191),(93,217,130,192),(94,218,131,193),(95,219,132,194),(96,220,133,195),(97,221,134,196),(98,222,135,197),(99,223,136,198),(100,224,137,181),(101,225,138,182),(102,226,139,183),(103,227,140,184),(104,228,141,185),(105,229,142,186),(106,230,143,187),(107,231,144,188),(108,232,127,189)], [(1,38,156,207),(2,39,157,208),(3,40,158,209),(4,41,159,210),(5,42,160,211),(6,43,161,212),(7,44,162,213),(8,45,145,214),(9,46,146,215),(10,47,147,216),(11,48,148,199),(12,49,149,200),(13,50,150,201),(14,51,151,202),(15,52,152,203),(16,53,153,204),(17,54,154,205),(18,37,155,206),(19,80,110,281),(20,81,111,282),(21,82,112,283),(22,83,113,284),(23,84,114,285),(24,85,115,286),(25,86,116,287),(26,87,117,288),(27,88,118,271),(28,89,119,272),(29,90,120,273),(30,73,121,274),(31,74,122,275),(32,75,123,276),(33,76,124,277),(34,77,125,278),(35,78,126,279),(36,79,109,280),(55,139,239,102),(56,140,240,103),(57,141,241,104),(58,142,242,105),(59,143,243,106),(60,144,244,107),(61,127,245,108),(62,128,246,91),(63,129,247,92),(64,130,248,93),(65,131,249,94),(66,132,250,95),(67,133,251,96),(68,134,252,97),(69,135,235,98),(70,136,236,99),(71,137,237,100),(72,138,238,101),(163,183,270,226),(164,184,253,227),(165,185,254,228),(166,186,255,229),(167,187,256,230),(168,188,257,231),(169,189,258,232),(170,190,259,233),(171,191,260,234),(172,192,261,217),(173,193,262,218),(174,194,263,219),(175,195,264,220),(176,196,265,221),(177,197,266,222),(178,198,267,223),(179,181,268,224),(180,182,269,225)], [(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)], [(1,227,10,218),(2,226,11,217),(3,225,12,234),(4,224,13,233),(5,223,14,232),(6,222,15,231),(7,221,16,230),(8,220,17,229),(9,219,18,228),(19,92,28,101),(20,91,29,100),(21,108,30,99),(22,107,31,98),(23,106,32,97),(24,105,33,96),(25,104,34,95),(26,103,35,94),(27,102,36,93),(37,57,46,66),(38,56,47,65),(39,55,48,64),(40,72,49,63),(41,71,50,62),(42,70,51,61),(43,69,52,60),(44,68,53,59),(45,67,54,58),(73,267,82,258),(74,266,83,257),(75,265,84,256),(76,264,85,255),(77,263,86,254),(78,262,87,253),(79,261,88,270),(80,260,89,269),(81,259,90,268),(109,130,118,139),(110,129,119,138),(111,128,120,137),(112,127,121,136),(113,144,122,135),(114,143,123,134),(115,142,124,133),(116,141,125,132),(117,140,126,131),(145,195,154,186),(146,194,155,185),(147,193,156,184),(148,192,157,183),(149,191,158,182),(150,190,159,181),(151,189,160,198),(152,188,161,197),(153,187,162,196),(163,280,172,271),(164,279,173,288),(165,278,174,287),(166,277,175,286),(167,276,176,285),(168,275,177,284),(169,274,178,283),(170,273,179,282),(171,272,180,281),(199,248,208,239),(200,247,209,238),(201,246,210,237),(202,245,211,236),(203,244,212,235),(204,243,213,252),(205,242,214,251),(206,241,215,250),(207,240,216,249)])

54 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F6A6B6C8A8B8C8D9A9B9C12A···12F18A···18I36A···36R
order12223444444666888899912···1218···1836···36
size1111222443636222181818182224···42···24···4

54 irreducible representations

dim11111222222222222224444
type++++++++--++-+--+
imageC1C2C2C2C4S3D4D4D6Dic3SD16Q16D9C3⋊D4C3⋊D4D18Dic9C9⋊D4C9⋊D4Q82S3C3⋊Q16C9⋊Q16Q82D9
kernelQ82Dic9C2×C9⋊C8C4⋊Dic9Q8×C18Q8×C9C6×Q8C36C2×C18C2×C12C3×Q8C18C18C2×Q8C12C2×C6C2×C4Q8C4C22C6C6C2C2
# reps11114111122232236661133

Matrix representation of Q82Dic9 in GL4(𝔽73) generated by

1000
0100
0001
00720
,
72000
07200
005219
001921
,
704200
312800
00720
00072
,
256200
374800
003217
001741
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] >;

Q82Dic9 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

׿
×
𝔽