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## G = Dic9⋊Q8order 288 = 25·32

### 2nd semidirect product of Dic9 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — Dic9⋊Q8
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C4×Dic9 — Dic9⋊Q8
 Lower central C9 — C2×C18 — Dic9⋊Q8
 Upper central C1 — C22 — C2×Q8

Generators and relations for Dic9⋊Q8
G = < a,b,c,d | a18=c4=1, b2=a9, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a9b, bd=db, dcd-1=c-1 >

Subgroups: 356 in 102 conjugacy classes, 48 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C9, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×Q8, C2×Q8, C18, C18, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4⋊Q8, Dic9, Dic9, C36, C36, C2×C18, C4×Dic3, Dic3⋊C4, C2×Dic6, C6×Q8, Dic18, C2×Dic9, C2×C36, C2×C36, Q8×C9, Dic3⋊Q8, C4×Dic9, Dic9⋊C4, C2×Dic18, Q8×C18, Dic9⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, D9, C3⋊D4, C22×S3, C4⋊Q8, D18, S3×Q8, C2×C3⋊D4, C9⋊D4, C22×D9, Dic3⋊Q8, Q8×D9, C2×C9⋊D4, Dic9⋊Q8

Smallest permutation representation of Dic9⋊Q8
Regular action on 288 points
Generators in S288
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162)(163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198)(199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234)(235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252)(253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270)(271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288)
(1 73 10 82)(2 90 11 81)(3 89 12 80)(4 88 13 79)(5 87 14 78)(6 86 15 77)(7 85 16 76)(8 84 17 75)(9 83 18 74)(19 284 28 275)(20 283 29 274)(21 282 30 273)(22 281 31 272)(23 280 32 271)(24 279 33 288)(25 278 34 287)(26 277 35 286)(27 276 36 285)(37 127 46 136)(38 144 47 135)(39 143 48 134)(40 142 49 133)(41 141 50 132)(42 140 51 131)(43 139 52 130)(44 138 53 129)(45 137 54 128)(55 160 64 151)(56 159 65 150)(57 158 66 149)(58 157 67 148)(59 156 68 147)(60 155 69 146)(61 154 70 145)(62 153 71 162)(63 152 72 161)(91 190 100 181)(92 189 101 198)(93 188 102 197)(94 187 103 196)(95 186 104 195)(96 185 105 194)(97 184 106 193)(98 183 107 192)(99 182 108 191)(109 250 118 241)(110 249 119 240)(111 248 120 239)(112 247 121 238)(113 246 122 237)(114 245 123 236)(115 244 124 235)(116 243 125 252)(117 242 126 251)(163 209 172 200)(164 208 173 199)(165 207 174 216)(166 206 175 215)(167 205 176 214)(168 204 177 213)(169 203 178 212)(170 202 179 211)(171 201 180 210)(217 270 226 261)(218 269 227 260)(219 268 228 259)(220 267 229 258)(221 266 230 257)(222 265 231 256)(223 264 232 255)(224 263 233 254)(225 262 234 253)
(1 162 164 251)(2 145 165 252)(3 146 166 235)(4 147 167 236)(5 148 168 237)(6 149 169 238)(7 150 170 239)(8 151 171 240)(9 152 172 241)(10 153 173 242)(11 154 174 243)(12 155 175 244)(13 156 176 245)(14 157 177 246)(15 158 178 247)(16 159 179 248)(17 160 180 249)(18 161 163 250)(19 106 266 42)(20 107 267 43)(21 108 268 44)(22 91 269 45)(23 92 270 46)(24 93 253 47)(25 94 254 48)(26 95 255 49)(27 96 256 50)(28 97 257 51)(29 98 258 52)(30 99 259 53)(31 100 260 54)(32 101 261 37)(33 102 262 38)(34 103 263 39)(35 104 264 40)(36 105 265 41)(55 210 110 75)(56 211 111 76)(57 212 112 77)(58 213 113 78)(59 214 114 79)(60 215 115 80)(61 216 116 81)(62 199 117 82)(63 200 118 83)(64 201 119 84)(65 202 120 85)(66 203 121 86)(67 204 122 87)(68 205 123 88)(69 206 124 89)(70 207 125 90)(71 208 126 73)(72 209 109 74)(127 280 198 226)(128 281 181 227)(129 282 182 228)(130 283 183 229)(131 284 184 230)(132 285 185 231)(133 286 186 232)(134 287 187 233)(135 288 188 234)(136 271 189 217)(137 272 190 218)(138 273 191 219)(139 274 192 220)(140 275 193 221)(141 276 194 222)(142 277 195 223)(143 278 196 224)(144 279 197 225)
(1 185 164 132)(2 186 165 133)(3 187 166 134)(4 188 167 135)(5 189 168 136)(6 190 169 137)(7 191 170 138)(8 192 171 139)(9 193 172 140)(10 194 173 141)(11 195 174 142)(12 196 175 143)(13 197 176 144)(14 198 177 127)(15 181 178 128)(16 182 179 129)(17 183 180 130)(18 184 163 131)(19 109 266 72)(20 110 267 55)(21 111 268 56)(22 112 269 57)(23 113 270 58)(24 114 253 59)(25 115 254 60)(26 116 255 61)(27 117 256 62)(28 118 257 63)(29 119 258 64)(30 120 259 65)(31 121 260 66)(32 122 261 67)(33 123 262 68)(34 124 263 69)(35 125 264 70)(36 126 265 71)(37 87 101 204)(38 88 102 205)(39 89 103 206)(40 90 104 207)(41 73 105 208)(42 74 106 209)(43 75 107 210)(44 76 108 211)(45 77 91 212)(46 78 92 213)(47 79 93 214)(48 80 94 215)(49 81 95 216)(50 82 96 199)(51 83 97 200)(52 84 98 201)(53 85 99 202)(54 86 100 203)(145 286 252 232)(146 287 235 233)(147 288 236 234)(148 271 237 217)(149 272 238 218)(150 273 239 219)(151 274 240 220)(152 275 241 221)(153 276 242 222)(154 277 243 223)(155 278 244 224)(156 279 245 225)(157 280 246 226)(158 281 247 227)(159 282 248 228)(160 283 249 229)(161 284 250 230)(162 285 251 231)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 4 4 18 18 18 18 36 36 2 2 2 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 4 4 type + + + + + + - + + + + - - image C1 C2 C2 C2 C2 S3 Q8 D4 D6 D9 C3⋊D4 D18 C9⋊D4 S3×Q8 Q8×D9 kernel Dic9⋊Q8 C4×Dic9 Dic9⋊C4 C2×Dic18 Q8×C18 C6×Q8 Dic9 C36 C2×C12 C2×Q8 C12 C2×C4 C4 C6 C2 # reps 1 1 4 1 1 1 4 2 3 3 4 9 12 2 6

Matrix representation of Dic9⋊Q8 in GL6(𝔽37)

 17 11 0 0 0 0 26 6 0 0 0 0 0 0 9 34 0 0 0 0 12 29 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 5 10 0 0 0 0 5 32 0 0 0 0 0 0 8 32 0 0 0 0 13 29 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 29 5 0 0 0 0 17 8 0 0 0 0 0 0 30 9 0 0 0 0 15 7
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 2 0 0 0 0 1 21

G:=sub<GL(6,GF(37))| [17,26,0,0,0,0,11,6,0,0,0,0,0,0,9,12,0,0,0,0,34,29,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,5,0,0,0,0,10,32,0,0,0,0,0,0,8,13,0,0,0,0,32,29,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,29,17,0,0,0,0,5,8,0,0,0,0,0,0,30,15,0,0,0,0,9,7],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,2,21] >;

Dic9⋊Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_9\rtimes Q_8
% in TeX

G:=Group("Dic9:Q8");
// GroupNames label

G:=SmallGroup(288,154);
// by ID

G=gap.SmallGroup(288,154);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,422,135,58,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^18=c^4=1,b^2=a^9,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽