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G = Dic9⋊Q8order 288 = 25·32

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic92Q8, C36.21D4, C93(C4⋊Q8), C2.8(Q8×D9), (C2×Q8).6D9, C6.38(S3×Q8), (C2×C4).18D18, C18.56(C2×D4), (Q8×C18).4C2, (C6×Q8).14S3, C18.15(C2×Q8), (C2×C12).218D6, C4.10(C9⋊D4), Dic9⋊C4.6C2, (C4×Dic9).3C2, C12.18(C3⋊D4), C3.(Dic3⋊Q8), (C2×C36).62C22, (C2×C18).56C23, (C2×Dic18).9C2, C22.63(C22×D9), (C2×Dic9).42C22, C2.20(C2×C9⋊D4), C6.104(C2×C3⋊D4), (C2×C6).213(C22×S3), SmallGroup(288,154)

Series: Derived Chief Lower central Upper central

C1C2×C18 — Dic9⋊Q8
C1C3C9C18C2×C18C2×Dic9C4×Dic9 — Dic9⋊Q8
C9C2×C18 — Dic9⋊Q8
C1C22C2×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 [×2], C3, C4 [×2], C4 [×8], C22, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C9, Dic3 [×6], C12 [×2], C12 [×2], C2×C6, C42, C4⋊C4 [×4], C2×Q8, C2×Q8, C18, C18 [×2], Dic6 [×2], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C3×Q8 [×2], C4⋊Q8, Dic9 [×4], Dic9 [×2], C36 [×2], C36 [×2], C2×C18, C4×Dic3, Dic3⋊C4 [×4], C2×Dic6, C6×Q8, Dic18 [×2], C2×Dic9 [×4], C2×C36, C2×C36 [×2], Q8×C9 [×2], Dic3⋊Q8, C4×Dic9, Dic9⋊C4 [×4], C2×Dic18, Q8×C18, Dic9⋊Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D6 [×3], C2×D4, C2×Q8 [×2], D9, C3⋊D4 [×2], C22×S3, C4⋊Q8, D18 [×3], S3×Q8 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, Dic3⋊Q8, Q8×D9 [×2], 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 63 10 72)(2 62 11 71)(3 61 12 70)(4 60 13 69)(5 59 14 68)(6 58 15 67)(7 57 16 66)(8 56 17 65)(9 55 18 64)(19 177 28 168)(20 176 29 167)(21 175 30 166)(22 174 31 165)(23 173 32 164)(24 172 33 163)(25 171 34 180)(26 170 35 179)(27 169 36 178)(37 239 46 248)(38 238 47 247)(39 237 48 246)(40 236 49 245)(41 235 50 244)(42 252 51 243)(43 251 52 242)(44 250 53 241)(45 249 54 240)(73 282 82 273)(74 281 83 272)(75 280 84 271)(76 279 85 288)(77 278 86 287)(78 277 87 286)(79 276 88 285)(80 275 89 284)(81 274 90 283)(91 265 100 256)(92 264 101 255)(93 263 102 254)(94 262 103 253)(95 261 104 270)(96 260 105 269)(97 259 106 268)(98 258 107 267)(99 257 108 266)(109 160 118 151)(110 159 119 150)(111 158 120 149)(112 157 121 148)(113 156 122 147)(114 155 123 146)(115 154 124 145)(116 153 125 162)(117 152 126 161)(127 201 136 210)(128 200 137 209)(129 199 138 208)(130 216 139 207)(131 215 140 206)(132 214 141 205)(133 213 142 204)(134 212 143 203)(135 211 144 202)(181 227 190 218)(182 226 191 217)(183 225 192 234)(184 224 193 233)(185 223 194 232)(186 222 195 231)(187 221 196 230)(188 220 197 229)(189 219 198 228)
(1 179 192 47)(2 180 193 48)(3 163 194 49)(4 164 195 50)(5 165 196 51)(6 166 197 52)(7 167 198 53)(8 168 181 54)(9 169 182 37)(10 170 183 38)(11 171 184 39)(12 172 185 40)(13 173 186 41)(14 174 187 42)(15 175 188 43)(16 176 189 44)(17 177 190 45)(18 178 191 46)(19 218 240 65)(20 219 241 66)(21 220 242 67)(22 221 243 68)(23 222 244 69)(24 223 245 70)(25 224 246 71)(26 225 247 72)(27 226 248 55)(28 227 249 56)(29 228 250 57)(30 229 251 58)(31 230 252 59)(32 231 235 60)(33 232 236 61)(34 233 237 62)(35 234 238 63)(36 217 239 64)(73 143 146 94)(74 144 147 95)(75 127 148 96)(76 128 149 97)(77 129 150 98)(78 130 151 99)(79 131 152 100)(80 132 153 101)(81 133 154 102)(82 134 155 103)(83 135 156 104)(84 136 157 105)(85 137 158 106)(86 138 159 107)(87 139 160 108)(88 140 161 91)(89 141 162 92)(90 142 145 93)(109 266 277 207)(110 267 278 208)(111 268 279 209)(112 269 280 210)(113 270 281 211)(114 253 282 212)(115 254 283 213)(116 255 284 214)(117 256 285 215)(118 257 286 216)(119 258 287 199)(120 259 288 200)(121 260 271 201)(122 261 272 202)(123 262 273 203)(124 263 274 204)(125 264 275 205)(126 265 276 206)
(1 206 192 265)(2 207 193 266)(3 208 194 267)(4 209 195 268)(5 210 196 269)(6 211 197 270)(7 212 198 253)(8 213 181 254)(9 214 182 255)(10 215 183 256)(11 216 184 257)(12 199 185 258)(13 200 186 259)(14 201 187 260)(15 202 188 261)(16 203 189 262)(17 204 190 263)(18 205 191 264)(19 81 240 154)(20 82 241 155)(21 83 242 156)(22 84 243 157)(23 85 244 158)(24 86 245 159)(25 87 246 160)(26 88 247 161)(27 89 248 162)(28 90 249 145)(29 73 250 146)(30 74 251 147)(31 75 252 148)(32 76 235 149)(33 77 236 150)(34 78 237 151)(35 79 238 152)(36 80 239 153)(37 116 169 284)(38 117 170 285)(39 118 171 286)(40 119 172 287)(41 120 173 288)(42 121 174 271)(43 122 175 272)(44 123 176 273)(45 124 177 274)(46 125 178 275)(47 126 179 276)(48 109 180 277)(49 110 163 278)(50 111 164 279)(51 112 165 280)(52 113 166 281)(53 114 167 282)(54 115 168 283)(55 141 226 92)(56 142 227 93)(57 143 228 94)(58 144 229 95)(59 127 230 96)(60 128 231 97)(61 129 232 98)(62 130 233 99)(63 131 234 100)(64 132 217 101)(65 133 218 102)(66 134 219 103)(67 135 220 104)(68 136 221 105)(69 137 222 106)(70 138 223 107)(71 139 224 108)(72 140 225 91)

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

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

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

54 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J6A6B6C9A9B9C12A···12F18A···18I36A···36R
order12223444444444466699912···1218···1836···36
size1111222441818181836362222224···42···24···4

54 irreducible representations

dim111112222222244
type++++++-++++--
imageC1C2C2C2C2S3Q8D4D6D9C3⋊D4D18C9⋊D4S3×Q8Q8×D9
kernelDic9⋊Q8C4×Dic9Dic9⋊C4C2×Dic18Q8×C18C6×Q8Dic9C36C2×C12C2×Q8C12C2×C4C4C6C2
# reps1141114233491226

Matrix representation of Dic9⋊Q8 in GL6(𝔽37)

17110000
2660000
0093400
00122900
000010
000001
,
5100000
5320000
0083200
00132900
0000360
0000036
,
3600000
0360000
0029500
0017800
0000309
0000157
,
3600000
0360000
001000
000100
0000162
0000121

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