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## G = Q8×Dic9order 288 = 25·32

### Direct product of Q8 and Dic9

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8×Dic9
G = < a,b,c,d | a4=c18=1, b2=a2, d2=c9, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 304 in 105 conjugacy classes, 70 normal (20 characteristic)
C1, C2 [×3], C3, C4 [×6], C4 [×5], C22, C6 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C9, Dic3 [×5], C12 [×6], C2×C6, C42 [×3], C4⋊C4 [×3], C2×Q8, C18 [×3], C2×Dic3 [×4], C2×C12 [×3], C3×Q8 [×4], C4×Q8, Dic9 [×2], Dic9 [×3], C36 [×6], C2×C18, C4×Dic3 [×3], C4⋊Dic3 [×3], C6×Q8, C2×Dic9, C2×Dic9 [×3], C2×C36 [×3], Q8×C9 [×4], Q8×Dic3, C4×Dic9 [×3], C4⋊Dic9 [×3], Q8×C18, Q8×Dic9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, Dic3 [×4], D6 [×3], C22×C4, C2×Q8, C4○D4, D9, C2×Dic3 [×6], C22×S3, C4×Q8, Dic9 [×4], D18 [×3], S3×Q8, Q83S3, C22×Dic3, C2×Dic9 [×6], C22×D9, Q8×Dic3, Q8×D9, Q83D9, C22×Dic9, Q8×Dic9

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 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 6 6 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 ··· 2 9 9 9 9 18 ··· 18 2 2 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + - + + - - + - + image C1 C2 C2 C2 C4 S3 Q8 D6 Dic3 C4○D4 D9 D18 Dic9 S3×Q8 Q8⋊3S3 Q8×D9 Q8⋊3D9 kernel Q8×Dic9 C4×Dic9 C4⋊Dic9 Q8×C18 Q8×C9 C6×Q8 Dic9 C2×C12 C3×Q8 C18 C2×Q8 C2×C4 Q8 C6 C6 C2 C2 # reps 1 3 3 1 8 1 2 3 4 2 3 9 12 1 1 3 3

Matrix representation of Q8×Dic9 in GL5(𝔽37)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 20 31 0 0 0 36 17
,
 36 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 3 2 0 0 0 32 34
,
 36 0 0 0 0 0 20 6 0 0 0 31 26 0 0 0 0 0 36 0 0 0 0 0 36
,
 31 0 0 0 0 0 26 17 0 0 0 6 11 0 0 0 0 0 31 0 0 0 0 0 31

G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,20,36,0,0,0,31,17],[36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,3,32,0,0,0,2,34],[36,0,0,0,0,0,20,31,0,0,0,6,26,0,0,0,0,0,36,0,0,0,0,0,36],[31,0,0,0,0,0,26,6,0,0,0,17,11,0,0,0,0,0,31,0,0,0,0,0,31] >;

Q8×Dic9 in GAP, Magma, Sage, TeX

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

G:=Group("Q8xDic9");
// GroupNames label

G:=SmallGroup(288,155);
// by ID

G=gap.SmallGroup(288,155);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,120,219,100,6725,292,9414]);
// Polycyclic

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

׿
×
𝔽