direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×Dic9, C9⋊3(C4×Q8), (Q8×C9)⋊3C4, C3.(Q8×Dic3), C2.3(Q8×D9), (C2×Q8).7D9, C6.39(S3×Q8), C36.14(C2×C4), (C2×C12).65D6, (C2×C4).56D18, (Q8×C18).5C2, (C6×Q8).15S3, C18.16(C2×Q8), C4.4(C2×Dic9), C4⋊Dic9.12C2, (C3×Q8).7Dic3, (C4×Dic9).4C2, C12.5(C2×Dic3), C18.35(C4○D4), (C2×C18).57C23, C18.26(C22×C4), (C2×C36).43C22, C2.3(Q8⋊3D9), C2.7(C22×Dic9), C6.43(Q8⋊3S3), C6.27(C22×Dic3), C22.26(C22×D9), (C2×Dic9).50C22, (C2×C6).214(C22×S3), SmallGroup(288,155)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C4, C22, C6, C2×C4, C2×C4, Q8, C9, Dic3, C12, C2×C6, C42, C4⋊C4, C2×Q8, C18, C2×Dic3, C2×C12, C3×Q8, C4×Q8, Dic9, Dic9, C36, C2×C18, C4×Dic3, C4⋊Dic3, C6×Q8, C2×Dic9, C2×Dic9, C2×C36, Q8×C9, Q8×Dic3, C4×Dic9, C4⋊Dic9, Q8×C18, Q8×Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, Dic3, D6, C22×C4, C2×Q8, C4○D4, D9, C2×Dic3, C22×S3, C4×Q8, Dic9, D18, S3×Q8, Q8⋊3S3, C22×Dic3, C2×Dic9, C22×D9, Q8×Dic3, Q8×D9, Q8⋊3D9, C22×Dic9, Q8×Dic9
►Regular action on 288 points
Generators in S288
(1 19 215 136)(2 20 216 137)(3 21 199 138)(4 22 200 139)(5 23 201 140)(6 24 202 141)(7 25 203 142)(8 26 204 143)(9 27 205 144)(10 28 206 127)(11 29 207 128)(12 30 208 129)(13 31 209 130)(14 32 210 131)(15 33 211 132)(16 34 212 133)(17 35 213 134)(18 36 214 135)(37 158 266 238)(38 159 267 239)(39 160 268 240)(40 161 269 241)(41 162 270 242)(42 145 253 243)(43 146 254 244)(44 147 255 245)(45 148 256 246)(46 149 257 247)(47 150 258 248)(48 151 259 249)(49 152 260 250)(50 153 261 251)(51 154 262 252)(52 155 263 235)(53 156 264 236)(54 157 265 237)(55 179 84 281)(56 180 85 282)(57 163 86 283)(58 164 87 284)(59 165 88 285)(60 166 89 286)(61 167 90 287)(62 168 73 288)(63 169 74 271)(64 170 75 272)(65 171 76 273)(66 172 77 274)(67 173 78 275)(68 174 79 276)(69 175 80 277)(70 176 81 278)(71 177 82 279)(72 178 83 280)(91 183 220 116)(92 184 221 117)(93 185 222 118)(94 186 223 119)(95 187 224 120)(96 188 225 121)(97 189 226 122)(98 190 227 123)(99 191 228 124)(100 192 229 125)(101 193 230 126)(102 194 231 109)(103 195 232 110)(104 196 233 111)(105 197 234 112)(106 198 217 113)(107 181 218 114)(108 182 219 115)
(1 232 215 103)(2 233 216 104)(3 234 199 105)(4 217 200 106)(5 218 201 107)(6 219 202 108)(7 220 203 91)(8 221 204 92)(9 222 205 93)(10 223 206 94)(11 224 207 95)(12 225 208 96)(13 226 209 97)(14 227 210 98)(15 228 211 99)(16 229 212 100)(17 230 213 101)(18 231 214 102)(19 195 136 110)(20 196 137 111)(21 197 138 112)(22 198 139 113)(23 181 140 114)(24 182 141 115)(25 183 142 116)(26 184 143 117)(27 185 144 118)(28 186 127 119)(29 187 128 120)(30 188 129 121)(31 189 130 122)(32 190 131 123)(33 191 132 124)(34 192 133 125)(35 193 134 126)(36 194 135 109)(37 76 266 65)(38 77 267 66)(39 78 268 67)(40 79 269 68)(41 80 270 69)(42 81 253 70)(43 82 254 71)(44 83 255 72)(45 84 256 55)(46 85 257 56)(47 86 258 57)(48 87 259 58)(49 88 260 59)(50 89 261 60)(51 90 262 61)(52 73 263 62)(53 74 264 63)(54 75 265 64)(145 176 243 278)(146 177 244 279)(147 178 245 280)(148 179 246 281)(149 180 247 282)(150 163 248 283)(151 164 249 284)(152 165 250 285)(153 166 251 286)(154 167 252 287)(155 168 235 288)(156 169 236 271)(157 170 237 272)(158 171 238 273)(159 172 239 274)(160 173 240 275)(161 174 241 276)(162 175 242 277)
(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)
(1 158 10 149)(2 157 11 148)(3 156 12 147)(4 155 13 146)(5 154 14 145)(6 153 15 162)(7 152 16 161)(8 151 17 160)(9 150 18 159)(19 266 28 257)(20 265 29 256)(21 264 30 255)(22 263 31 254)(23 262 32 253)(24 261 33 270)(25 260 34 269)(26 259 35 268)(27 258 36 267)(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)(55 196 64 187)(56 195 65 186)(57 194 66 185)(58 193 67 184)(59 192 68 183)(60 191 69 182)(61 190 70 181)(62 189 71 198)(63 188 72 197)(73 122 82 113)(74 121 83 112)(75 120 84 111)(76 119 85 110)(77 118 86 109)(78 117 87 126)(79 116 88 125)(80 115 89 124)(81 114 90 123)(91 285 100 276)(92 284 101 275)(93 283 102 274)(94 282 103 273)(95 281 104 272)(96 280 105 271)(97 279 106 288)(98 278 107 287)(99 277 108 286)(163 231 172 222)(164 230 173 221)(165 229 174 220)(166 228 175 219)(167 227 176 218)(168 226 177 217)(169 225 178 234)(170 224 179 233)(171 223 180 232)(199 236 208 245)(200 235 209 244)(201 252 210 243)(202 251 211 242)(203 250 212 241)(204 249 213 240)(205 248 214 239)(206 247 215 238)(207 246 216 237)
G:=sub<Sym(288)| (1,19,215,136)(2,20,216,137)(3,21,199,138)(4,22,200,139)(5,23,201,140)(6,24,202,141)(7,25,203,142)(8,26,204,143)(9,27,205,144)(10,28,206,127)(11,29,207,128)(12,30,208,129)(13,31,209,130)(14,32,210,131)(15,33,211,132)(16,34,212,133)(17,35,213,134)(18,36,214,135)(37,158,266,238)(38,159,267,239)(39,160,268,240)(40,161,269,241)(41,162,270,242)(42,145,253,243)(43,146,254,244)(44,147,255,245)(45,148,256,246)(46,149,257,247)(47,150,258,248)(48,151,259,249)(49,152,260,250)(50,153,261,251)(51,154,262,252)(52,155,263,235)(53,156,264,236)(54,157,265,237)(55,179,84,281)(56,180,85,282)(57,163,86,283)(58,164,87,284)(59,165,88,285)(60,166,89,286)(61,167,90,287)(62,168,73,288)(63,169,74,271)(64,170,75,272)(65,171,76,273)(66,172,77,274)(67,173,78,275)(68,174,79,276)(69,175,80,277)(70,176,81,278)(71,177,82,279)(72,178,83,280)(91,183,220,116)(92,184,221,117)(93,185,222,118)(94,186,223,119)(95,187,224,120)(96,188,225,121)(97,189,226,122)(98,190,227,123)(99,191,228,124)(100,192,229,125)(101,193,230,126)(102,194,231,109)(103,195,232,110)(104,196,233,111)(105,197,234,112)(106,198,217,113)(107,181,218,114)(108,182,219,115), (1,232,215,103)(2,233,216,104)(3,234,199,105)(4,217,200,106)(5,218,201,107)(6,219,202,108)(7,220,203,91)(8,221,204,92)(9,222,205,93)(10,223,206,94)(11,224,207,95)(12,225,208,96)(13,226,209,97)(14,227,210,98)(15,228,211,99)(16,229,212,100)(17,230,213,101)(18,231,214,102)(19,195,136,110)(20,196,137,111)(21,197,138,112)(22,198,139,113)(23,181,140,114)(24,182,141,115)(25,183,142,116)(26,184,143,117)(27,185,144,118)(28,186,127,119)(29,187,128,120)(30,188,129,121)(31,189,130,122)(32,190,131,123)(33,191,132,124)(34,192,133,125)(35,193,134,126)(36,194,135,109)(37,76,266,65)(38,77,267,66)(39,78,268,67)(40,79,269,68)(41,80,270,69)(42,81,253,70)(43,82,254,71)(44,83,255,72)(45,84,256,55)(46,85,257,56)(47,86,258,57)(48,87,259,58)(49,88,260,59)(50,89,261,60)(51,90,262,61)(52,73,263,62)(53,74,264,63)(54,75,265,64)(145,176,243,278)(146,177,244,279)(147,178,245,280)(148,179,246,281)(149,180,247,282)(150,163,248,283)(151,164,249,284)(152,165,250,285)(153,166,251,286)(154,167,252,287)(155,168,235,288)(156,169,236,271)(157,170,237,272)(158,171,238,273)(159,172,239,274)(160,173,240,275)(161,174,241,276)(162,175,242,277), (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), (1,158,10,149)(2,157,11,148)(3,156,12,147)(4,155,13,146)(5,154,14,145)(6,153,15,162)(7,152,16,161)(8,151,17,160)(9,150,18,159)(19,266,28,257)(20,265,29,256)(21,264,30,255)(22,263,31,254)(23,262,32,253)(24,261,33,270)(25,260,34,269)(26,259,35,268)(27,258,36,267)(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)(55,196,64,187)(56,195,65,186)(57,194,66,185)(58,193,67,184)(59,192,68,183)(60,191,69,182)(61,190,70,181)(62,189,71,198)(63,188,72,197)(73,122,82,113)(74,121,83,112)(75,120,84,111)(76,119,85,110)(77,118,86,109)(78,117,87,126)(79,116,88,125)(80,115,89,124)(81,114,90,123)(91,285,100,276)(92,284,101,275)(93,283,102,274)(94,282,103,273)(95,281,104,272)(96,280,105,271)(97,279,106,288)(98,278,107,287)(99,277,108,286)(163,231,172,222)(164,230,173,221)(165,229,174,220)(166,228,175,219)(167,227,176,218)(168,226,177,217)(169,225,178,234)(170,224,179,233)(171,223,180,232)(199,236,208,245)(200,235,209,244)(201,252,210,243)(202,251,211,242)(203,250,212,241)(204,249,213,240)(205,248,214,239)(206,247,215,238)(207,246,216,237)>;
G:=Group( (1,19,215,136)(2,20,216,137)(3,21,199,138)(4,22,200,139)(5,23,201,140)(6,24,202,141)(7,25,203,142)(8,26,204,143)(9,27,205,144)(10,28,206,127)(11,29,207,128)(12,30,208,129)(13,31,209,130)(14,32,210,131)(15,33,211,132)(16,34,212,133)(17,35,213,134)(18,36,214,135)(37,158,266,238)(38,159,267,239)(39,160,268,240)(40,161,269,241)(41,162,270,242)(42,145,253,243)(43,146,254,244)(44,147,255,245)(45,148,256,246)(46,149,257,247)(47,150,258,248)(48,151,259,249)(49,152,260,250)(50,153,261,251)(51,154,262,252)(52,155,263,235)(53,156,264,236)(54,157,265,237)(55,179,84,281)(56,180,85,282)(57,163,86,283)(58,164,87,284)(59,165,88,285)(60,166,89,286)(61,167,90,287)(62,168,73,288)(63,169,74,271)(64,170,75,272)(65,171,76,273)(66,172,77,274)(67,173,78,275)(68,174,79,276)(69,175,80,277)(70,176,81,278)(71,177,82,279)(72,178,83,280)(91,183,220,116)(92,184,221,117)(93,185,222,118)(94,186,223,119)(95,187,224,120)(96,188,225,121)(97,189,226,122)(98,190,227,123)(99,191,228,124)(100,192,229,125)(101,193,230,126)(102,194,231,109)(103,195,232,110)(104,196,233,111)(105,197,234,112)(106,198,217,113)(107,181,218,114)(108,182,219,115), (1,232,215,103)(2,233,216,104)(3,234,199,105)(4,217,200,106)(5,218,201,107)(6,219,202,108)(7,220,203,91)(8,221,204,92)(9,222,205,93)(10,223,206,94)(11,224,207,95)(12,225,208,96)(13,226,209,97)(14,227,210,98)(15,228,211,99)(16,229,212,100)(17,230,213,101)(18,231,214,102)(19,195,136,110)(20,196,137,111)(21,197,138,112)(22,198,139,113)(23,181,140,114)(24,182,141,115)(25,183,142,116)(26,184,143,117)(27,185,144,118)(28,186,127,119)(29,187,128,120)(30,188,129,121)(31,189,130,122)(32,190,131,123)(33,191,132,124)(34,192,133,125)(35,193,134,126)(36,194,135,109)(37,76,266,65)(38,77,267,66)(39,78,268,67)(40,79,269,68)(41,80,270,69)(42,81,253,70)(43,82,254,71)(44,83,255,72)(45,84,256,55)(46,85,257,56)(47,86,258,57)(48,87,259,58)(49,88,260,59)(50,89,261,60)(51,90,262,61)(52,73,263,62)(53,74,264,63)(54,75,265,64)(145,176,243,278)(146,177,244,279)(147,178,245,280)(148,179,246,281)(149,180,247,282)(150,163,248,283)(151,164,249,284)(152,165,250,285)(153,166,251,286)(154,167,252,287)(155,168,235,288)(156,169,236,271)(157,170,237,272)(158,171,238,273)(159,172,239,274)(160,173,240,275)(161,174,241,276)(162,175,242,277), (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), (1,158,10,149)(2,157,11,148)(3,156,12,147)(4,155,13,146)(5,154,14,145)(6,153,15,162)(7,152,16,161)(8,151,17,160)(9,150,18,159)(19,266,28,257)(20,265,29,256)(21,264,30,255)(22,263,31,254)(23,262,32,253)(24,261,33,270)(25,260,34,269)(26,259,35,268)(27,258,36,267)(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)(55,196,64,187)(56,195,65,186)(57,194,66,185)(58,193,67,184)(59,192,68,183)(60,191,69,182)(61,190,70,181)(62,189,71,198)(63,188,72,197)(73,122,82,113)(74,121,83,112)(75,120,84,111)(76,119,85,110)(77,118,86,109)(78,117,87,126)(79,116,88,125)(80,115,89,124)(81,114,90,123)(91,285,100,276)(92,284,101,275)(93,283,102,274)(94,282,103,273)(95,281,104,272)(96,280,105,271)(97,279,106,288)(98,278,107,287)(99,277,108,286)(163,231,172,222)(164,230,173,221)(165,229,174,220)(166,228,175,219)(167,227,176,218)(168,226,177,217)(169,225,178,234)(170,224,179,233)(171,223,180,232)(199,236,208,245)(200,235,209,244)(201,252,210,243)(202,251,211,242)(203,250,212,241)(204,249,213,240)(205,248,214,239)(206,247,215,238)(207,246,216,237) );
G=PermutationGroup([[(1,19,215,136),(2,20,216,137),(3,21,199,138),(4,22,200,139),(5,23,201,140),(6,24,202,141),(7,25,203,142),(8,26,204,143),(9,27,205,144),(10,28,206,127),(11,29,207,128),(12,30,208,129),(13,31,209,130),(14,32,210,131),(15,33,211,132),(16,34,212,133),(17,35,213,134),(18,36,214,135),(37,158,266,238),(38,159,267,239),(39,160,268,240),(40,161,269,241),(41,162,270,242),(42,145,253,243),(43,146,254,244),(44,147,255,245),(45,148,256,246),(46,149,257,247),(47,150,258,248),(48,151,259,249),(49,152,260,250),(50,153,261,251),(51,154,262,252),(52,155,263,235),(53,156,264,236),(54,157,265,237),(55,179,84,281),(56,180,85,282),(57,163,86,283),(58,164,87,284),(59,165,88,285),(60,166,89,286),(61,167,90,287),(62,168,73,288),(63,169,74,271),(64,170,75,272),(65,171,76,273),(66,172,77,274),(67,173,78,275),(68,174,79,276),(69,175,80,277),(70,176,81,278),(71,177,82,279),(72,178,83,280),(91,183,220,116),(92,184,221,117),(93,185,222,118),(94,186,223,119),(95,187,224,120),(96,188,225,121),(97,189,226,122),(98,190,227,123),(99,191,228,124),(100,192,229,125),(101,193,230,126),(102,194,231,109),(103,195,232,110),(104,196,233,111),(105,197,234,112),(106,198,217,113),(107,181,218,114),(108,182,219,115)], [(1,232,215,103),(2,233,216,104),(3,234,199,105),(4,217,200,106),(5,218,201,107),(6,219,202,108),(7,220,203,91),(8,221,204,92),(9,222,205,93),(10,223,206,94),(11,224,207,95),(12,225,208,96),(13,226,209,97),(14,227,210,98),(15,228,211,99),(16,229,212,100),(17,230,213,101),(18,231,214,102),(19,195,136,110),(20,196,137,111),(21,197,138,112),(22,198,139,113),(23,181,140,114),(24,182,141,115),(25,183,142,116),(26,184,143,117),(27,185,144,118),(28,186,127,119),(29,187,128,120),(30,188,129,121),(31,189,130,122),(32,190,131,123),(33,191,132,124),(34,192,133,125),(35,193,134,126),(36,194,135,109),(37,76,266,65),(38,77,267,66),(39,78,268,67),(40,79,269,68),(41,80,270,69),(42,81,253,70),(43,82,254,71),(44,83,255,72),(45,84,256,55),(46,85,257,56),(47,86,258,57),(48,87,259,58),(49,88,260,59),(50,89,261,60),(51,90,262,61),(52,73,263,62),(53,74,264,63),(54,75,265,64),(145,176,243,278),(146,177,244,279),(147,178,245,280),(148,179,246,281),(149,180,247,282),(150,163,248,283),(151,164,249,284),(152,165,250,285),(153,166,251,286),(154,167,252,287),(155,168,235,288),(156,169,236,271),(157,170,237,272),(158,171,238,273),(159,172,239,274),(160,173,240,275),(161,174,241,276),(162,175,242,277)], [(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)], [(1,158,10,149),(2,157,11,148),(3,156,12,147),(4,155,13,146),(5,154,14,145),(6,153,15,162),(7,152,16,161),(8,151,17,160),(9,150,18,159),(19,266,28,257),(20,265,29,256),(21,264,30,255),(22,263,31,254),(23,262,32,253),(24,261,33,270),(25,260,34,269),(26,259,35,268),(27,258,36,267),(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),(55,196,64,187),(56,195,65,186),(57,194,66,185),(58,193,67,184),(59,192,68,183),(60,191,69,182),(61,190,70,181),(62,189,71,198),(63,188,72,197),(73,122,82,113),(74,121,83,112),(75,120,84,111),(76,119,85,110),(77,118,86,109),(78,117,87,126),(79,116,88,125),(80,115,89,124),(81,114,90,123),(91,285,100,276),(92,284,101,275),(93,283,102,274),(94,282,103,273),(95,281,104,272),(96,280,105,271),(97,279,106,288),(98,278,107,287),(99,277,108,286),(163,231,172,222),(164,230,173,221),(165,229,174,220),(166,228,175,219),(167,227,176,218),(168,226,177,217),(169,225,178,234),(170,224,179,233),(171,223,180,232),(199,236,208,245),(200,235,209,244),(201,252,210,243),(202,251,211,242),(203,250,212,241),(204,249,213,240),(205,248,214,239),(206,247,215,238),(207,246,216,237)]])
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