Copied to
clipboard

## G = Q8×C2×C18order 288 = 25·32

### Direct product of C2×C18 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C2×C18
 Chief series C1 — C3 — C6 — C18 — C36 — Q8×C9 — Q8×C18 — Q8×C2×C18
 Lower central C1 — C2 — Q8×C2×C18
 Upper central C1 — C22×C18 — Q8×C2×C18

Generators and relations for Q8×C2×C18
G = < a,b,c,d | a2=b18=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 234, all normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, Q8, C23, C9, C12, C2×C6, C22×C4, C2×Q8, C18, C18, C2×C12, C3×Q8, C22×C6, C22×Q8, C36, C2×C18, C22×C12, C6×Q8, C2×C36, Q8×C9, C22×C18, Q8×C2×C6, C22×C36, Q8×C18, Q8×C2×C18
Quotients: C1, C2, C3, C22, C6, Q8, C23, C9, C2×C6, C2×Q8, C24, C18, C3×Q8, C22×C6, C22×Q8, C2×C18, C6×Q8, C23×C6, Q8×C9, C22×C18, Q8×C2×C6, Q8×C18, C23×C18, Q8×C2×C18

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

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

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

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

180 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4L 6A ··· 6N 9A ··· 9F 12A ··· 12X 18A ··· 18AP 36A ··· 36BT order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 ··· 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 type + + + - image C1 C2 C2 C3 C6 C6 C9 C18 C18 Q8 C3×Q8 Q8×C9 kernel Q8×C2×C18 C22×C36 Q8×C18 Q8×C2×C6 C22×C12 C6×Q8 C22×Q8 C22×C4 C2×Q8 C2×C18 C2×C6 C22 # reps 1 3 12 2 6 24 6 18 72 4 8 24

Matrix representation of Q8×C2×C18 in GL4(𝔽37) generated by

 36 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 36 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 36 0 0 0 0 4 35 0 0 27 33
,
 1 0 0 0 0 1 0 0 0 0 3 10 0 0 36 34
G:=sub<GL(4,GF(37))| [36,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,36,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,36,0,0,0,0,4,27,0,0,35,33],[1,0,0,0,0,1,0,0,0,0,3,36,0,0,10,34] >;

Q8×C2×C18 in GAP, Magma, Sage, TeX

Q_8\times C_2\times C_{18}
% in TeX

G:=Group("Q8xC2xC18");
// GroupNames label

G:=SmallGroup(288,369);
// by ID

G=gap.SmallGroup(288,369);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,336,701,344,242]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^18=c^4=1,d^2=c^2,a*b=b*a,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