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G = Q8×C2×C18order 288 = 25·32

Direct product of C2×C18 and Q8

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

Aliases: Q8×C2×C18, C36.50C23, C18.17C24, C6.23(C6×Q8), (C6×Q8).28C6, C6.17(C23×C6), C2.2(C23×C18), (C22×C4).9C18, C4.7(C22×C18), C12.51(C22×C6), C23.17(C2×C18), (C2×C18).84C23, (C22×C36).17C2, (C22×C12).32C6, (C2×C36).131C22, C22.8(C22×C18), (C22×C18).50C22, C3.(Q8×C2×C6), (Q8×C2×C6).4C3, (C2×C6).13(C3×Q8), (C2×C4).30(C2×C18), (C3×Q8).32(C2×C6), (C2×C12).155(C2×C6), (C22×C6).76(C2×C6), (C2×C6).89(C22×C6), SmallGroup(288,369)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C2×C18
C1C3C6C18C36Q8×C9Q8×C18 — Q8×C2×C18
C1C2 — Q8×C2×C18
C1C22×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 [×6], C3, C4 [×12], C22 [×7], C6, C6 [×6], C2×C4 [×18], Q8 [×16], C23, C9, C12 [×12], C2×C6 [×7], C22×C4 [×3], C2×Q8 [×12], C18, C18 [×6], C2×C12 [×18], C3×Q8 [×16], C22×C6, C22×Q8, C36 [×12], C2×C18 [×7], C22×C12 [×3], C6×Q8 [×12], C2×C36 [×18], Q8×C9 [×16], C22×C18, Q8×C2×C6, C22×C36 [×3], Q8×C18 [×12], Q8×C2×C18
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], Q8 [×4], C23 [×15], C9, C2×C6 [×35], C2×Q8 [×6], C24, C18 [×15], C3×Q8 [×4], C22×C6 [×15], C22×Q8, C2×C18 [×35], C6×Q8 [×6], C23×C6, Q8×C9 [×4], C22×C18 [×15], Q8×C2×C6, Q8×C18 [×6], C23×C18, Q8×C2×C18

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

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

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

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

180 conjugacy classes

class 1 2A···2G3A3B4A···4L6A···6N9A···9F12A···12X18A···18AP36A···36BT
order12···2334···46···69···912···1218···1836···36
size11···1112···21···11···12···21···12···2

180 irreducible representations

dim111111111222
type+++-
imageC1C2C2C3C6C6C9C18C18Q8C3×Q8Q8×C9
kernelQ8×C2×C18C22×C36Q8×C18Q8×C2×C6C22×C12C6×Q8C22×Q8C22×C4C2×Q8C2×C18C2×C6C22
# reps13122624618724824

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

36000
0100
0010
0001
,
1000
03600
0030
0003
,
1000
03600
00435
002733
,
1000
0100
00310
003634
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