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G = C22×C9⋊C8order 288 = 25·32

Direct product of C22 and C9⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C22×C9⋊C8
 Chief series C1 — C3 — C9 — C18 — C36 — C9⋊C8 — C2×C9⋊C8 — C22×C9⋊C8
 Lower central C9 — C22×C9⋊C8
 Upper central C1 — C22×C4

Generators and relations for C22×C9⋊C8
G = < a,b,c,d | a2=b2=c9=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 224 in 114 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×3], C22 [×7], C6, C6 [×6], C8 [×4], C2×C4 [×6], C23, C9, C12, C12 [×3], C2×C6 [×7], C2×C8 [×6], C22×C4, C18, C18 [×6], C3⋊C8 [×4], C2×C12 [×6], C22×C6, C22×C8, C36, C36 [×3], C2×C18 [×7], C2×C3⋊C8 [×6], C22×C12, C9⋊C8 [×4], C2×C36 [×6], C22×C18, C22×C3⋊C8, C2×C9⋊C8 [×6], C22×C36, C22×C9⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C8 [×4], C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C2×C8 [×6], C22×C4, D9, C3⋊C8 [×4], C2×Dic3 [×6], C22×S3, C22×C8, Dic9 [×4], D18 [×3], C2×C3⋊C8 [×6], C22×Dic3, C9⋊C8 [×4], C2×Dic9 [×6], C22×D9, C22×C3⋊C8, C2×C9⋊C8 [×6], C22×Dic9, C22×C9⋊C8

Smallest permutation representation of C22×C9⋊C8
Regular action on 288 points
Generators in S288
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 73)(8 74)(9 75)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)(145 217)(146 218)(147 219)(148 220)(149 221)(150 222)(151 223)(152 224)(153 225)(154 226)(155 227)(156 228)(157 229)(158 230)(159 231)(160 232)(161 233)(162 234)(163 235)(164 236)(165 237)(166 238)(167 239)(168 240)(169 241)(170 242)(171 243)(172 244)(173 245)(174 246)(175 247)(176 248)(177 249)(178 250)(179 251)(180 252)(181 253)(182 254)(183 255)(184 256)(185 257)(186 258)(187 259)(188 260)(189 261)(190 262)(191 263)(192 264)(193 265)(194 266)(195 267)(196 268)(197 269)(198 270)(199 271)(200 272)(201 273)(202 274)(203 275)(204 276)(205 277)(206 278)(207 279)(208 280)(209 281)(210 282)(211 283)(212 284)(213 285)(214 286)(215 287)(216 288)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 46)(8 47)(9 48)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 136)(92 137)(93 138)(94 139)(95 140)(96 141)(97 142)(98 143)(99 144)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)(145 190)(146 191)(147 192)(148 193)(149 194)(150 195)(151 196)(152 197)(153 198)(154 181)(155 182)(156 183)(157 184)(158 185)(159 186)(160 187)(161 188)(162 189)(163 208)(164 209)(165 210)(166 211)(167 212)(168 213)(169 214)(170 215)(171 216)(172 199)(173 200)(174 201)(175 202)(176 203)(177 204)(178 205)(179 206)(180 207)(217 262)(218 263)(219 264)(220 265)(221 266)(222 267)(223 268)(224 269)(225 270)(226 253)(227 254)(228 255)(229 256)(230 257)(231 258)(232 259)(233 260)(234 261)(235 280)(236 281)(237 282)(238 283)(239 284)(240 285)(241 286)(242 287)(243 288)(244 271)(245 272)(246 273)(247 274)(248 275)(249 276)(250 277)(251 278)(252 279)
(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)
(1 212 31 194 13 203 22 185)(2 211 32 193 14 202 23 184)(3 210 33 192 15 201 24 183)(4 209 34 191 16 200 25 182)(5 208 35 190 17 199 26 181)(6 216 36 198 18 207 27 189)(7 215 28 197 10 206 19 188)(8 214 29 196 11 205 20 187)(9 213 30 195 12 204 21 186)(37 179 64 161 46 170 55 152)(38 178 65 160 47 169 56 151)(39 177 66 159 48 168 57 150)(40 176 67 158 49 167 58 149)(41 175 68 157 50 166 59 148)(42 174 69 156 51 165 60 147)(43 173 70 155 52 164 61 146)(44 172 71 154 53 163 62 145)(45 180 72 162 54 171 63 153)(73 287 100 269 82 278 91 260)(74 286 101 268 83 277 92 259)(75 285 102 267 84 276 93 258)(76 284 103 266 85 275 94 257)(77 283 104 265 86 274 95 256)(78 282 105 264 87 273 96 255)(79 281 106 263 88 272 97 254)(80 280 107 262 89 271 98 253)(81 288 108 270 90 279 99 261)(109 251 136 233 118 242 127 224)(110 250 137 232 119 241 128 223)(111 249 138 231 120 240 129 222)(112 248 139 230 121 239 130 221)(113 247 140 229 122 238 131 220)(114 246 141 228 123 237 132 219)(115 245 142 227 124 236 133 218)(116 244 143 226 125 235 134 217)(117 252 144 234 126 243 135 225)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 6A ··· 6G 8A ··· 8P 9A 9B 9C 12A ··· 12H 18A ··· 18U 36A ··· 36X order 1 2 ··· 2 3 4 ··· 4 6 ··· 6 8 ··· 8 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 ··· 1 2 1 ··· 1 2 ··· 2 9 ··· 9 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + - + - image C1 C2 C2 C4 C4 C8 S3 Dic3 D6 Dic3 D9 C3⋊C8 Dic9 D18 Dic9 C9⋊C8 kernel C22×C9⋊C8 C2×C9⋊C8 C22×C36 C2×C36 C22×C18 C2×C18 C22×C12 C2×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 16 1 3 3 1 3 8 9 9 3 24

Matrix representation of C22×C9⋊C8 in GL5(𝔽73)

 1 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 72 0 0 0 1 72 0 0 0 0 0 42 28 0 0 0 45 70
,
 63 0 0 0 0 0 65 47 0 0 0 39 8 0 0 0 0 0 2 18 0 0 0 20 71

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,72,72,0,0,0,0,0,42,45,0,0,0,28,70],[63,0,0,0,0,0,65,39,0,0,0,47,8,0,0,0,0,0,2,20,0,0,0,18,71] >;

C22×C9⋊C8 in GAP, Magma, Sage, TeX

C_2^2\times C_9\rtimes C_8
% in TeX

G:=Group("C2^2xC9:C8");
// GroupNames label

G:=SmallGroup(288,130);
// by ID

G=gap.SmallGroup(288,130);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^9=d^8=1,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