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## G = C22×C9⋊C9order 324 = 22·34

### Direct product of C22 and C9⋊C9

direct product, metacyclic, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C22×C9⋊C9
 Chief series C1 — C3 — C32 — C3×C9 — C9⋊C9 — C2×C9⋊C9 — C22×C9⋊C9
 Lower central C1 — C3 — C22×C9⋊C9
 Upper central C1 — C62 — C22×C9⋊C9

Generators and relations for C22×C9⋊C9
G = < a,b,c,d | a2=b2=c9=d9=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c7 >

Subgroups: 115 in 85 conjugacy classes, 70 normal (10 characteristic)
C1, C2, C3, C3, C22, C6, C9, C9, C32, C2×C6, C2×C6, C18, C18, C3×C6, C3×C9, C3×C9, C2×C18, C2×C18, C62, C3×C18, C9⋊C9, C6×C18, C6×C18, C2×C9⋊C9, C22×C9⋊C9
Quotients: C1, C2, C3, C22, C6, C9, C32, C2×C6, C18, C3×C6, C3×C9, 3- 1+2, C2×C18, C62, C3×C18, C2×3- 1+2, C9⋊C9, C6×C18, C22×3- 1+2, C2×C9⋊C9, C22×C9⋊C9

Smallest permutation representation of C22×C9⋊C9
Regular action on 324 points
Generators in S324
(1 195)(2 196)(3 197)(4 198)(5 190)(6 191)(7 192)(8 193)(9 194)(10 171)(11 163)(12 164)(13 165)(14 166)(15 167)(16 168)(17 169)(18 170)(19 180)(20 172)(21 173)(22 174)(23 175)(24 176)(25 177)(26 178)(27 179)(28 184)(29 185)(30 186)(31 187)(32 188)(33 189)(34 181)(35 182)(36 183)(37 199)(38 200)(39 201)(40 202)(41 203)(42 204)(43 205)(44 206)(45 207)(46 208)(47 209)(48 210)(49 211)(50 212)(51 213)(52 214)(53 215)(54 216)(55 217)(56 218)(57 219)(58 220)(59 221)(60 222)(61 223)(62 224)(63 225)(64 226)(65 227)(66 228)(67 229)(68 230)(69 231)(70 232)(71 233)(72 234)(73 235)(74 236)(75 237)(76 238)(77 239)(78 240)(79 241)(80 242)(81 243)(82 244)(83 245)(84 246)(85 247)(86 248)(87 249)(88 250)(89 251)(90 252)(91 253)(92 254)(93 255)(94 256)(95 257)(96 258)(97 259)(98 260)(99 261)(100 262)(101 263)(102 264)(103 265)(104 266)(105 267)(106 268)(107 269)(108 270)(109 271)(110 272)(111 273)(112 274)(113 275)(114 276)(115 277)(116 278)(117 279)(118 280)(119 281)(120 282)(121 283)(122 284)(123 285)(124 286)(125 287)(126 288)(127 289)(128 290)(129 291)(130 292)(131 293)(132 294)(133 295)(134 296)(135 297)(136 298)(137 299)(138 300)(139 301)(140 302)(141 303)(142 304)(143 305)(144 306)(145 307)(146 308)(147 309)(148 310)(149 311)(150 312)(151 313)(152 314)(153 315)(154 316)(155 317)(156 318)(157 319)(158 320)(159 321)(160 322)(161 323)(162 324)
(1 114)(2 115)(3 116)(4 117)(5 109)(6 110)(7 111)(8 112)(9 113)(10 252)(11 244)(12 245)(13 246)(14 247)(15 248)(16 249)(17 250)(18 251)(19 261)(20 253)(21 254)(22 255)(23 256)(24 257)(25 258)(26 259)(27 260)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 100)(35 101)(36 102)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)(82 163)(83 164)(84 165)(85 166)(86 167)(87 168)(88 169)(89 170)(90 171)(91 172)(92 173)(93 174)(94 175)(95 176)(96 177)(97 178)(98 179)(99 180)(181 262)(182 263)(183 264)(184 265)(185 266)(186 267)(187 268)(188 269)(189 270)(190 271)(191 272)(192 273)(193 274)(194 275)(195 276)(196 277)(197 278)(198 279)(199 280)(200 281)(201 282)(202 283)(203 284)(204 285)(205 286)(206 287)(207 288)(208 289)(209 290)(210 291)(211 292)(212 293)(213 294)(214 295)(215 296)(216 297)(217 298)(218 299)(219 300)(220 301)(221 302)(222 303)(223 304)(224 305)(225 306)(226 307)(227 308)(228 309)(229 310)(230 311)(231 312)(232 313)(233 314)(234 315)(235 316)(236 317)(237 318)(238 319)(239 320)(240 321)(241 322)(242 323)(243 324)
(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)(289 290 291 292 293 294 295 296 297)(298 299 300 301 302 303 304 305 306)(307 308 309 310 311 312 313 314 315)(316 317 318 319 320 321 322 323 324)
(1 78 51 30 93 66 39 87 60)(2 73 49 31 97 64 40 82 58)(3 77 47 32 92 71 41 86 56)(4 81 54 33 96 69 42 90 63)(5 76 52 34 91 67 43 85 61)(6 80 50 35 95 65 44 89 59)(7 75 48 36 99 72 45 84 57)(8 79 46 28 94 70 37 88 55)(9 74 53 29 98 68 38 83 62)(10 306 279 324 297 270 25 312 285)(11 301 277 316 292 268 26 307 283)(12 305 275 317 296 266 27 311 281)(13 300 273 318 291 264 19 315 288)(14 304 271 319 295 262 20 310 286)(15 299 278 320 290 269 21 314 284)(16 303 276 321 294 267 22 309 282)(17 298 274 322 289 265 23 313 280)(18 302 272 323 293 263 24 308 287)(100 172 148 124 166 142 109 157 133)(101 176 146 125 170 140 110 161 131)(102 180 153 126 165 138 111 156 129)(103 175 151 118 169 136 112 160 127)(104 179 149 119 164 143 113 155 134)(105 174 147 120 168 141 114 159 132)(106 178 145 121 163 139 115 154 130)(107 173 152 122 167 137 116 158 128)(108 177 150 123 171 144 117 162 135)(181 253 229 205 247 223 190 238 214)(182 257 227 206 251 221 191 242 212)(183 261 234 207 246 219 192 237 210)(184 256 232 199 250 217 193 241 208)(185 260 230 200 245 224 194 236 215)(186 255 228 201 249 222 195 240 213)(187 259 226 202 244 220 196 235 211)(188 254 233 203 248 218 197 239 209)(189 258 231 204 252 225 198 243 216)

G:=sub<Sym(324)| (1,195)(2,196)(3,197)(4,198)(5,190)(6,191)(7,192)(8,193)(9,194)(10,171)(11,163)(12,164)(13,165)(14,166)(15,167)(16,168)(17,169)(18,170)(19,180)(20,172)(21,173)(22,174)(23,175)(24,176)(25,177)(26,178)(27,179)(28,184)(29,185)(30,186)(31,187)(32,188)(33,189)(34,181)(35,182)(36,183)(37,199)(38,200)(39,201)(40,202)(41,203)(42,204)(43,205)(44,206)(45,207)(46,208)(47,209)(48,210)(49,211)(50,212)(51,213)(52,214)(53,215)(54,216)(55,217)(56,218)(57,219)(58,220)(59,221)(60,222)(61,223)(62,224)(63,225)(64,226)(65,227)(66,228)(67,229)(68,230)(69,231)(70,232)(71,233)(72,234)(73,235)(74,236)(75,237)(76,238)(77,239)(78,240)(79,241)(80,242)(81,243)(82,244)(83,245)(84,246)(85,247)(86,248)(87,249)(88,250)(89,251)(90,252)(91,253)(92,254)(93,255)(94,256)(95,257)(96,258)(97,259)(98,260)(99,261)(100,262)(101,263)(102,264)(103,265)(104,266)(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)(112,274)(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)(144,306)(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)(161,323)(162,324), (1,114)(2,115)(3,116)(4,117)(5,109)(6,110)(7,111)(8,112)(9,113)(10,252)(11,244)(12,245)(13,246)(14,247)(15,248)(16,249)(17,250)(18,251)(19,261)(20,253)(21,254)(22,255)(23,256)(24,257)(25,258)(26,259)(27,260)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,100)(35,101)(36,102)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,161)(81,162)(82,163)(83,164)(84,165)(85,166)(86,167)(87,168)(88,169)(89,170)(90,171)(91,172)(92,173)(93,174)(94,175)(95,176)(96,177)(97,178)(98,179)(99,180)(181,262)(182,263)(183,264)(184,265)(185,266)(186,267)(187,268)(188,269)(189,270)(190,271)(191,272)(192,273)(193,274)(194,275)(195,276)(196,277)(197,278)(198,279)(199,280)(200,281)(201,282)(202,283)(203,284)(204,285)(205,286)(206,287)(207,288)(208,289)(209,290)(210,291)(211,292)(212,293)(213,294)(214,295)(215,296)(216,297)(217,298)(218,299)(219,300)(220,301)(221,302)(222,303)(223,304)(224,305)(225,306)(226,307)(227,308)(228,309)(229,310)(230,311)(231,312)(232,313)(233,314)(234,315)(235,316)(236,317)(237,318)(238,319)(239,320)(240,321)(241,322)(242,323)(243,324), (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)(289,290,291,292,293,294,295,296,297)(298,299,300,301,302,303,304,305,306)(307,308,309,310,311,312,313,314,315)(316,317,318,319,320,321,322,323,324), (1,78,51,30,93,66,39,87,60)(2,73,49,31,97,64,40,82,58)(3,77,47,32,92,71,41,86,56)(4,81,54,33,96,69,42,90,63)(5,76,52,34,91,67,43,85,61)(6,80,50,35,95,65,44,89,59)(7,75,48,36,99,72,45,84,57)(8,79,46,28,94,70,37,88,55)(9,74,53,29,98,68,38,83,62)(10,306,279,324,297,270,25,312,285)(11,301,277,316,292,268,26,307,283)(12,305,275,317,296,266,27,311,281)(13,300,273,318,291,264,19,315,288)(14,304,271,319,295,262,20,310,286)(15,299,278,320,290,269,21,314,284)(16,303,276,321,294,267,22,309,282)(17,298,274,322,289,265,23,313,280)(18,302,272,323,293,263,24,308,287)(100,172,148,124,166,142,109,157,133)(101,176,146,125,170,140,110,161,131)(102,180,153,126,165,138,111,156,129)(103,175,151,118,169,136,112,160,127)(104,179,149,119,164,143,113,155,134)(105,174,147,120,168,141,114,159,132)(106,178,145,121,163,139,115,154,130)(107,173,152,122,167,137,116,158,128)(108,177,150,123,171,144,117,162,135)(181,253,229,205,247,223,190,238,214)(182,257,227,206,251,221,191,242,212)(183,261,234,207,246,219,192,237,210)(184,256,232,199,250,217,193,241,208)(185,260,230,200,245,224,194,236,215)(186,255,228,201,249,222,195,240,213)(187,259,226,202,244,220,196,235,211)(188,254,233,203,248,218,197,239,209)(189,258,231,204,252,225,198,243,216)>;

G:=Group( (1,195)(2,196)(3,197)(4,198)(5,190)(6,191)(7,192)(8,193)(9,194)(10,171)(11,163)(12,164)(13,165)(14,166)(15,167)(16,168)(17,169)(18,170)(19,180)(20,172)(21,173)(22,174)(23,175)(24,176)(25,177)(26,178)(27,179)(28,184)(29,185)(30,186)(31,187)(32,188)(33,189)(34,181)(35,182)(36,183)(37,199)(38,200)(39,201)(40,202)(41,203)(42,204)(43,205)(44,206)(45,207)(46,208)(47,209)(48,210)(49,211)(50,212)(51,213)(52,214)(53,215)(54,216)(55,217)(56,218)(57,219)(58,220)(59,221)(60,222)(61,223)(62,224)(63,225)(64,226)(65,227)(66,228)(67,229)(68,230)(69,231)(70,232)(71,233)(72,234)(73,235)(74,236)(75,237)(76,238)(77,239)(78,240)(79,241)(80,242)(81,243)(82,244)(83,245)(84,246)(85,247)(86,248)(87,249)(88,250)(89,251)(90,252)(91,253)(92,254)(93,255)(94,256)(95,257)(96,258)(97,259)(98,260)(99,261)(100,262)(101,263)(102,264)(103,265)(104,266)(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)(112,274)(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)(144,306)(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)(161,323)(162,324), (1,114)(2,115)(3,116)(4,117)(5,109)(6,110)(7,111)(8,112)(9,113)(10,252)(11,244)(12,245)(13,246)(14,247)(15,248)(16,249)(17,250)(18,251)(19,261)(20,253)(21,254)(22,255)(23,256)(24,257)(25,258)(26,259)(27,260)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,100)(35,101)(36,102)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,161)(81,162)(82,163)(83,164)(84,165)(85,166)(86,167)(87,168)(88,169)(89,170)(90,171)(91,172)(92,173)(93,174)(94,175)(95,176)(96,177)(97,178)(98,179)(99,180)(181,262)(182,263)(183,264)(184,265)(185,266)(186,267)(187,268)(188,269)(189,270)(190,271)(191,272)(192,273)(193,274)(194,275)(195,276)(196,277)(197,278)(198,279)(199,280)(200,281)(201,282)(202,283)(203,284)(204,285)(205,286)(206,287)(207,288)(208,289)(209,290)(210,291)(211,292)(212,293)(213,294)(214,295)(215,296)(216,297)(217,298)(218,299)(219,300)(220,301)(221,302)(222,303)(223,304)(224,305)(225,306)(226,307)(227,308)(228,309)(229,310)(230,311)(231,312)(232,313)(233,314)(234,315)(235,316)(236,317)(237,318)(238,319)(239,320)(240,321)(241,322)(242,323)(243,324), (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)(289,290,291,292,293,294,295,296,297)(298,299,300,301,302,303,304,305,306)(307,308,309,310,311,312,313,314,315)(316,317,318,319,320,321,322,323,324), (1,78,51,30,93,66,39,87,60)(2,73,49,31,97,64,40,82,58)(3,77,47,32,92,71,41,86,56)(4,81,54,33,96,69,42,90,63)(5,76,52,34,91,67,43,85,61)(6,80,50,35,95,65,44,89,59)(7,75,48,36,99,72,45,84,57)(8,79,46,28,94,70,37,88,55)(9,74,53,29,98,68,38,83,62)(10,306,279,324,297,270,25,312,285)(11,301,277,316,292,268,26,307,283)(12,305,275,317,296,266,27,311,281)(13,300,273,318,291,264,19,315,288)(14,304,271,319,295,262,20,310,286)(15,299,278,320,290,269,21,314,284)(16,303,276,321,294,267,22,309,282)(17,298,274,322,289,265,23,313,280)(18,302,272,323,293,263,24,308,287)(100,172,148,124,166,142,109,157,133)(101,176,146,125,170,140,110,161,131)(102,180,153,126,165,138,111,156,129)(103,175,151,118,169,136,112,160,127)(104,179,149,119,164,143,113,155,134)(105,174,147,120,168,141,114,159,132)(106,178,145,121,163,139,115,154,130)(107,173,152,122,167,137,116,158,128)(108,177,150,123,171,144,117,162,135)(181,253,229,205,247,223,190,238,214)(182,257,227,206,251,221,191,242,212)(183,261,234,207,246,219,192,237,210)(184,256,232,199,250,217,193,241,208)(185,260,230,200,245,224,194,236,215)(186,255,228,201,249,222,195,240,213)(187,259,226,202,244,220,196,235,211)(188,254,233,203,248,218,197,239,209)(189,258,231,204,252,225,198,243,216) );

G=PermutationGroup([[(1,195),(2,196),(3,197),(4,198),(5,190),(6,191),(7,192),(8,193),(9,194),(10,171),(11,163),(12,164),(13,165),(14,166),(15,167),(16,168),(17,169),(18,170),(19,180),(20,172),(21,173),(22,174),(23,175),(24,176),(25,177),(26,178),(27,179),(28,184),(29,185),(30,186),(31,187),(32,188),(33,189),(34,181),(35,182),(36,183),(37,199),(38,200),(39,201),(40,202),(41,203),(42,204),(43,205),(44,206),(45,207),(46,208),(47,209),(48,210),(49,211),(50,212),(51,213),(52,214),(53,215),(54,216),(55,217),(56,218),(57,219),(58,220),(59,221),(60,222),(61,223),(62,224),(63,225),(64,226),(65,227),(66,228),(67,229),(68,230),(69,231),(70,232),(71,233),(72,234),(73,235),(74,236),(75,237),(76,238),(77,239),(78,240),(79,241),(80,242),(81,243),(82,244),(83,245),(84,246),(85,247),(86,248),(87,249),(88,250),(89,251),(90,252),(91,253),(92,254),(93,255),(94,256),(95,257),(96,258),(97,259),(98,260),(99,261),(100,262),(101,263),(102,264),(103,265),(104,266),(105,267),(106,268),(107,269),(108,270),(109,271),(110,272),(111,273),(112,274),(113,275),(114,276),(115,277),(116,278),(117,279),(118,280),(119,281),(120,282),(121,283),(122,284),(123,285),(124,286),(125,287),(126,288),(127,289),(128,290),(129,291),(130,292),(131,293),(132,294),(133,295),(134,296),(135,297),(136,298),(137,299),(138,300),(139,301),(140,302),(141,303),(142,304),(143,305),(144,306),(145,307),(146,308),(147,309),(148,310),(149,311),(150,312),(151,313),(152,314),(153,315),(154,316),(155,317),(156,318),(157,319),(158,320),(159,321),(160,322),(161,323),(162,324)], [(1,114),(2,115),(3,116),(4,117),(5,109),(6,110),(7,111),(8,112),(9,113),(10,252),(11,244),(12,245),(13,246),(14,247),(15,248),(16,249),(17,250),(18,251),(19,261),(20,253),(21,254),(22,255),(23,256),(24,257),(25,258),(26,259),(27,260),(28,103),(29,104),(30,105),(31,106),(32,107),(33,108),(34,100),(35,101),(36,102),(37,118),(38,119),(39,120),(40,121),(41,122),(42,123),(43,124),(44,125),(45,126),(46,127),(47,128),(48,129),(49,130),(50,131),(51,132),(52,133),(53,134),(54,135),(55,136),(56,137),(57,138),(58,139),(59,140),(60,141),(61,142),(62,143),(63,144),(64,145),(65,146),(66,147),(67,148),(68,149),(69,150),(70,151),(71,152),(72,153),(73,154),(74,155),(75,156),(76,157),(77,158),(78,159),(79,160),(80,161),(81,162),(82,163),(83,164),(84,165),(85,166),(86,167),(87,168),(88,169),(89,170),(90,171),(91,172),(92,173),(93,174),(94,175),(95,176),(96,177),(97,178),(98,179),(99,180),(181,262),(182,263),(183,264),(184,265),(185,266),(186,267),(187,268),(188,269),(189,270),(190,271),(191,272),(192,273),(193,274),(194,275),(195,276),(196,277),(197,278),(198,279),(199,280),(200,281),(201,282),(202,283),(203,284),(204,285),(205,286),(206,287),(207,288),(208,289),(209,290),(210,291),(211,292),(212,293),(213,294),(214,295),(215,296),(216,297),(217,298),(218,299),(219,300),(220,301),(221,302),(222,303),(223,304),(224,305),(225,306),(226,307),(227,308),(228,309),(229,310),(230,311),(231,312),(232,313),(233,314),(234,315),(235,316),(236,317),(237,318),(238,319),(239,320),(240,321),(241,322),(242,323),(243,324)], [(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),(289,290,291,292,293,294,295,296,297),(298,299,300,301,302,303,304,305,306),(307,308,309,310,311,312,313,314,315),(316,317,318,319,320,321,322,323,324)], [(1,78,51,30,93,66,39,87,60),(2,73,49,31,97,64,40,82,58),(3,77,47,32,92,71,41,86,56),(4,81,54,33,96,69,42,90,63),(5,76,52,34,91,67,43,85,61),(6,80,50,35,95,65,44,89,59),(7,75,48,36,99,72,45,84,57),(8,79,46,28,94,70,37,88,55),(9,74,53,29,98,68,38,83,62),(10,306,279,324,297,270,25,312,285),(11,301,277,316,292,268,26,307,283),(12,305,275,317,296,266,27,311,281),(13,300,273,318,291,264,19,315,288),(14,304,271,319,295,262,20,310,286),(15,299,278,320,290,269,21,314,284),(16,303,276,321,294,267,22,309,282),(17,298,274,322,289,265,23,313,280),(18,302,272,323,293,263,24,308,287),(100,172,148,124,166,142,109,157,133),(101,176,146,125,170,140,110,161,131),(102,180,153,126,165,138,111,156,129),(103,175,151,118,169,136,112,160,127),(104,179,149,119,164,143,113,155,134),(105,174,147,120,168,141,114,159,132),(106,178,145,121,163,139,115,154,130),(107,173,152,122,167,137,116,158,128),(108,177,150,123,171,144,117,162,135),(181,253,229,205,247,223,190,238,214),(182,257,227,206,251,221,191,242,212),(183,261,234,207,246,219,192,237,210),(184,256,232,199,250,217,193,241,208),(185,260,230,200,245,224,194,236,215),(186,255,228,201,249,222,195,240,213),(187,259,226,202,244,220,196,235,211),(188,254,233,203,248,218,197,239,209),(189,258,231,204,252,225,198,243,216)]])

132 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 6A ··· 6X 9A ··· 9X 18A ··· 18BT order 1 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 1 ··· 1 1 ··· 1 3 ··· 3 3 ··· 3

132 irreducible representations

 dim 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C6 C9 C18 3- 1+2 C2×3- 1+2 kernel C22×C9⋊C9 C2×C9⋊C9 C6×C18 C3×C18 C2×C18 C18 C2×C6 C6 # reps 1 3 8 24 18 54 6 18

Matrix representation of C22×C9⋊C9 in GL5(𝔽19)

 18 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 7 0 0 0 0 0 1 0 0 0 0 0 7 12 0 0 0 0 5 9 0 0 11 9 7
,
 5 0 0 0 0 0 1 0 0 0 0 0 8 3 15 0 0 15 4 16 0 0 16 12 7

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[7,0,0,0,0,0,1,0,0,0,0,0,7,0,11,0,0,12,5,9,0,0,0,9,7],[5,0,0,0,0,0,1,0,0,0,0,0,8,15,16,0,0,3,4,12,0,0,15,16,7] >;

C22×C9⋊C9 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xC9:C9");
// GroupNames label

G:=SmallGroup(324,83);
// by ID

G=gap.SmallGroup(324,83);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,500,303,93]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^9=d^9=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^7>;
// generators/relations

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