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G = C32⋊5Dic9order 324 = 22·34

2nd semidirect product of C32 and Dic9 acting via Dic9/C18=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C9 — C32⋊5Dic9
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C32×C18 — C32⋊5Dic9
 Lower central C32×C9 — C32⋊5Dic9
 Upper central C1 — C2

Generators and relations for C325Dic9
G = < a,b,c,d | a3=b3=c18=1, d2=c9, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 770 in 150 conjugacy classes, 101 normal (9 characteristic)
C1, C2, C3, C3, C4, C6, C6, C9, C32, Dic3, C18, C3×C6, C3×C9, C33, Dic9, C3⋊Dic3, C3×C18, C32×C6, C32×C9, C9⋊Dic3, C335C4, C32×C18, C325Dic9
Quotients: C1, C2, C4, S3, Dic3, D9, C3⋊S3, Dic9, C3⋊Dic3, C9⋊S3, C33⋊C2, C9⋊Dic3, C335C4, C324D9, C325Dic9

Smallest permutation representation of C325Dic9
Regular action on 324 points
Generators in S324
(1 80 269)(2 81 270)(3 82 253)(4 83 254)(5 84 255)(6 85 256)(7 86 257)(8 87 258)(9 88 259)(10 89 260)(11 90 261)(12 73 262)(13 74 263)(14 75 264)(15 76 265)(16 77 266)(17 78 267)(18 79 268)(19 286 240)(20 287 241)(21 288 242)(22 271 243)(23 272 244)(24 273 245)(25 274 246)(26 275 247)(27 276 248)(28 277 249)(29 278 250)(30 279 251)(31 280 252)(32 281 235)(33 282 236)(34 283 237)(35 284 238)(36 285 239)(37 141 226)(38 142 227)(39 143 228)(40 144 229)(41 127 230)(42 128 231)(43 129 232)(44 130 233)(45 131 234)(46 132 217)(47 133 218)(48 134 219)(49 135 220)(50 136 221)(51 137 222)(52 138 223)(53 139 224)(54 140 225)(55 204 300)(56 205 301)(57 206 302)(58 207 303)(59 208 304)(60 209 305)(61 210 306)(62 211 289)(63 212 290)(64 213 291)(65 214 292)(66 215 293)(67 216 294)(68 199 295)(69 200 296)(70 201 297)(71 202 298)(72 203 299)(91 169 318)(92 170 319)(93 171 320)(94 172 321)(95 173 322)(96 174 323)(97 175 324)(98 176 307)(99 177 308)(100 178 309)(101 179 310)(102 180 311)(103 163 312)(104 164 313)(105 165 314)(106 166 315)(107 167 316)(108 168 317)(109 162 197)(110 145 198)(111 146 181)(112 147 182)(113 148 183)(114 149 184)(115 150 185)(116 151 186)(117 152 187)(118 153 188)(119 154 189)(120 155 190)(121 156 191)(122 157 192)(123 158 193)(124 159 194)(125 160 195)(126 161 196)
(1 183 49)(2 184 50)(3 185 51)(4 186 52)(5 187 53)(6 188 54)(7 189 37)(8 190 38)(9 191 39)(10 192 40)(11 193 41)(12 194 42)(13 195 43)(14 196 44)(15 197 45)(16 198 46)(17 181 47)(18 182 48)(19 176 58)(20 177 59)(21 178 60)(22 179 61)(23 180 62)(24 163 63)(25 164 64)(26 165 65)(27 166 66)(28 167 67)(29 168 68)(30 169 69)(31 170 70)(32 171 71)(33 172 72)(34 173 55)(35 174 56)(36 175 57)(73 124 128)(74 125 129)(75 126 130)(76 109 131)(77 110 132)(78 111 133)(79 112 134)(80 113 135)(81 114 136)(82 115 137)(83 116 138)(84 117 139)(85 118 140)(86 119 141)(87 120 142)(88 121 143)(89 122 144)(90 123 127)(91 296 251)(92 297 252)(93 298 235)(94 299 236)(95 300 237)(96 301 238)(97 302 239)(98 303 240)(99 304 241)(100 305 242)(101 306 243)(102 289 244)(103 290 245)(104 291 246)(105 292 247)(106 293 248)(107 294 249)(108 295 250)(145 217 266)(146 218 267)(147 219 268)(148 220 269)(149 221 270)(150 222 253)(151 223 254)(152 224 255)(153 225 256)(154 226 257)(155 227 258)(156 228 259)(157 229 260)(158 230 261)(159 231 262)(160 232 263)(161 233 264)(162 234 265)(199 278 317)(200 279 318)(201 280 319)(202 281 320)(203 282 321)(204 283 322)(205 284 323)(206 285 324)(207 286 307)(208 287 308)(209 288 309)(210 271 310)(211 272 311)(212 273 312)(213 274 313)(214 275 314)(215 276 315)(216 277 316)
(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)
(1 105 10 96)(2 104 11 95)(3 103 12 94)(4 102 13 93)(5 101 14 92)(6 100 15 91)(7 99 16 108)(8 98 17 107)(9 97 18 106)(19 146 28 155)(20 145 29 154)(21 162 30 153)(22 161 31 152)(23 160 32 151)(24 159 33 150)(25 158 34 149)(26 157 35 148)(27 156 36 147)(37 304 46 295)(38 303 47 294)(39 302 48 293)(40 301 49 292)(41 300 50 291)(42 299 51 290)(43 298 52 289)(44 297 53 306)(45 296 54 305)(55 221 64 230)(56 220 65 229)(57 219 66 228)(58 218 67 227)(59 217 68 226)(60 234 69 225)(61 233 70 224)(62 232 71 223)(63 231 72 222)(73 321 82 312)(74 320 83 311)(75 319 84 310)(76 318 85 309)(77 317 86 308)(78 316 87 307)(79 315 88 324)(80 314 89 323)(81 313 90 322)(109 279 118 288)(110 278 119 287)(111 277 120 286)(112 276 121 285)(113 275 122 284)(114 274 123 283)(115 273 124 282)(116 272 125 281)(117 271 126 280)(127 204 136 213)(128 203 137 212)(129 202 138 211)(130 201 139 210)(131 200 140 209)(132 199 141 208)(133 216 142 207)(134 215 143 206)(135 214 144 205)(163 262 172 253)(164 261 173 270)(165 260 174 269)(166 259 175 268)(167 258 176 267)(168 257 177 266)(169 256 178 265)(170 255 179 264)(171 254 180 263)(181 249 190 240)(182 248 191 239)(183 247 192 238)(184 246 193 237)(185 245 194 236)(186 244 195 235)(187 243 196 252)(188 242 197 251)(189 241 198 250)

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

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

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

84 conjugacy classes

 class 1 2 3A ··· 3M 4A 4B 6A ··· 6M 9A ··· 9AA 18A ··· 18AA order 1 2 3 ··· 3 4 4 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 2 ··· 2 81 81 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + - - + - image C1 C2 C4 S3 S3 Dic3 Dic3 D9 Dic9 kernel C32⋊5Dic9 C32×C18 C32×C9 C3×C18 C32×C6 C3×C9 C33 C3×C6 C32 # reps 1 1 2 12 1 12 1 27 27

Matrix representation of C325Dic9 in GL6(𝔽37)

 0 1 0 0 0 0 36 36 0 0 0 0 0 0 36 36 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 16 26
,
 26 6 0 0 0 0 31 20 0 0 0 0 0 0 20 26 0 0 0 0 11 31 0 0 0 0 0 0 12 0 0 0 0 0 22 34
,
 5 10 0 0 0 0 5 32 0 0 0 0 0 0 31 31 0 0 0 0 0 6 0 0 0 0 0 0 9 9 0 0 0 0 24 28

G:=sub<GL(6,GF(37))| [0,36,0,0,0,0,1,36,0,0,0,0,0,0,36,1,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,16,0,0,0,0,0,26],[26,31,0,0,0,0,6,20,0,0,0,0,0,0,20,11,0,0,0,0,26,31,0,0,0,0,0,0,12,22,0,0,0,0,0,34],[5,5,0,0,0,0,10,32,0,0,0,0,0,0,31,0,0,0,0,0,31,6,0,0,0,0,0,0,9,24,0,0,0,0,9,28] >;

C325Dic9 in GAP, Magma, Sage, TeX

C_3^2\rtimes_5{\rm Dic}_9
% in TeX

G:=Group("C3^2:5Dic9");
// GroupNames label

G:=SmallGroup(324,103);
// by ID

G=gap.SmallGroup(324,103);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,2090,986,579,2164,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^18=1,d^2=c^9,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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