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G = C33×C12order 324 = 22·34

Abelian group of type [3,3,3,12]

direct product, abelian, monomial, 3-elementary

Aliases: C33×C12, SmallGroup(324,159)

Series: Derived Chief Lower central Upper central

C1 — C33×C12
C1C2C6C3×C6C32×C6C33×C6 — C33×C12
C1 — C33×C12
C1 — C33×C12

Generators and relations for C33×C12
 G = < a,b,c,d | a3=b3=c3=d12=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 636, all normal (6 characteristic)
C1, C2, C3 [×40], C4, C6 [×40], C32 [×130], C12 [×40], C3×C6 [×130], C33 [×40], C3×C12 [×130], C32×C6 [×40], C34, C32×C12 [×40], C33×C6, C33×C12
Quotients: C1, C2, C3 [×40], C4, C6 [×40], C32 [×130], C12 [×40], C3×C6 [×130], C33 [×40], C3×C12 [×130], C32×C6 [×40], C34, C32×C12 [×40], C33×C6, C33×C12

Smallest permutation representation of C33×C12
Regular action on 324 points
Generators in S324
(1 176 124)(2 177 125)(3 178 126)(4 179 127)(5 180 128)(6 169 129)(7 170 130)(8 171 131)(9 172 132)(10 173 121)(11 174 122)(12 175 123)(13 133 109)(14 134 110)(15 135 111)(16 136 112)(17 137 113)(18 138 114)(19 139 115)(20 140 116)(21 141 117)(22 142 118)(23 143 119)(24 144 120)(25 89 226)(26 90 227)(27 91 228)(28 92 217)(29 93 218)(30 94 219)(31 95 220)(32 96 221)(33 85 222)(34 86 223)(35 87 224)(36 88 225)(37 302 295)(38 303 296)(39 304 297)(40 305 298)(41 306 299)(42 307 300)(43 308 289)(44 309 290)(45 310 291)(46 311 292)(47 312 293)(48 301 294)(49 283 257)(50 284 258)(51 285 259)(52 286 260)(53 287 261)(54 288 262)(55 277 263)(56 278 264)(57 279 253)(58 280 254)(59 281 255)(60 282 256)(61 265 313)(62 266 314)(63 267 315)(64 268 316)(65 269 317)(66 270 318)(67 271 319)(68 272 320)(69 273 321)(70 274 322)(71 275 323)(72 276 324)(73 108 243)(74 97 244)(75 98 245)(76 99 246)(77 100 247)(78 101 248)(79 102 249)(80 103 250)(81 104 251)(82 105 252)(83 106 241)(84 107 242)(145 181 166)(146 182 167)(147 183 168)(148 184 157)(149 185 158)(150 186 159)(151 187 160)(152 188 161)(153 189 162)(154 190 163)(155 191 164)(156 192 165)(193 230 210)(194 231 211)(195 232 212)(196 233 213)(197 234 214)(198 235 215)(199 236 216)(200 237 205)(201 238 206)(202 239 207)(203 240 208)(204 229 209)
(1 259 305)(2 260 306)(3 261 307)(4 262 308)(5 263 309)(6 264 310)(7 253 311)(8 254 312)(9 255 301)(10 256 302)(11 257 303)(12 258 304)(13 100 196)(14 101 197)(15 102 198)(16 103 199)(17 104 200)(18 105 201)(19 106 202)(20 107 203)(21 108 204)(22 97 193)(23 98 194)(24 99 195)(25 149 72)(26 150 61)(27 151 62)(28 152 63)(29 153 64)(30 154 65)(31 155 66)(32 156 67)(33 145 68)(34 146 69)(35 147 70)(36 148 71)(37 121 282)(38 122 283)(39 123 284)(40 124 285)(41 125 286)(42 126 287)(43 127 288)(44 128 277)(45 129 278)(46 130 279)(47 131 280)(48 132 281)(49 296 174)(50 297 175)(51 298 176)(52 299 177)(53 300 178)(54 289 179)(55 290 180)(56 291 169)(57 292 170)(58 293 171)(59 294 172)(60 295 173)(73 209 117)(74 210 118)(75 211 119)(76 212 120)(77 213 109)(78 214 110)(79 215 111)(80 216 112)(81 205 113)(82 206 114)(83 207 115)(84 208 116)(85 181 272)(86 182 273)(87 183 274)(88 184 275)(89 185 276)(90 186 265)(91 187 266)(92 188 267)(93 189 268)(94 190 269)(95 191 270)(96 192 271)(133 247 233)(134 248 234)(135 249 235)(136 250 236)(137 251 237)(138 252 238)(139 241 239)(140 242 240)(141 243 229)(142 244 230)(143 245 231)(144 246 232)(157 323 225)(158 324 226)(159 313 227)(160 314 228)(161 315 217)(162 316 218)(163 317 219)(164 318 220)(165 319 221)(166 320 222)(167 321 223)(168 322 224)
(1 69 24)(2 70 13)(3 71 14)(4 72 15)(5 61 16)(6 62 17)(7 63 18)(8 64 19)(9 65 20)(10 66 21)(11 67 22)(12 68 23)(25 102 262)(26 103 263)(27 104 264)(28 105 253)(29 106 254)(30 107 255)(31 108 256)(32 97 257)(33 98 258)(34 99 259)(35 100 260)(36 101 261)(37 164 209)(38 165 210)(39 166 211)(40 167 212)(41 168 213)(42 157 214)(43 158 215)(44 159 216)(45 160 205)(46 161 206)(47 162 207)(48 163 208)(49 96 244)(50 85 245)(51 86 246)(52 87 247)(53 88 248)(54 89 249)(55 90 250)(56 91 251)(57 92 252)(58 93 241)(59 94 242)(60 95 243)(73 282 220)(74 283 221)(75 284 222)(76 285 223)(77 286 224)(78 287 225)(79 288 226)(80 277 227)(81 278 228)(82 279 217)(83 280 218)(84 281 219)(109 125 322)(110 126 323)(111 127 324)(112 128 313)(113 129 314)(114 130 315)(115 131 316)(116 132 317)(117 121 318)(118 122 319)(119 123 320)(120 124 321)(133 177 274)(134 178 275)(135 179 276)(136 180 265)(137 169 266)(138 170 267)(139 171 268)(140 172 269)(141 173 270)(142 174 271)(143 175 272)(144 176 273)(145 194 304)(146 195 305)(147 196 306)(148 197 307)(149 198 308)(150 199 309)(151 200 310)(152 201 311)(153 202 312)(154 203 301)(155 204 302)(156 193 303)(181 231 297)(182 232 298)(183 233 299)(184 234 300)(185 235 289)(186 236 290)(187 237 291)(188 238 292)(189 239 293)(190 240 294)(191 229 295)(192 230 296)
(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)

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

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

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

324 conjugacy classes

class 1  2 3A···3CB4A4B6A···6CB12A···12FD
order123···3446···612···12
size111···1111···11···1

324 irreducible representations

dim111111
type++
imageC1C2C3C4C6C12
kernelC33×C12C33×C6C32×C12C34C32×C6C33
# reps1180280160

Matrix representation of C33×C12 in GL4(𝔽13) generated by

9000
0900
0010
0009
,
9000
0100
0010
0001
,
1000
0900
0090
0001
,
8000
0600
0020
0007
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,9],[9,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,9,0,0,0,0,9,0,0,0,0,1],[8,0,0,0,0,6,0,0,0,0,2,0,0,0,0,7] >;

C33×C12 in GAP, Magma, Sage, TeX

C_3^3\times C_{12}
% in TeX

G:=Group("C3^3xC12");
// GroupNames label

G:=SmallGroup(324,159);
// by ID

G=gap.SmallGroup(324,159);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-2,972]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^12=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

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