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## G = C33×C12order 324 = 22·34

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

Aliases: C33×C12, SmallGroup(324,159)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33×C12
 Chief series C1 — C2 — C6 — C3×C6 — C32×C6 — C33×C6 — C33×C12
 Lower central C1 — C33×C12
 Upper central 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, C4, C6, C32, C12, C3×C6, C33, C3×C12, C32×C6, C34, C32×C12, C33×C6, C33×C12
Quotients: C1, C2, C3, C4, C6, C32, C12, C3×C6, C33, C3×C12, C32×C6, C34, C32×C12, C33×C6, C33×C12

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

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

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

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

324 conjugacy classes

 class 1 2 3A ··· 3CB 4A 4B 6A ··· 6CB 12A ··· 12FD order 1 2 3 ··· 3 4 4 6 ··· 6 12 ··· 12 size 1 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1

324 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C12 kernel C33×C12 C33×C6 C32×C12 C34 C32×C6 C33 # reps 1 1 80 2 80 160

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

 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
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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