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G = C34⋊8C4  order 324 = 22·34

4th semidirect product of C34 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, A-group

Aliases: C34⋊8C4, C33⋊11Dic3, C3⋊(C33⋊5C4), C2.(C34⋊C2), (C33×C6).4C2, (C32×C6).23S3, C6.3(C33⋊C2), C32⋊5(C3⋊Dic3), (C3×C6).24(C3⋊S3), SmallGroup(324,158)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C34 — C34⋊8C4
Chief series
C1 — C3 — C32 — C33 — C34 — C33×C6 — C34⋊8C4
Lower central
C34 — C34⋊8C4
Upper central
C1 — C2

Generators and relations for C34⋊8C4
 G = < a,b,c,d,e | a3=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, eae-1=a-1, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 2876 in 636 conjugacy classes, 425 normal (5 characteristic)
C1, C2, C3, C4, C6, C32, Dic3, C3×C6, C33, C3⋊Dic3, C32×C6, C34, C33⋊5C4, C33×C6, C34⋊8C4
Quotients: C1, C2, C4, S3, Dic3, C3⋊S3, C3⋊Dic3, C33⋊C2, C33⋊5C4, C34⋊C2, C34⋊8C4

Smallest permutation representation of C34⋊8C4
►Regular action on 324 points
Generators in S324
(1 39 171)(2 172 40)(3 37 169)(4 170 38)(5 139 270)(6 271 140)(7 137 272)(8 269 138)(9 143 274)(10 275 144)(11 141 276)(12 273 142)(13 117 227)(14 228 118)(15 119 225)(16 226 120)(17 149 281)(18 282 150)(19 151 283)(20 284 152)(21 153 285)(22 286 154)(23 155 287)(24 288 156)(25 159 291)(26 292 160)(27 157 289)(28 290 158)(29 83 295)(30 296 84)(31 81 293)(32 294 82)(33 248 299)(34 300 245)(35 246 297)(36 298 247)(41 173 305)(42 306 174)(43 175 307)(44 308 176)(45 310 260)(46 257 311)(47 312 258)(48 259 309)(49 314 182)(50 183 315)(51 316 184)(52 181 313)(53 318 188)(54 185 319)(55 320 186)(56 187 317)(57 242 192)(58 189 243)(59 244 190)(60 191 241)(61 85 196)(62 193 86)(63 87 194)(64 195 88)(65 91 198)(66 199 92)(67 89 200)(68 197 90)(69 95 202)(70 203 96)(71 93 204)(72 201 94)(73 99 205)(74 206 100)(75 97 207)(76 208 98)(77 304 209)(78 210 301)(79 302 211)(80 212 303)(101 234 127)(102 128 235)(103 236 125)(104 126 233)(105 213 324)(106 321 214)(107 215 322)(108 323 216)(109 217 165)(110 166 218)(111 219 167)(112 168 220)(113 221 250)(114 251 222)(115 223 252)(116 249 224)(121 231 179)(122 180 232)(123 229 177)(124 178 230)(129 264 240)(130 237 261)(131 262 238)(132 239 263)(133 268 164)(134 161 265)(135 266 162)(136 163 267)(145 278 253)(146 254 279)(147 280 255)(148 256 277)

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

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

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

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

84 conjugacy classes

class 1  2 3A···3AN4A4B6A···6AN
order123···3446···6
size112···281812···2

84 irreducible representations

dim11122
type+++-
imageC1C2C4S3Dic3
kernelC34⋊8C4C33×C6C34C32×C6C33
# reps1124040

Matrix representation of C34⋊8C4 ►in GL8(𝔽13)

30000000
19000000
00950000
00030000
00001000
00000100
00000030
00000069
,
30000000
19000000
00950000
00030000
00003000
00008900
00000030
00000069
,
30000000
19000000
00950000
00030000
00003000
00008900
00000090
00000073
,
30000000
19000000
00100000
00010000
00009000
00005300
00000030
00000069
,
108000000
23000000
001050000
00130000
00002800
000011100
00000022
000000411

G:=sub<GL(8,GF(13))| [3,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,5,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,9],[3,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,5,3,0,0,0,0,0,0,0,0,3,8,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,9],[3,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,5,3,0,0,0,0,0,0,0,0,3,8,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,7,0,0,0,0,0,0,0,3],[3,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,5,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,9],[10,2,0,0,0,0,0,0,8,3,0,0,0,0,0,0,0,0,10,1,0,0,0,0,0,0,5,3,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,8,11,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,2,11] >;
 

C34⋊8C4 in GAP, Magma, Sage, TeX 

C_3^4\rtimes_8C_4
 
% in TeX
 
G:=Group("C3^4:8C4");
 
// GroupNames label
 
G:=SmallGroup(324,158);
 
// by ID
 
G=gap.SmallGroup(324,158);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,146,579,2164,7781]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

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