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G = C32×C62order 324 = 22·34

Abelian group of type [3,3,6,6]

direct product, abelian, monomial

Aliases: C32×C62, SmallGroup(324,176)

Series: Derived Chief Lower central Upper central

C1 — C32×C62
C1C3C32C33C34C33×C6 — C32×C62
C1 — C32×C62
C1 — C32×C62

Generators and relations for C32×C62
 G = < a,b,c,d | a3=b3=c6=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 1060, all normal (4 characteristic)
C1, C2 [×3], C3 [×40], C22, C6 [×120], C32 [×130], C2×C6 [×40], C3×C6 [×390], C33 [×40], C62 [×130], C32×C6 [×120], C34, C3×C62 [×40], C33×C6 [×3], C32×C62
Quotients: C1, C2 [×3], C3 [×40], C22, C6 [×120], C32 [×130], C2×C6 [×40], C3×C6 [×390], C33 [×40], C62 [×130], C32×C6 [×120], C34, C3×C62 [×40], C33×C6 [×3], C32×C62

Smallest permutation representation of C32×C62
Regular action on 324 points
Generators in S324
(1 42 31)(2 37 32)(3 38 33)(4 39 34)(5 40 35)(6 41 36)(7 229 138)(8 230 133)(9 231 134)(10 232 135)(11 233 136)(12 234 137)(13 238 58)(14 239 59)(15 240 60)(16 235 55)(17 236 56)(18 237 57)(19 47 199)(20 48 200)(21 43 201)(22 44 202)(23 45 203)(24 46 204)(25 183 52)(26 184 53)(27 185 54)(28 186 49)(29 181 50)(30 182 51)(61 219 88)(62 220 89)(63 221 90)(64 222 85)(65 217 86)(66 218 87)(67 249 78)(68 250 73)(69 251 74)(70 252 75)(71 247 76)(72 248 77)(79 155 228)(80 156 223)(81 151 224)(82 152 225)(83 153 226)(84 154 227)(91 119 192)(92 120 187)(93 115 188)(94 116 189)(95 117 190)(96 118 191)(97 129 194)(98 130 195)(99 131 196)(100 132 197)(101 127 198)(102 128 193)(103 294 279)(104 289 280)(105 290 281)(106 291 282)(107 292 277)(108 293 278)(109 180 210)(110 175 205)(111 176 206)(112 177 207)(113 178 208)(114 179 209)(121 267 258)(122 268 253)(123 269 254)(124 270 255)(125 265 256)(126 266 257)(139 170 315)(140 171 316)(141 172 317)(142 173 318)(143 174 313)(144 169 314)(145 216 246)(146 211 241)(147 212 242)(148 213 243)(149 214 244)(150 215 245)(157 303 163)(158 304 164)(159 305 165)(160 306 166)(161 301 167)(162 302 168)(259 283 274)(260 284 275)(261 285 276)(262 286 271)(263 287 272)(264 288 273)(295 319 310)(296 320 311)(297 321 312)(298 322 307)(299 323 308)(300 324 309)
(1 24 186)(2 19 181)(3 20 182)(4 21 183)(5 22 184)(6 23 185)(7 241 80)(8 242 81)(9 243 82)(10 244 83)(11 245 84)(12 246 79)(13 89 247)(14 90 248)(15 85 249)(16 86 250)(17 87 251)(18 88 252)(25 34 201)(26 35 202)(27 36 203)(28 31 204)(29 32 199)(30 33 200)(37 47 50)(38 48 51)(39 43 52)(40 44 53)(41 45 54)(42 46 49)(55 217 68)(56 218 69)(57 219 70)(58 220 71)(59 221 72)(60 222 67)(61 75 237)(62 76 238)(63 77 239)(64 78 240)(65 73 235)(66 74 236)(91 101 180)(92 102 175)(93 97 176)(94 98 177)(95 99 178)(96 100 179)(103 261 258)(104 262 253)(105 263 254)(106 264 255)(107 259 256)(108 260 257)(109 192 198)(110 187 193)(111 188 194)(112 189 195)(113 190 196)(114 191 197)(115 129 206)(116 130 207)(117 131 208)(118 132 209)(119 127 210)(120 128 205)(121 294 285)(122 289 286)(123 290 287)(124 291 288)(125 292 283)(126 293 284)(133 212 224)(134 213 225)(135 214 226)(136 215 227)(137 216 228)(138 211 223)(139 297 303)(140 298 304)(141 299 305)(142 300 306)(143 295 301)(144 296 302)(145 155 234)(146 156 229)(147 151 230)(148 152 231)(149 153 232)(150 154 233)(157 315 312)(158 316 307)(159 317 308)(160 318 309)(161 313 310)(162 314 311)(163 170 321)(164 171 322)(165 172 323)(166 173 324)(167 174 319)(168 169 320)(265 277 274)(266 278 275)(267 279 276)(268 280 271)(269 281 272)(270 282 273)
(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)
(1 212 256 222 93 295)(2 213 257 217 94 296)(3 214 258 218 95 297)(4 215 253 219 96 298)(5 216 254 220 91 299)(6 211 255 221 92 300)(7 288 14 205 173 54)(8 283 15 206 174 49)(9 284 16 207 169 50)(10 285 17 208 170 51)(11 286 18 209 171 52)(12 287 13 210 172 53)(19 225 108 68 98 302)(20 226 103 69 99 303)(21 227 104 70 100 304)(22 228 105 71 101 305)(23 223 106 72 102 306)(24 224 107 67 97 301)(25 233 271 237 114 316)(26 234 272 238 109 317)(27 229 273 239 110 318)(28 230 274 240 111 313)(29 231 275 235 112 314)(30 232 276 236 113 315)(31 147 265 64 188 310)(32 148 266 65 189 311)(33 149 267 66 190 312)(34 150 268 61 191 307)(35 145 269 62 192 308)(36 146 270 63 187 309)(37 243 126 86 116 320)(38 244 121 87 117 321)(39 245 122 88 118 322)(40 246 123 89 119 323)(41 241 124 90 120 324)(42 242 125 85 115 319)(43 84 289 252 132 164)(44 79 290 247 127 165)(45 80 291 248 128 166)(46 81 292 249 129 167)(47 82 293 250 130 168)(48 83 294 251 131 163)(55 177 144 181 134 260)(56 178 139 182 135 261)(57 179 140 183 136 262)(58 180 141 184 137 263)(59 175 142 185 138 264)(60 176 143 186 133 259)(73 195 162 199 152 278)(74 196 157 200 153 279)(75 197 158 201 154 280)(76 198 159 202 155 281)(77 193 160 203 156 282)(78 194 161 204 151 277)

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

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

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

324 conjugacy classes

class 1 2A2B2C3A···3CB6A···6IF
order12223···36···6
size11111···11···1

324 irreducible representations

dim1111
type++
imageC1C2C3C6
kernelC32×C62C33×C6C3×C62C32×C6
# reps1380240

Matrix representation of C32×C62 in GL4(𝔽7) generated by

1000
0100
0020
0004
,
2000
0200
0020
0004
,
6000
0300
0020
0002
,
3000
0400
0010
0003
G:=sub<GL(4,GF(7))| [1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,4],[2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,4],[6,0,0,0,0,3,0,0,0,0,2,0,0,0,0,2],[3,0,0,0,0,4,0,0,0,0,1,0,0,0,0,3] >;

C32×C62 in GAP, Magma, Sage, TeX

C_3^2\times C_6^2
% in TeX

G:=Group("C3^2xC6^2");
// GroupNames label

G:=SmallGroup(324,176);
// by ID

G=gap.SmallGroup(324,176);
# by ID

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

G:=Group<a,b,c,d|a^3=b^3=c^6=d^6=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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