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## G = C32×C62order 324 = 22·34

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

Aliases: C32×C62, SmallGroup(324,176)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C62
 Chief series C1 — C3 — C32 — C33 — C34 — C33×C6 — C32×C62
 Lower central C1 — C32×C62
 Upper central 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, C3, C22, C6, C32, C2×C6, C3×C6, C33, C62, C32×C6, C34, C3×C62, C33×C6, C32×C62
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, C33, C62, C32×C6, C34, C3×C62, C33×C6, C32×C62

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

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

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

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

324 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3CB 6A ··· 6IF order 1 2 2 2 3 ··· 3 6 ··· 6 size 1 1 1 1 1 ··· 1 1 ··· 1

324 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C3 C6 kernel C32×C62 C33×C6 C3×C62 C32×C6 # reps 1 3 80 240

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

 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
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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