metabelian, supersoluble, monomial
Aliases: C62.231C23, C6.42(S3×Q8), C32⋊12(C4×Q8), C12.37(C4×S3), C3⋊Dic3⋊10Q8, (C2×C12).210D6, C32⋊4Q8⋊11C4, C3⋊4(Dic6⋊C4), C6.98(D4⋊2S3), (C6×C12).252C22, C6.Dic6.9C2, C2.3(C12.D6), C4.4(C4×C3⋊S3), C6.68(S3×C2×C4), C2.1(Q8×C3⋊S3), C4⋊C4.7(C3⋊S3), (C3×C4⋊C4).20S3, (C3×C6).70(C2×Q8), (C3×C12).71(C2×C4), (C32×C4⋊C4).11C2, C3⋊Dic3.37(C2×C4), (C4×C3⋊Dic3).22C2, (C3×C6).99(C22×C4), (C3×C6).145(C4○D4), (C2×C6).248(C22×S3), C22.15(C22×C3⋊S3), (C2×C32⋊4Q8).14C2, (C2×C3⋊Dic3).156C22, C2.10(C2×C4×C3⋊S3), (C2×C4).30(C2×C3⋊S3), SmallGroup(288,744)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C4×C3⋊Dic3 — C62.231C23 |
Generators and relations for C62.231C23
G = < a,b,c,d,e | a6=b6=1, c2=b3, d2=a3, e2=a3b3, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=a3d >
Subgroups: 596 in 210 conjugacy classes, 87 normal (19 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×C12, C4×Q8, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C4×Dic3, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, C32⋊4Q8, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, Dic6⋊C4, C4×C3⋊Dic3, C4×C3⋊Dic3, C6.Dic6, C32×C4⋊C4, C2×C32⋊4Q8, C62.231C23
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, C3⋊S3, C4×S3, C22×S3, C4×Q8, C2×C3⋊S3, S3×C2×C4, D4⋊2S3, S3×Q8, C4×C3⋊S3, C22×C3⋊S3, Dic6⋊C4, C2×C4×C3⋊S3, C12.D6, Q8×C3⋊S3, C62.231C23
►Regular action on 288 points
Generators in S288
(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)
(1 15 58 33 39 61)(2 16 59 34 40 62)(3 17 60 35 41 63)(4 18 55 36 42 64)(5 13 56 31 37 65)(6 14 57 32 38 66)(7 280 267 22 29 286)(8 281 268 23 30 287)(9 282 269 24 25 288)(10 277 270 19 26 283)(11 278 265 20 27 284)(12 279 266 21 28 285)(43 92 67 73 101 50)(44 93 68 74 102 51)(45 94 69 75 97 52)(46 95 70 76 98 53)(47 96 71 77 99 54)(48 91 72 78 100 49)(79 128 103 109 137 86)(80 129 104 110 138 87)(81 130 105 111 133 88)(82 131 106 112 134 89)(83 132 107 113 135 90)(84 127 108 114 136 85)(115 168 139 149 155 122)(116 163 140 150 156 123)(117 164 141 145 151 124)(118 165 142 146 152 125)(119 166 143 147 153 126)(120 167 144 148 154 121)(157 190 203 176 184 170)(158 191 204 177 185 171)(159 192 199 178 186 172)(160 187 200 179 181 173)(161 188 201 180 182 174)(162 189 202 175 183 169)(193 226 239 212 220 206)(194 227 240 213 221 207)(195 228 235 214 222 208)(196 223 236 215 217 209)(197 224 237 216 218 210)(198 225 238 211 219 205)(229 262 275 248 256 242)(230 263 276 249 257 243)(231 264 271 250 258 244)(232 259 272 251 253 245)(233 260 273 252 254 246)(234 261 274 247 255 241)
(1 177 33 158)(2 176 34 157)(3 175 35 162)(4 180 36 161)(5 179 31 160)(6 178 32 159)(7 163 22 156)(8 168 23 155)(9 167 24 154)(10 166 19 153)(11 165 20 152)(12 164 21 151)(13 200 37 173)(14 199 38 172)(15 204 39 171)(16 203 40 170)(17 202 41 169)(18 201 42 174)(25 148 282 120)(26 147 277 119)(27 146 278 118)(28 145 279 117)(29 150 280 116)(30 149 281 115)(43 236 73 209)(44 235 74 208)(45 240 75 207)(46 239 76 206)(47 238 77 205)(48 237 78 210)(49 216 72 197)(50 215 67 196)(51 214 68 195)(52 213 69 194)(53 212 70 193)(54 211 71 198)(55 188 64 182)(56 187 65 181)(57 192 66 186)(58 191 61 185)(59 190 62 184)(60 189 63 183)(79 272 109 245)(80 271 110 244)(81 276 111 243)(82 275 112 242)(83 274 113 241)(84 273 114 246)(85 252 108 233)(86 251 103 232)(87 250 104 231)(88 249 105 230)(89 248 106 229)(90 247 107 234)(91 224 100 218)(92 223 101 217)(93 228 102 222)(94 227 97 221)(95 226 98 220)(96 225 99 219)(121 288 144 269)(122 287 139 268)(123 286 140 267)(124 285 141 266)(125 284 142 265)(126 283 143 270)(127 260 136 254)(128 259 137 253)(129 264 138 258)(130 263 133 257)(131 262 134 256)(132 261 135 255)
(1 85 4 88)(2 86 5 89)(3 87 6 90)(7 222 10 219)(8 217 11 220)(9 218 12 221)(13 82 16 79)(14 83 17 80)(15 84 18 81)(19 225 22 228)(20 226 23 223)(21 227 24 224)(25 237 28 240)(26 238 29 235)(27 239 30 236)(31 106 34 103)(32 107 35 104)(33 108 36 105)(37 112 40 109)(38 113 41 110)(39 114 42 111)(43 118 46 115)(44 119 47 116)(45 120 48 117)(49 124 52 121)(50 125 53 122)(51 126 54 123)(55 130 58 127)(56 131 59 128)(57 132 60 129)(61 136 64 133)(62 137 65 134)(63 138 66 135)(67 142 70 139)(68 143 71 140)(69 144 72 141)(73 146 76 149)(74 147 77 150)(75 148 78 145)(91 164 94 167)(92 165 95 168)(93 166 96 163)(97 154 100 151)(98 155 101 152)(99 156 102 153)(157 232 160 229)(158 233 161 230)(159 234 162 231)(169 244 172 241)(170 245 173 242)(171 246 174 243)(175 250 178 247)(176 251 179 248)(177 252 180 249)(181 256 184 253)(182 257 185 254)(183 258 186 255)(187 262 190 259)(188 263 191 260)(189 264 192 261)(193 268 196 265)(194 269 197 266)(195 270 198 267)(199 274 202 271)(200 275 203 272)(201 276 204 273)(205 280 208 277)(206 281 209 278)(207 282 210 279)(211 286 214 283)(212 287 215 284)(213 288 216 285)
(1 72 36 52)(2 67 31 53)(3 68 32 54)(4 69 33 49)(5 70 34 50)(6 71 35 51)(7 255 19 264)(8 256 20 259)(9 257 21 260)(10 258 22 261)(11 253 23 262)(12 254 24 263)(13 76 40 43)(14 77 41 44)(15 78 42 45)(16 73 37 46)(17 74 38 47)(18 75 39 48)(25 276 279 246)(26 271 280 241)(27 272 281 242)(28 273 282 243)(29 274 277 244)(30 275 278 245)(55 97 61 91)(56 98 62 92)(57 99 63 93)(58 100 64 94)(59 101 65 95)(60 102 66 96)(79 149 112 118)(80 150 113 119)(81 145 114 120)(82 146 109 115)(83 147 110 116)(84 148 111 117)(85 144 105 124)(86 139 106 125)(87 140 107 126)(88 141 108 121)(89 142 103 122)(90 143 104 123)(127 154 133 164)(128 155 134 165)(129 156 135 166)(130 151 136 167)(131 152 137 168)(132 153 138 163)(157 215 179 193)(158 216 180 194)(159 211 175 195)(160 212 176 196)(161 213 177 197)(162 214 178 198)(169 235 199 205)(170 236 200 206)(171 237 201 207)(172 238 202 208)(173 239 203 209)(174 240 204 210)(181 226 190 217)(182 227 191 218)(183 228 192 219)(184 223 187 220)(185 224 188 221)(186 225 189 222)(229 284 251 268)(230 285 252 269)(231 286 247 270)(232 287 248 265)(233 288 249 266)(234 283 250 267)
G:=sub<Sym(288)| (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), (1,15,58,33,39,61)(2,16,59,34,40,62)(3,17,60,35,41,63)(4,18,55,36,42,64)(5,13,56,31,37,65)(6,14,57,32,38,66)(7,280,267,22,29,286)(8,281,268,23,30,287)(9,282,269,24,25,288)(10,277,270,19,26,283)(11,278,265,20,27,284)(12,279,266,21,28,285)(43,92,67,73,101,50)(44,93,68,74,102,51)(45,94,69,75,97,52)(46,95,70,76,98,53)(47,96,71,77,99,54)(48,91,72,78,100,49)(79,128,103,109,137,86)(80,129,104,110,138,87)(81,130,105,111,133,88)(82,131,106,112,134,89)(83,132,107,113,135,90)(84,127,108,114,136,85)(115,168,139,149,155,122)(116,163,140,150,156,123)(117,164,141,145,151,124)(118,165,142,146,152,125)(119,166,143,147,153,126)(120,167,144,148,154,121)(157,190,203,176,184,170)(158,191,204,177,185,171)(159,192,199,178,186,172)(160,187,200,179,181,173)(161,188,201,180,182,174)(162,189,202,175,183,169)(193,226,239,212,220,206)(194,227,240,213,221,207)(195,228,235,214,222,208)(196,223,236,215,217,209)(197,224,237,216,218,210)(198,225,238,211,219,205)(229,262,275,248,256,242)(230,263,276,249,257,243)(231,264,271,250,258,244)(232,259,272,251,253,245)(233,260,273,252,254,246)(234,261,274,247,255,241), (1,177,33,158)(2,176,34,157)(3,175,35,162)(4,180,36,161)(5,179,31,160)(6,178,32,159)(7,163,22,156)(8,168,23,155)(9,167,24,154)(10,166,19,153)(11,165,20,152)(12,164,21,151)(13,200,37,173)(14,199,38,172)(15,204,39,171)(16,203,40,170)(17,202,41,169)(18,201,42,174)(25,148,282,120)(26,147,277,119)(27,146,278,118)(28,145,279,117)(29,150,280,116)(30,149,281,115)(43,236,73,209)(44,235,74,208)(45,240,75,207)(46,239,76,206)(47,238,77,205)(48,237,78,210)(49,216,72,197)(50,215,67,196)(51,214,68,195)(52,213,69,194)(53,212,70,193)(54,211,71,198)(55,188,64,182)(56,187,65,181)(57,192,66,186)(58,191,61,185)(59,190,62,184)(60,189,63,183)(79,272,109,245)(80,271,110,244)(81,276,111,243)(82,275,112,242)(83,274,113,241)(84,273,114,246)(85,252,108,233)(86,251,103,232)(87,250,104,231)(88,249,105,230)(89,248,106,229)(90,247,107,234)(91,224,100,218)(92,223,101,217)(93,228,102,222)(94,227,97,221)(95,226,98,220)(96,225,99,219)(121,288,144,269)(122,287,139,268)(123,286,140,267)(124,285,141,266)(125,284,142,265)(126,283,143,270)(127,260,136,254)(128,259,137,253)(129,264,138,258)(130,263,133,257)(131,262,134,256)(132,261,135,255), (1,85,4,88)(2,86,5,89)(3,87,6,90)(7,222,10,219)(8,217,11,220)(9,218,12,221)(13,82,16,79)(14,83,17,80)(15,84,18,81)(19,225,22,228)(20,226,23,223)(21,227,24,224)(25,237,28,240)(26,238,29,235)(27,239,30,236)(31,106,34,103)(32,107,35,104)(33,108,36,105)(37,112,40,109)(38,113,41,110)(39,114,42,111)(43,118,46,115)(44,119,47,116)(45,120,48,117)(49,124,52,121)(50,125,53,122)(51,126,54,123)(55,130,58,127)(56,131,59,128)(57,132,60,129)(61,136,64,133)(62,137,65,134)(63,138,66,135)(67,142,70,139)(68,143,71,140)(69,144,72,141)(73,146,76,149)(74,147,77,150)(75,148,78,145)(91,164,94,167)(92,165,95,168)(93,166,96,163)(97,154,100,151)(98,155,101,152)(99,156,102,153)(157,232,160,229)(158,233,161,230)(159,234,162,231)(169,244,172,241)(170,245,173,242)(171,246,174,243)(175,250,178,247)(176,251,179,248)(177,252,180,249)(181,256,184,253)(182,257,185,254)(183,258,186,255)(187,262,190,259)(188,263,191,260)(189,264,192,261)(193,268,196,265)(194,269,197,266)(195,270,198,267)(199,274,202,271)(200,275,203,272)(201,276,204,273)(205,280,208,277)(206,281,209,278)(207,282,210,279)(211,286,214,283)(212,287,215,284)(213,288,216,285), (1,72,36,52)(2,67,31,53)(3,68,32,54)(4,69,33,49)(5,70,34,50)(6,71,35,51)(7,255,19,264)(8,256,20,259)(9,257,21,260)(10,258,22,261)(11,253,23,262)(12,254,24,263)(13,76,40,43)(14,77,41,44)(15,78,42,45)(16,73,37,46)(17,74,38,47)(18,75,39,48)(25,276,279,246)(26,271,280,241)(27,272,281,242)(28,273,282,243)(29,274,277,244)(30,275,278,245)(55,97,61,91)(56,98,62,92)(57,99,63,93)(58,100,64,94)(59,101,65,95)(60,102,66,96)(79,149,112,118)(80,150,113,119)(81,145,114,120)(82,146,109,115)(83,147,110,116)(84,148,111,117)(85,144,105,124)(86,139,106,125)(87,140,107,126)(88,141,108,121)(89,142,103,122)(90,143,104,123)(127,154,133,164)(128,155,134,165)(129,156,135,166)(130,151,136,167)(131,152,137,168)(132,153,138,163)(157,215,179,193)(158,216,180,194)(159,211,175,195)(160,212,176,196)(161,213,177,197)(162,214,178,198)(169,235,199,205)(170,236,200,206)(171,237,201,207)(172,238,202,208)(173,239,203,209)(174,240,204,210)(181,226,190,217)(182,227,191,218)(183,228,192,219)(184,223,187,220)(185,224,188,221)(186,225,189,222)(229,284,251,268)(230,285,252,269)(231,286,247,270)(232,287,248,265)(233,288,249,266)(234,283,250,267)>;
G:=Group( (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), (1,15,58,33,39,61)(2,16,59,34,40,62)(3,17,60,35,41,63)(4,18,55,36,42,64)(5,13,56,31,37,65)(6,14,57,32,38,66)(7,280,267,22,29,286)(8,281,268,23,30,287)(9,282,269,24,25,288)(10,277,270,19,26,283)(11,278,265,20,27,284)(12,279,266,21,28,285)(43,92,67,73,101,50)(44,93,68,74,102,51)(45,94,69,75,97,52)(46,95,70,76,98,53)(47,96,71,77,99,54)(48,91,72,78,100,49)(79,128,103,109,137,86)(80,129,104,110,138,87)(81,130,105,111,133,88)(82,131,106,112,134,89)(83,132,107,113,135,90)(84,127,108,114,136,85)(115,168,139,149,155,122)(116,163,140,150,156,123)(117,164,141,145,151,124)(118,165,142,146,152,125)(119,166,143,147,153,126)(120,167,144,148,154,121)(157,190,203,176,184,170)(158,191,204,177,185,171)(159,192,199,178,186,172)(160,187,200,179,181,173)(161,188,201,180,182,174)(162,189,202,175,183,169)(193,226,239,212,220,206)(194,227,240,213,221,207)(195,228,235,214,222,208)(196,223,236,215,217,209)(197,224,237,216,218,210)(198,225,238,211,219,205)(229,262,275,248,256,242)(230,263,276,249,257,243)(231,264,271,250,258,244)(232,259,272,251,253,245)(233,260,273,252,254,246)(234,261,274,247,255,241), (1,177,33,158)(2,176,34,157)(3,175,35,162)(4,180,36,161)(5,179,31,160)(6,178,32,159)(7,163,22,156)(8,168,23,155)(9,167,24,154)(10,166,19,153)(11,165,20,152)(12,164,21,151)(13,200,37,173)(14,199,38,172)(15,204,39,171)(16,203,40,170)(17,202,41,169)(18,201,42,174)(25,148,282,120)(26,147,277,119)(27,146,278,118)(28,145,279,117)(29,150,280,116)(30,149,281,115)(43,236,73,209)(44,235,74,208)(45,240,75,207)(46,239,76,206)(47,238,77,205)(48,237,78,210)(49,216,72,197)(50,215,67,196)(51,214,68,195)(52,213,69,194)(53,212,70,193)(54,211,71,198)(55,188,64,182)(56,187,65,181)(57,192,66,186)(58,191,61,185)(59,190,62,184)(60,189,63,183)(79,272,109,245)(80,271,110,244)(81,276,111,243)(82,275,112,242)(83,274,113,241)(84,273,114,246)(85,252,108,233)(86,251,103,232)(87,250,104,231)(88,249,105,230)(89,248,106,229)(90,247,107,234)(91,224,100,218)(92,223,101,217)(93,228,102,222)(94,227,97,221)(95,226,98,220)(96,225,99,219)(121,288,144,269)(122,287,139,268)(123,286,140,267)(124,285,141,266)(125,284,142,265)(126,283,143,270)(127,260,136,254)(128,259,137,253)(129,264,138,258)(130,263,133,257)(131,262,134,256)(132,261,135,255), (1,85,4,88)(2,86,5,89)(3,87,6,90)(7,222,10,219)(8,217,11,220)(9,218,12,221)(13,82,16,79)(14,83,17,80)(15,84,18,81)(19,225,22,228)(20,226,23,223)(21,227,24,224)(25,237,28,240)(26,238,29,235)(27,239,30,236)(31,106,34,103)(32,107,35,104)(33,108,36,105)(37,112,40,109)(38,113,41,110)(39,114,42,111)(43,118,46,115)(44,119,47,116)(45,120,48,117)(49,124,52,121)(50,125,53,122)(51,126,54,123)(55,130,58,127)(56,131,59,128)(57,132,60,129)(61,136,64,133)(62,137,65,134)(63,138,66,135)(67,142,70,139)(68,143,71,140)(69,144,72,141)(73,146,76,149)(74,147,77,150)(75,148,78,145)(91,164,94,167)(92,165,95,168)(93,166,96,163)(97,154,100,151)(98,155,101,152)(99,156,102,153)(157,232,160,229)(158,233,161,230)(159,234,162,231)(169,244,172,241)(170,245,173,242)(171,246,174,243)(175,250,178,247)(176,251,179,248)(177,252,180,249)(181,256,184,253)(182,257,185,254)(183,258,186,255)(187,262,190,259)(188,263,191,260)(189,264,192,261)(193,268,196,265)(194,269,197,266)(195,270,198,267)(199,274,202,271)(200,275,203,272)(201,276,204,273)(205,280,208,277)(206,281,209,278)(207,282,210,279)(211,286,214,283)(212,287,215,284)(213,288,216,285), (1,72,36,52)(2,67,31,53)(3,68,32,54)(4,69,33,49)(5,70,34,50)(6,71,35,51)(7,255,19,264)(8,256,20,259)(9,257,21,260)(10,258,22,261)(11,253,23,262)(12,254,24,263)(13,76,40,43)(14,77,41,44)(15,78,42,45)(16,73,37,46)(17,74,38,47)(18,75,39,48)(25,276,279,246)(26,271,280,241)(27,272,281,242)(28,273,282,243)(29,274,277,244)(30,275,278,245)(55,97,61,91)(56,98,62,92)(57,99,63,93)(58,100,64,94)(59,101,65,95)(60,102,66,96)(79,149,112,118)(80,150,113,119)(81,145,114,120)(82,146,109,115)(83,147,110,116)(84,148,111,117)(85,144,105,124)(86,139,106,125)(87,140,107,126)(88,141,108,121)(89,142,103,122)(90,143,104,123)(127,154,133,164)(128,155,134,165)(129,156,135,166)(130,151,136,167)(131,152,137,168)(132,153,138,163)(157,215,179,193)(158,216,180,194)(159,211,175,195)(160,212,176,196)(161,213,177,197)(162,214,178,198)(169,235,199,205)(170,236,200,206)(171,237,201,207)(172,238,202,208)(173,239,203,209)(174,240,204,210)(181,226,190,217)(182,227,191,218)(183,228,192,219)(184,223,187,220)(185,224,188,221)(186,225,189,222)(229,284,251,268)(230,285,252,269)(231,286,247,270)(232,287,248,265)(233,288,249,266)(234,283,250,267) );
G=PermutationGroup([[(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)], [(1,15,58,33,39,61),(2,16,59,34,40,62),(3,17,60,35,41,63),(4,18,55,36,42,64),(5,13,56,31,37,65),(6,14,57,32,38,66),(7,280,267,22,29,286),(8,281,268,23,30,287),(9,282,269,24,25,288),(10,277,270,19,26,283),(11,278,265,20,27,284),(12,279,266,21,28,285),(43,92,67,73,101,50),(44,93,68,74,102,51),(45,94,69,75,97,52),(46,95,70,76,98,53),(47,96,71,77,99,54),(48,91,72,78,100,49),(79,128,103,109,137,86),(80,129,104,110,138,87),(81,130,105,111,133,88),(82,131,106,112,134,89),(83,132,107,113,135,90),(84,127,108,114,136,85),(115,168,139,149,155,122),(116,163,140,150,156,123),(117,164,141,145,151,124),(118,165,142,146,152,125),(119,166,143,147,153,126),(120,167,144,148,154,121),(157,190,203,176,184,170),(158,191,204,177,185,171),(159,192,199,178,186,172),(160,187,200,179,181,173),(161,188,201,180,182,174),(162,189,202,175,183,169),(193,226,239,212,220,206),(194,227,240,213,221,207),(195,228,235,214,222,208),(196,223,236,215,217,209),(197,224,237,216,218,210),(198,225,238,211,219,205),(229,262,275,248,256,242),(230,263,276,249,257,243),(231,264,271,250,258,244),(232,259,272,251,253,245),(233,260,273,252,254,246),(234,261,274,247,255,241)], [(1,177,33,158),(2,176,34,157),(3,175,35,162),(4,180,36,161),(5,179,31,160),(6,178,32,159),(7,163,22,156),(8,168,23,155),(9,167,24,154),(10,166,19,153),(11,165,20,152),(12,164,21,151),(13,200,37,173),(14,199,38,172),(15,204,39,171),(16,203,40,170),(17,202,41,169),(18,201,42,174),(25,148,282,120),(26,147,277,119),(27,146,278,118),(28,145,279,117),(29,150,280,116),(30,149,281,115),(43,236,73,209),(44,235,74,208),(45,240,75,207),(46,239,76,206),(47,238,77,205),(48,237,78,210),(49,216,72,197),(50,215,67,196),(51,214,68,195),(52,213,69,194),(53,212,70,193),(54,211,71,198),(55,188,64,182),(56,187,65,181),(57,192,66,186),(58,191,61,185),(59,190,62,184),(60,189,63,183),(79,272,109,245),(80,271,110,244),(81,276,111,243),(82,275,112,242),(83,274,113,241),(84,273,114,246),(85,252,108,233),(86,251,103,232),(87,250,104,231),(88,249,105,230),(89,248,106,229),(90,247,107,234),(91,224,100,218),(92,223,101,217),(93,228,102,222),(94,227,97,221),(95,226,98,220),(96,225,99,219),(121,288,144,269),(122,287,139,268),(123,286,140,267),(124,285,141,266),(125,284,142,265),(126,283,143,270),(127,260,136,254),(128,259,137,253),(129,264,138,258),(130,263,133,257),(131,262,134,256),(132,261,135,255)], [(1,85,4,88),(2,86,5,89),(3,87,6,90),(7,222,10,219),(8,217,11,220),(9,218,12,221),(13,82,16,79),(14,83,17,80),(15,84,18,81),(19,225,22,228),(20,226,23,223),(21,227,24,224),(25,237,28,240),(26,238,29,235),(27,239,30,236),(31,106,34,103),(32,107,35,104),(33,108,36,105),(37,112,40,109),(38,113,41,110),(39,114,42,111),(43,118,46,115),(44,119,47,116),(45,120,48,117),(49,124,52,121),(50,125,53,122),(51,126,54,123),(55,130,58,127),(56,131,59,128),(57,132,60,129),(61,136,64,133),(62,137,65,134),(63,138,66,135),(67,142,70,139),(68,143,71,140),(69,144,72,141),(73,146,76,149),(74,147,77,150),(75,148,78,145),(91,164,94,167),(92,165,95,168),(93,166,96,163),(97,154,100,151),(98,155,101,152),(99,156,102,153),(157,232,160,229),(158,233,161,230),(159,234,162,231),(169,244,172,241),(170,245,173,242),(171,246,174,243),(175,250,178,247),(176,251,179,248),(177,252,180,249),(181,256,184,253),(182,257,185,254),(183,258,186,255),(187,262,190,259),(188,263,191,260),(189,264,192,261),(193,268,196,265),(194,269,197,266),(195,270,198,267),(199,274,202,271),(200,275,203,272),(201,276,204,273),(205,280,208,277),(206,281,209,278),(207,282,210,279),(211,286,214,283),(212,287,215,284),(213,288,216,285)], [(1,72,36,52),(2,67,31,53),(3,68,32,54),(4,69,33,49),(5,70,34,50),(6,71,35,51),(7,255,19,264),(8,256,20,259),(9,257,21,260),(10,258,22,261),(11,253,23,262),(12,254,24,263),(13,76,40,43),(14,77,41,44),(15,78,42,45),(16,73,37,46),(17,74,38,47),(18,75,39,48),(25,276,279,246),(26,271,280,241),(27,272,281,242),(28,273,282,243),(29,274,277,244),(30,275,278,245),(55,97,61,91),(56,98,62,92),(57,99,63,93),(58,100,64,94),(59,101,65,95),(60,102,66,96),(79,149,112,118),(80,150,113,119),(81,145,114,120),(82,146,109,115),(83,147,110,116),(84,148,111,117),(85,144,105,124),(86,139,106,125),(87,140,107,126),(88,141,108,121),(89,142,103,122),(90,143,104,123),(127,154,133,164),(128,155,134,165),(129,156,135,166),(130,151,136,167),(131,152,137,168),(132,153,138,163),(157,215,179,193),(158,216,180,194),(159,211,175,195),(160,212,176,196),(161,213,177,197),(162,214,178,198),(169,235,199,205),(170,236,200,206),(171,237,201,207),(172,238,202,208),(173,239,203,209),(174,240,204,210),(181,226,190,217),(182,227,191,218),(183,228,192,219),(184,223,187,220),(185,224,188,221),(186,225,189,222),(229,284,251,268),(230,285,252,269),(231,286,247,270),(232,287,248,265),(233,288,249,266),(234,283,250,267)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 6A | ··· | 6L | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D6 | C4○D4 | C4×S3 | D4⋊2S3 | S3×Q8 |
| kernel | C62.231C23 | C4×C3⋊Dic3 | C6.Dic6 | C32×C4⋊C4 | C2×C32⋊4Q8 | C32⋊4Q8 | C3×C4⋊C4 | C3⋊Dic3 | C2×C12 | C3×C6 | C12 | C6 | C6 |
| # reps | 1 | 3 | 2 | 1 | 1 | 8 | 4 | 2 | 12 | 2 | 16 | 4 | 4 |
Matrix representation of C62.231C23 ►in GL6(𝔽13)
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 2 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 5 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 5 |
| 0 | 0 | 0 | 0 | 7 | 9 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 10 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,9,2,0,0,0,0,0,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,7,3,0,0,0,0,5,6,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,4,7,0,0,0,0,5,9],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,10,12] >;
C62.231C23 in GAP, Magma, Sage, TeX
C_6^2._{231}C_2^3 % in TeX
G:=Group("C6^2.231C2^3"); // GroupNames label
G:=SmallGroup(288,744);
// by ID
G=gap.SmallGroup(288,744);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,135,142,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=b^3,d^2=a^3,e^2=a^3*b^3,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^3*d>;
// generators/relations