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G = C62.234C23order 288 = 25·32

79th non-split extension by C62 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial

Aliases: C12.13Dic6, C62.234C23, (C2×C12).33D6, (C3×C12).16Q8, C6.43(C2×Dic6), C35(C4.Dic6), C4.3(C324Q8), (C6×C12).135C22, C6.47(Q83S3), C12⋊Dic3.8C2, C6.Dic6.3C2, C6.100(D42S3), C3214(C42.C2), C2.4(C12.26D6), C2.12(C12.D6), C4⋊C4.6(C3⋊S3), (C3×C4⋊C4).23S3, (C3×C6).57(C2×Q8), (C4×C3⋊Dic3).6C2, C2.8(C2×C324Q8), (C32×C4⋊C4).14C2, (C3×C6).147(C4○D4), (C2×C6).251(C22×S3), C22.48(C22×C3⋊S3), (C2×C3⋊Dic3).84C22, (C2×C4).10(C2×C3⋊S3), SmallGroup(288,747)

Series: Derived Chief Lower central Upper central

C1C62 — C62.234C23
C1C3C32C3×C6C62C2×C3⋊Dic3C4×C3⋊Dic3 — C62.234C23
C32C62 — C62.234C23
C1C22C4⋊C4

Generators and relations for C62.234C23
 G = < a,b,c,d,e | a6=b6=1, c2=e2=a3, d2=b3, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=a3b3c, ede-1=b3d >

Subgroups: 508 in 168 conjugacy classes, 77 normal (19 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×6], C22, C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×4], C32, Dic3 [×16], C12 [×8], C12 [×8], C2×C6 [×4], C42, C4⋊C4, C4⋊C4 [×5], C3×C6 [×3], C2×Dic3 [×16], C2×C12 [×12], C42.C2, C3⋊Dic3 [×4], C3×C12 [×2], C3×C12 [×2], C62, C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×12], C3×C4⋊C4 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, C6×C12 [×2], C4.Dic6 [×4], C4×C3⋊Dic3, C6.Dic6 [×2], C12⋊Dic3, C12⋊Dic3 [×2], C32×C4⋊C4, C62.234C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], Q8 [×2], C23, D6 [×12], C2×Q8, C4○D4 [×2], C3⋊S3, Dic6 [×8], C22×S3 [×4], C42.C2, C2×C3⋊S3 [×3], C2×Dic6 [×4], D42S3 [×4], Q83S3 [×4], C324Q8 [×2], C22×C3⋊S3, C4.Dic6 [×4], C2×C324Q8, C12.D6, C12.26D6, C62.234C23

Smallest permutation representation of C62.234C23
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 35 38 57 13 52)(2 36 39 58 14 53)(3 31 40 59 15 54)(4 32 41 60 16 49)(5 33 42 55 17 50)(6 34 37 56 18 51)(7 286 27 22 267 281)(8 287 28 23 268 282)(9 288 29 24 269 277)(10 283 30 19 270 278)(11 284 25 20 265 279)(12 285 26 21 266 280)(43 78 91 63 86 71)(44 73 92 64 87 72)(45 74 93 65 88 67)(46 75 94 66 89 68)(47 76 95 61 90 69)(48 77 96 62 85 70)(79 114 127 99 122 107)(80 109 128 100 123 108)(81 110 129 101 124 103)(82 111 130 102 125 104)(83 112 131 97 126 105)(84 113 132 98 121 106)(115 165 145 135 155 143)(116 166 146 136 156 144)(117 167 147 137 151 139)(118 168 148 138 152 140)(119 163 149 133 153 141)(120 164 150 134 154 142)(157 171 184 176 204 190)(158 172 185 177 199 191)(159 173 186 178 200 192)(160 174 181 179 201 187)(161 169 182 180 202 188)(162 170 183 175 203 189)(193 207 220 212 240 226)(194 208 221 213 235 227)(195 209 222 214 236 228)(196 210 217 215 237 223)(197 205 218 216 238 224)(198 206 219 211 239 225)(229 243 256 248 276 262)(230 244 257 249 271 263)(231 245 258 250 272 264)(232 246 253 251 273 259)(233 241 254 252 274 260)(234 242 255 247 275 261)
(1 202 4 199)(2 201 5 204)(3 200 6 203)(7 166 10 163)(8 165 11 168)(9 164 12 167)(13 161 16 158)(14 160 17 157)(15 159 18 162)(19 153 22 156)(20 152 23 155)(21 151 24 154)(25 140 28 143)(26 139 29 142)(27 144 30 141)(31 178 34 175)(32 177 35 180)(33 176 36 179)(37 183 40 186)(38 182 41 185)(39 181 42 184)(43 212 46 215)(44 211 47 214)(45 216 48 213)(49 191 52 188)(50 190 53 187)(51 189 54 192)(55 171 58 174)(56 170 59 173)(57 169 60 172)(61 195 64 198)(62 194 65 197)(63 193 66 196)(67 238 70 235)(68 237 71 240)(69 236 72 239)(73 219 76 222)(74 218 77 221)(75 217 78 220)(79 248 82 251)(80 247 83 250)(81 252 84 249)(85 227 88 224)(86 226 89 223)(87 225 90 228)(91 207 94 210)(92 206 95 209)(93 205 96 208)(97 231 100 234)(98 230 101 233)(99 229 102 232)(103 274 106 271)(104 273 107 276)(105 272 108 275)(109 255 112 258)(110 254 113 257)(111 253 114 256)(115 284 118 287)(116 283 119 286)(117 288 120 285)(121 263 124 260)(122 262 125 259)(123 261 126 264)(127 243 130 246)(128 242 131 245)(129 241 132 244)(133 267 136 270)(134 266 137 269)(135 265 138 268)(145 279 148 282)(146 278 149 281)(147 277 150 280)
(1 129 57 103)(2 130 58 104)(3 131 59 105)(4 132 60 106)(5 127 55 107)(6 128 56 108)(7 222 22 228)(8 217 23 223)(9 218 24 224)(10 219 19 225)(11 220 20 226)(12 221 21 227)(13 81 35 101)(14 82 36 102)(15 83 31 97)(16 84 32 98)(17 79 33 99)(18 80 34 100)(25 240 279 207)(26 235 280 208)(27 236 281 209)(28 237 282 210)(29 238 277 205)(30 239 278 206)(37 123 51 109)(38 124 52 110)(39 125 53 111)(40 126 54 112)(41 121 49 113)(42 122 50 114)(43 135 63 115)(44 136 64 116)(45 137 65 117)(46 138 66 118)(47 133 61 119)(48 134 62 120)(67 147 93 139)(68 148 94 140)(69 149 95 141)(70 150 96 142)(71 145 91 143)(72 146 92 144)(73 156 87 166)(74 151 88 167)(75 152 89 168)(76 153 90 163)(77 154 85 164)(78 155 86 165)(157 248 176 229)(158 249 177 230)(159 250 178 231)(160 251 179 232)(161 252 180 233)(162 247 175 234)(169 274 202 241)(170 275 203 242)(171 276 204 243)(172 271 199 244)(173 272 200 245)(174 273 201 246)(181 259 187 253)(182 260 188 254)(183 261 189 255)(184 262 190 256)(185 263 191 257)(186 264 192 258)(193 284 212 265)(194 285 213 266)(195 286 214 267)(196 287 215 268)(197 288 216 269)(198 283 211 270)
(1 70 4 67)(2 71 5 68)(3 72 6 69)(7 255 10 258)(8 256 11 253)(9 257 12 254)(13 62 16 65)(14 63 17 66)(15 64 18 61)(19 264 22 261)(20 259 23 262)(21 260 24 263)(25 273 28 276)(26 274 29 271)(27 275 30 272)(31 44 34 47)(32 45 35 48)(33 46 36 43)(37 76 40 73)(38 77 41 74)(39 78 42 75)(49 88 52 85)(50 89 53 86)(51 90 54 87)(55 94 58 91)(56 95 59 92)(57 96 60 93)(79 138 82 135)(80 133 83 136)(81 134 84 137)(97 116 100 119)(98 117 101 120)(99 118 102 115)(103 150 106 147)(104 145 107 148)(105 146 108 149)(109 153 112 156)(110 154 113 151)(111 155 114 152)(121 167 124 164)(122 168 125 165)(123 163 126 166)(127 140 130 143)(128 141 131 144)(129 142 132 139)(157 212 160 215)(158 213 161 216)(159 214 162 211)(169 238 172 235)(170 239 173 236)(171 240 174 237)(175 198 178 195)(176 193 179 196)(177 194 180 197)(181 223 184 226)(182 224 185 227)(183 225 186 228)(187 217 190 220)(188 218 191 221)(189 219 192 222)(199 208 202 205)(200 209 203 206)(201 210 204 207)(229 265 232 268)(230 266 233 269)(231 267 234 270)(241 277 244 280)(242 278 245 281)(243 279 246 282)(247 283 250 286)(248 284 251 287)(249 285 252 288)

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

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

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

54 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F4G4H4I4J6A···6L12A···12X
order1222333344444444446···612···12
size1111222222441818181836362···24···4

54 irreducible representations

dim111112222244
type++++++-+--+
imageC1C2C2C2C2S3Q8D6C4○D4Dic6D42S3Q83S3
kernelC62.234C23C4×C3⋊Dic3C6.Dic6C12⋊Dic3C32×C4⋊C4C3×C4⋊C4C3×C12C2×C12C3×C6C12C6C6
# reps11231421241644

Matrix representation of C62.234C23 in GL6(𝔽13)

1200000
0120000
0001200
0011200
000001
0000121
,
1200000
0120000
001000
000100
0000012
0000112
,
800000
080000
003300
0061000
00001111
000092
,
050000
500000
001000
000100
0000120
0000012
,
010000
1200000
001000
000100
0000106
000073

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,3,6,0,0,0,0,3,10,0,0,0,0,0,0,11,9,0,0,0,0,11,2],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,7,0,0,0,0,6,3] >;

C62.234C23 in GAP, Magma, Sage, TeX

C_6^2._{234}C_2^3
% in TeX

G:=Group("C6^2.234C2^3");
// GroupNames label

G:=SmallGroup(288,747);
// by ID

G=gap.SmallGroup(288,747);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,120,590,219,58,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=e^2=a^3,d^2=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,e*c*e^-1=a^3*b^3*c,e*d*e^-1=b^3*d>;
// generators/relations

׿
×
𝔽