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## G = C62.259C23order 288 = 25·32

### 104th non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.259C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C4×C3⋊Dic3 — C62.259C23
 Lower central C32 — C62 — C62.259C23
 Upper central C1 — C22 — C2×Q8

Generators and relations for C62.259C23
G = < a,b,c,d,e | a6=b6=1, c2=b3, d2=e2=a3, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=a3d >

Subgroups: 620 in 204 conjugacy classes, 81 normal (13 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×8], C22, C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C32, Dic3 [×24], C12 [×8], C12 [×8], C2×C6 [×4], C42, C4⋊C4 [×4], C2×Q8, C2×Q8, C3×C6, C3×C6 [×2], Dic6 [×8], C2×Dic3 [×16], C2×C12 [×12], C3×Q8 [×8], C4⋊Q8, C3⋊Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C3×C12 [×2], C62, C4×Dic3 [×4], Dic3⋊C4 [×16], C2×Dic6 [×4], C6×Q8 [×4], C324Q8 [×2], C2×C3⋊Dic3 [×4], C6×C12, C6×C12 [×2], Q8×C32 [×2], Dic3⋊Q8 [×4], C4×C3⋊Dic3, C6.Dic6 [×4], C2×C324Q8, Q8×C3×C6, C62.259C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], Q8 [×4], C23, D6 [×12], C2×D4, C2×Q8 [×2], C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C4⋊Q8, C2×C3⋊S3 [×3], S3×Q8 [×8], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, Dic3⋊Q8 [×4], Q8×C3⋊S3 [×2], C2×C327D4, C62.259C23

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6L 12A ··· 12X order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 2 2 4 4 18 18 18 18 36 36 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + - + + - image C1 C2 C2 C2 C2 S3 Q8 D4 D6 C3⋊D4 S3×Q8 kernel C62.259C23 C4×C3⋊Dic3 C6.Dic6 C2×C32⋊4Q8 Q8×C3×C6 C6×Q8 C3⋊Dic3 C3×C12 C2×C12 C12 C6 # reps 1 1 4 1 1 4 4 2 12 16 8

Matrix representation of C62.259C23 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 3 0 0 0 0 0 8 9
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 10 0 0 0 0 0 5 4
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 11 4 0 0 0 0 2 2 0 0 0 0 0 0 1 4 0 0 0 0 6 12
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 11 4 0 0 0 0 9 2 0 0 0 0 0 0 1 0 0 0 0 0 6 12

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,3,8,0,0,0,0,0,9],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,10,5,0,0,0,0,0,4],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,11,2,0,0,0,0,4,2,0,0,0,0,0,0,1,6,0,0,0,0,4,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,1,6,0,0,0,0,0,12] >;`

C62.259C23 in GAP, Magma, Sage, TeX

`C_6^2._{259}C_2^3`
`% in TeX`

`G:=Group("C6^2.259C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,801);`
`// by ID`

`G=gap.SmallGroup(288,801);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,422,135,58,2693,9414]);`
`// Polycyclic`

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

׿
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