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G = C62.234C23  order 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), C3⋊5(C4.Dic6), C4.3(C32⋊4Q8), (C6×C12).135C22, C6.47(Q8⋊3S3), C12⋊Dic3.8C2, C6.Dic6.3C2, C6.100(D4⋊2S3), C32⋊14(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×C32⋊4Q8), (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

Derived series
C1 — C62 — C62.234C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C4×C3⋊Dic3 — C62.234C23
Lower central
C32 — C62 — C62.234C23
Upper central
C1 — C22 — C4⋊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, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C3×C6, C2×Dic3, C2×C12, C42.C2, C3⋊Dic3, C3×C12, C3×C12, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C4.Dic6, C4×C3⋊Dic3, C6.Dic6, C12⋊Dic3, C12⋊Dic3, C32×C4⋊C4, C62.234C23
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C3⋊S3, Dic6, C22×S3, C42.C2, C2×C3⋊S3, C2×Dic6, D4⋊2S3, Q8⋊3S3, C32⋊4Q8, C22×C3⋊S3, C4.Dic6, C2×C32⋊4Q8, 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 32 58 13 39 61)(2 33 59 14 40 62)(3 34 60 15 41 63)(4 35 55 16 42 64)(5 36 56 17 37 65)(6 31 57 18 38 66)(7 29 282 269 24 286)(8 30 277 270 19 287)(9 25 278 265 20 288)(10 26 279 266 21 283)(11 27 280 267 22 284)(12 28 281 268 23 285)(43 71 91 53 78 100)(44 72 92 54 73 101)(45 67 93 49 74 102)(46 68 94 50 75 97)(47 69 95 51 76 98)(48 70 96 52 77 99)(79 107 127 89 114 136)(80 108 128 90 109 137)(81 103 129 85 110 138)(82 104 130 86 111 133)(83 105 131 87 112 134)(84 106 132 88 113 135)(115 143 149 125 166 154)(116 144 150 126 167 155)(117 139 145 121 168 156)(118 140 146 122 163 151)(119 141 147 123 164 152)(120 142 148 124 165 153)(157 190 180 200 183 170)(158 191 175 201 184 171)(159 192 176 202 185 172)(160 187 177 203 186 173)(161 188 178 204 181 174)(162 189 179 199 182 169)(193 226 216 236 219 206)(194 227 211 237 220 207)(195 228 212 238 221 208)(196 223 213 239 222 209)(197 224 214 240 217 210)(198 225 215 235 218 205)(229 262 252 272 255 242)(230 263 247 273 256 243)(231 264 248 274 257 244)(232 259 249 275 258 245)(233 260 250 276 253 246)(234 261 251 271 254 241)

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

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

(1 43 4 46)(2 44 5 47)(3 45 6 48)(7 273 10 276)(8 274 11 271)(9 275 12 272)(13 53 16 50)(14 54 17 51)(15 49 18 52)(19 264 22 261)(20 259 23 262)(21 260 24 263)(25 258 28 255)(26 253 29 256)(27 254 30 257)(31 70 34 67)(32 71 35 68)(33 72 36 69)(37 76 40 73)(38 77 41 74)(39 78 42 75)(55 94 58 91)(56 95 59 92)(57 96 60 93)(61 100 64 97)(62 101 65 98)(63 102 66 99)(79 122 82 125)(80 123 83 126)(81 124 84 121)(85 120 88 117)(86 115 89 118)(87 116 90 119)(103 165 106 168)(104 166 107 163)(105 167 108 164)(109 141 112 144)(110 142 113 139)(111 143 114 140)(127 151 130 154)(128 152 131 155)(129 153 132 156)(133 149 136 146)(134 150 137 147)(135 145 138 148)(157 236 160 239)(158 237 161 240)(159 238 162 235)(169 215 172 212)(170 216 173 213)(171 211 174 214)(175 207 178 210)(176 208 179 205)(177 209 180 206)(181 224 184 227)(182 225 185 228)(183 226 186 223)(187 222 190 219)(188 217 191 220)(189 218 192 221)(193 203 196 200)(194 204 197 201)(195 199 198 202)(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), (1,32,58,13,39,61)(2,33,59,14,40,62)(3,34,60,15,41,63)(4,35,55,16,42,64)(5,36,56,17,37,65)(6,31,57,18,38,66)(7,29,282,269,24,286)(8,30,277,270,19,287)(9,25,278,265,20,288)(10,26,279,266,21,283)(11,27,280,267,22,284)(12,28,281,268,23,285)(43,71,91,53,78,100)(44,72,92,54,73,101)(45,67,93,49,74,102)(46,68,94,50,75,97)(47,69,95,51,76,98)(48,70,96,52,77,99)(79,107,127,89,114,136)(80,108,128,90,109,137)(81,103,129,85,110,138)(82,104,130,86,111,133)(83,105,131,87,112,134)(84,106,132,88,113,135)(115,143,149,125,166,154)(116,144,150,126,167,155)(117,139,145,121,168,156)(118,140,146,122,163,151)(119,141,147,123,164,152)(120,142,148,124,165,153)(157,190,180,200,183,170)(158,191,175,201,184,171)(159,192,176,202,185,172)(160,187,177,203,186,173)(161,188,178,204,181,174)(162,189,179,199,182,169)(193,226,216,236,219,206)(194,227,211,237,220,207)(195,228,212,238,221,208)(196,223,213,239,222,209)(197,224,214,240,217,210)(198,225,215,235,218,205)(229,262,252,272,255,242)(230,263,247,273,256,243)(231,264,248,274,257,244)(232,259,249,275,258,245)(233,260,250,276,253,246)(234,261,251,271,254,241), (1,161,4,158)(2,160,5,157)(3,159,6,162)(7,125,10,122)(8,124,11,121)(9,123,12,126)(13,204,16,201)(14,203,17,200)(15,202,18,199)(19,153,22,156)(20,152,23,155)(21,151,24,154)(25,147,28,150)(26,146,29,149)(27,145,30,148)(31,169,34,172)(32,174,35,171)(33,173,36,170)(37,180,40,177)(38,179,41,176)(39,178,42,175)(43,194,46,197)(44,193,47,196)(45,198,48,195)(49,235,52,238)(50,240,53,237)(51,239,54,236)(55,184,58,181)(56,183,59,186)(57,182,60,185)(61,188,64,191)(62,187,65,190)(63,192,66,189)(67,205,70,208)(68,210,71,207)(69,209,72,206)(73,216,76,213)(74,215,77,212)(75,214,78,211)(79,230,82,233)(80,229,83,232)(81,234,84,231)(85,271,88,274)(86,276,89,273)(87,275,90,272)(91,220,94,217)(92,219,95,222)(93,218,96,221)(97,224,100,227)(98,223,101,226)(99,228,102,225)(103,241,106,244)(104,246,107,243)(105,245,108,242)(109,252,112,249)(110,251,113,248)(111,250,114,247)(115,266,118,269)(116,265,119,268)(117,270,120,267)(127,256,130,253)(128,255,131,258)(129,254,132,257)(133,260,136,263)(134,259,137,262)(135,264,138,261)(139,277,142,280)(140,282,143,279)(141,281,144,278)(163,286,166,283)(164,285,167,288)(165,284,168,287), (1,86,13,82)(2,87,14,83)(3,88,15,84)(4,89,16,79)(5,90,17,80)(6,85,18,81)(7,240,269,197)(8,235,270,198)(9,236,265,193)(10,237,266,194)(11,238,267,195)(12,239,268,196)(19,225,30,218)(20,226,25,219)(21,227,26,220)(22,228,27,221)(23,223,28,222)(24,224,29,217)(31,110,38,103)(32,111,39,104)(33,112,40,105)(34,113,41,106)(35,114,42,107)(36,109,37,108)(43,125,53,115)(44,126,54,116)(45,121,49,117)(46,122,50,118)(47,123,51,119)(48,124,52,120)(55,136,64,127)(56,137,65,128)(57,138,66,129)(58,133,61,130)(59,134,62,131)(60,135,63,132)(67,168,74,139)(68,163,75,140)(69,164,76,141)(70,165,77,142)(71,166,78,143)(72,167,73,144)(91,154,100,149)(92,155,101,150)(93,156,102,145)(94,151,97,146)(95,152,98,147)(96,153,99,148)(157,272,200,229)(158,273,201,230)(159,274,202,231)(160,275,203,232)(161,276,204,233)(162,271,199,234)(169,251,179,241)(170,252,180,242)(171,247,175,243)(172,248,176,244)(173,249,177,245)(174,250,178,246)(181,260,188,253)(182,261,189,254)(183,262,190,255)(184,263,191,256)(185,264,192,257)(186,259,187,258)(205,287,215,277)(206,288,216,278)(207,283,211,279)(208,284,212,280)(209,285,213,281)(210,286,214,282), (1,43,4,46)(2,44,5,47)(3,45,6,48)(7,273,10,276)(8,274,11,271)(9,275,12,272)(13,53,16,50)(14,54,17,51)(15,49,18,52)(19,264,22,261)(20,259,23,262)(21,260,24,263)(25,258,28,255)(26,253,29,256)(27,254,30,257)(31,70,34,67)(32,71,35,68)(33,72,36,69)(37,76,40,73)(38,77,41,74)(39,78,42,75)(55,94,58,91)(56,95,59,92)(57,96,60,93)(61,100,64,97)(62,101,65,98)(63,102,66,99)(79,122,82,125)(80,123,83,126)(81,124,84,121)(85,120,88,117)(86,115,89,118)(87,116,90,119)(103,165,106,168)(104,166,107,163)(105,167,108,164)(109,141,112,144)(110,142,113,139)(111,143,114,140)(127,151,130,154)(128,152,131,155)(129,153,132,156)(133,149,136,146)(134,150,137,147)(135,145,138,148)(157,236,160,239)(158,237,161,240)(159,238,162,235)(169,215,172,212)(170,216,173,213)(171,211,174,214)(175,207,178,210)(176,208,179,205)(177,209,180,206)(181,224,184,227)(182,225,185,228)(183,226,186,223)(187,222,190,219)(188,217,191,220)(189,218,192,221)(193,203,196,200)(194,204,197,201)(195,199,198,202)(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), (1,32,58,13,39,61)(2,33,59,14,40,62)(3,34,60,15,41,63)(4,35,55,16,42,64)(5,36,56,17,37,65)(6,31,57,18,38,66)(7,29,282,269,24,286)(8,30,277,270,19,287)(9,25,278,265,20,288)(10,26,279,266,21,283)(11,27,280,267,22,284)(12,28,281,268,23,285)(43,71,91,53,78,100)(44,72,92,54,73,101)(45,67,93,49,74,102)(46,68,94,50,75,97)(47,69,95,51,76,98)(48,70,96,52,77,99)(79,107,127,89,114,136)(80,108,128,90,109,137)(81,103,129,85,110,138)(82,104,130,86,111,133)(83,105,131,87,112,134)(84,106,132,88,113,135)(115,143,149,125,166,154)(116,144,150,126,167,155)(117,139,145,121,168,156)(118,140,146,122,163,151)(119,141,147,123,164,152)(120,142,148,124,165,153)(157,190,180,200,183,170)(158,191,175,201,184,171)(159,192,176,202,185,172)(160,187,177,203,186,173)(161,188,178,204,181,174)(162,189,179,199,182,169)(193,226,216,236,219,206)(194,227,211,237,220,207)(195,228,212,238,221,208)(196,223,213,239,222,209)(197,224,214,240,217,210)(198,225,215,235,218,205)(229,262,252,272,255,242)(230,263,247,273,256,243)(231,264,248,274,257,244)(232,259,249,275,258,245)(233,260,250,276,253,246)(234,261,251,271,254,241), (1,161,4,158)(2,160,5,157)(3,159,6,162)(7,125,10,122)(8,124,11,121)(9,123,12,126)(13,204,16,201)(14,203,17,200)(15,202,18,199)(19,153,22,156)(20,152,23,155)(21,151,24,154)(25,147,28,150)(26,146,29,149)(27,145,30,148)(31,169,34,172)(32,174,35,171)(33,173,36,170)(37,180,40,177)(38,179,41,176)(39,178,42,175)(43,194,46,197)(44,193,47,196)(45,198,48,195)(49,235,52,238)(50,240,53,237)(51,239,54,236)(55,184,58,181)(56,183,59,186)(57,182,60,185)(61,188,64,191)(62,187,65,190)(63,192,66,189)(67,205,70,208)(68,210,71,207)(69,209,72,206)(73,216,76,213)(74,215,77,212)(75,214,78,211)(79,230,82,233)(80,229,83,232)(81,234,84,231)(85,271,88,274)(86,276,89,273)(87,275,90,272)(91,220,94,217)(92,219,95,222)(93,218,96,221)(97,224,100,227)(98,223,101,226)(99,228,102,225)(103,241,106,244)(104,246,107,243)(105,245,108,242)(109,252,112,249)(110,251,113,248)(111,250,114,247)(115,266,118,269)(116,265,119,268)(117,270,120,267)(127,256,130,253)(128,255,131,258)(129,254,132,257)(133,260,136,263)(134,259,137,262)(135,264,138,261)(139,277,142,280)(140,282,143,279)(141,281,144,278)(163,286,166,283)(164,285,167,288)(165,284,168,287), (1,86,13,82)(2,87,14,83)(3,88,15,84)(4,89,16,79)(5,90,17,80)(6,85,18,81)(7,240,269,197)(8,235,270,198)(9,236,265,193)(10,237,266,194)(11,238,267,195)(12,239,268,196)(19,225,30,218)(20,226,25,219)(21,227,26,220)(22,228,27,221)(23,223,28,222)(24,224,29,217)(31,110,38,103)(32,111,39,104)(33,112,40,105)(34,113,41,106)(35,114,42,107)(36,109,37,108)(43,125,53,115)(44,126,54,116)(45,121,49,117)(46,122,50,118)(47,123,51,119)(48,124,52,120)(55,136,64,127)(56,137,65,128)(57,138,66,129)(58,133,61,130)(59,134,62,131)(60,135,63,132)(67,168,74,139)(68,163,75,140)(69,164,76,141)(70,165,77,142)(71,166,78,143)(72,167,73,144)(91,154,100,149)(92,155,101,150)(93,156,102,145)(94,151,97,146)(95,152,98,147)(96,153,99,148)(157,272,200,229)(158,273,201,230)(159,274,202,231)(160,275,203,232)(161,276,204,233)(162,271,199,234)(169,251,179,241)(170,252,180,242)(171,247,175,243)(172,248,176,244)(173,249,177,245)(174,250,178,246)(181,260,188,253)(182,261,189,254)(183,262,190,255)(184,263,191,256)(185,264,192,257)(186,259,187,258)(205,287,215,277)(206,288,216,278)(207,283,211,279)(208,284,212,280)(209,285,213,281)(210,286,214,282), (1,43,4,46)(2,44,5,47)(3,45,6,48)(7,273,10,276)(8,274,11,271)(9,275,12,272)(13,53,16,50)(14,54,17,51)(15,49,18,52)(19,264,22,261)(20,259,23,262)(21,260,24,263)(25,258,28,255)(26,253,29,256)(27,254,30,257)(31,70,34,67)(32,71,35,68)(33,72,36,69)(37,76,40,73)(38,77,41,74)(39,78,42,75)(55,94,58,91)(56,95,59,92)(57,96,60,93)(61,100,64,97)(62,101,65,98)(63,102,66,99)(79,122,82,125)(80,123,83,126)(81,124,84,121)(85,120,88,117)(86,115,89,118)(87,116,90,119)(103,165,106,168)(104,166,107,163)(105,167,108,164)(109,141,112,144)(110,142,113,139)(111,143,114,140)(127,151,130,154)(128,152,131,155)(129,153,132,156)(133,149,136,146)(134,150,137,147)(135,145,138,148)(157,236,160,239)(158,237,161,240)(159,238,162,235)(169,215,172,212)(170,216,173,213)(171,211,174,214)(175,207,178,210)(176,208,179,205)(177,209,180,206)(181,224,184,227)(182,225,185,228)(183,226,186,223)(187,222,190,219)(188,217,191,220)(189,218,192,221)(193,203,196,200)(194,204,197,201)(195,199,198,202)(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)], [(1,32,58,13,39,61),(2,33,59,14,40,62),(3,34,60,15,41,63),(4,35,55,16,42,64),(5,36,56,17,37,65),(6,31,57,18,38,66),(7,29,282,269,24,286),(8,30,277,270,19,287),(9,25,278,265,20,288),(10,26,279,266,21,283),(11,27,280,267,22,284),(12,28,281,268,23,285),(43,71,91,53,78,100),(44,72,92,54,73,101),(45,67,93,49,74,102),(46,68,94,50,75,97),(47,69,95,51,76,98),(48,70,96,52,77,99),(79,107,127,89,114,136),(80,108,128,90,109,137),(81,103,129,85,110,138),(82,104,130,86,111,133),(83,105,131,87,112,134),(84,106,132,88,113,135),(115,143,149,125,166,154),(116,144,150,126,167,155),(117,139,145,121,168,156),(118,140,146,122,163,151),(119,141,147,123,164,152),(120,142,148,124,165,153),(157,190,180,200,183,170),(158,191,175,201,184,171),(159,192,176,202,185,172),(160,187,177,203,186,173),(161,188,178,204,181,174),(162,189,179,199,182,169),(193,226,216,236,219,206),(194,227,211,237,220,207),(195,228,212,238,221,208),(196,223,213,239,222,209),(197,224,214,240,217,210),(198,225,215,235,218,205),(229,262,252,272,255,242),(230,263,247,273,256,243),(231,264,248,274,257,244),(232,259,249,275,258,245),(233,260,250,276,253,246),(234,261,251,271,254,241)], [(1,161,4,158),(2,160,5,157),(3,159,6,162),(7,125,10,122),(8,124,11,121),(9,123,12,126),(13,204,16,201),(14,203,17,200),(15,202,18,199),(19,153,22,156),(20,152,23,155),(21,151,24,154),(25,147,28,150),(26,146,29,149),(27,145,30,148),(31,169,34,172),(32,174,35,171),(33,173,36,170),(37,180,40,177),(38,179,41,176),(39,178,42,175),(43,194,46,197),(44,193,47,196),(45,198,48,195),(49,235,52,238),(50,240,53,237),(51,239,54,236),(55,184,58,181),(56,183,59,186),(57,182,60,185),(61,188,64,191),(62,187,65,190),(63,192,66,189),(67,205,70,208),(68,210,71,207),(69,209,72,206),(73,216,76,213),(74,215,77,212),(75,214,78,211),(79,230,82,233),(80,229,83,232),(81,234,84,231),(85,271,88,274),(86,276,89,273),(87,275,90,272),(91,220,94,217),(92,219,95,222),(93,218,96,221),(97,224,100,227),(98,223,101,226),(99,228,102,225),(103,241,106,244),(104,246,107,243),(105,245,108,242),(109,252,112,249),(110,251,113,248),(111,250,114,247),(115,266,118,269),(116,265,119,268),(117,270,120,267),(127,256,130,253),(128,255,131,258),(129,254,132,257),(133,260,136,263),(134,259,137,262),(135,264,138,261),(139,277,142,280),(140,282,143,279),(141,281,144,278),(163,286,166,283),(164,285,167,288),(165,284,168,287)], [(1,86,13,82),(2,87,14,83),(3,88,15,84),(4,89,16,79),(5,90,17,80),(6,85,18,81),(7,240,269,197),(8,235,270,198),(9,236,265,193),(10,237,266,194),(11,238,267,195),(12,239,268,196),(19,225,30,218),(20,226,25,219),(21,227,26,220),(22,228,27,221),(23,223,28,222),(24,224,29,217),(31,110,38,103),(32,111,39,104),(33,112,40,105),(34,113,41,106),(35,114,42,107),(36,109,37,108),(43,125,53,115),(44,126,54,116),(45,121,49,117),(46,122,50,118),(47,123,51,119),(48,124,52,120),(55,136,64,127),(56,137,65,128),(57,138,66,129),(58,133,61,130),(59,134,62,131),(60,135,63,132),(67,168,74,139),(68,163,75,140),(69,164,76,141),(70,165,77,142),(71,166,78,143),(72,167,73,144),(91,154,100,149),(92,155,101,150),(93,156,102,145),(94,151,97,146),(95,152,98,147),(96,153,99,148),(157,272,200,229),(158,273,201,230),(159,274,202,231),(160,275,203,232),(161,276,204,233),(162,271,199,234),(169,251,179,241),(170,252,180,242),(171,247,175,243),(172,248,176,244),(173,249,177,245),(174,250,178,246),(181,260,188,253),(182,261,189,254),(183,262,190,255),(184,263,191,256),(185,264,192,257),(186,259,187,258),(205,287,215,277),(206,288,216,278),(207,283,211,279),(208,284,212,280),(209,285,213,281),(210,286,214,282)], [(1,43,4,46),(2,44,5,47),(3,45,6,48),(7,273,10,276),(8,274,11,271),(9,275,12,272),(13,53,16,50),(14,54,17,51),(15,49,18,52),(19,264,22,261),(20,259,23,262),(21,260,24,263),(25,258,28,255),(26,253,29,256),(27,254,30,257),(31,70,34,67),(32,71,35,68),(33,72,36,69),(37,76,40,73),(38,77,41,74),(39,78,42,75),(55,94,58,91),(56,95,59,92),(57,96,60,93),(61,100,64,97),(62,101,65,98),(63,102,66,99),(79,122,82,125),(80,123,83,126),(81,124,84,121),(85,120,88,117),(86,115,89,118),(87,116,90,119),(103,165,106,168),(104,166,107,163),(105,167,108,164),(109,141,112,144),(110,142,113,139),(111,143,114,140),(127,151,130,154),(128,152,131,155),(129,153,132,156),(133,149,136,146),(134,150,137,147),(135,145,138,148),(157,236,160,239),(158,237,161,240),(159,238,162,235),(169,215,172,212),(170,216,173,213),(171,211,174,214),(175,207,178,210),(176,208,179,205),(177,209,180,206),(181,224,184,227),(182,225,185,228),(183,226,186,223),(187,222,190,219),(188,217,191,220),(189,218,192,221),(193,203,196,200),(194,204,197,201),(195,199,198,202),(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○D4Dic6D4⋊2S3Q8⋊3S3
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
 

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