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G = C2×C4×C3⋊Dic3order 288 = 25·32

Direct product of C2×C4 and C3⋊Dic3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C4×C3⋊Dic3
G = < a,b,c,d,e | a2=b4=c3=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 756 in 324 conjugacy classes, 189 normal (11 characteristic)
C1, C2, C2 [×6], C3 [×4], C4 [×4], C4 [×8], C22, C22 [×6], C6 [×28], C2×C4 [×6], C2×C4 [×12], C23, C32, Dic3 [×32], C12 [×16], C2×C6 [×28], C42 [×4], C22×C4, C22×C4 [×2], C3×C6, C3×C6 [×6], C2×Dic3 [×48], C2×C12 [×24], C22×C6 [×4], C2×C42, C3⋊Dic3 [×8], C3×C12 [×4], C62, C62 [×6], C4×Dic3 [×16], C22×Dic3 [×8], C22×C12 [×4], C2×C3⋊Dic3 [×12], C6×C12 [×6], C2×C62, C2×C4×Dic3 [×4], C4×C3⋊Dic3 [×4], C22×C3⋊Dic3 [×2], C2×C6×C12, C2×C4×C3⋊Dic3
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3 [×4], C2×C4 [×18], C23, Dic3 [×16], D6 [×12], C42 [×4], C22×C4 [×3], C3⋊S3, C4×S3 [×16], C2×Dic3 [×24], C22×S3 [×4], C2×C42, C3⋊Dic3 [×4], C2×C3⋊S3 [×3], C4×Dic3 [×16], S3×C2×C4 [×8], C22×Dic3 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3 [×6], C22×C3⋊S3, C2×C4×Dic3 [×4], C4×C3⋊Dic3 [×4], C2×C4×C3⋊S3 [×2], C22×C3⋊Dic3, C2×C4×C3⋊Dic3

Smallest permutation representation of C2×C4×C3⋊Dic3
Regular action on 288 points
Generators in S288
(1 127)(2 128)(3 129)(4 130)(5 131)(6 132)(7 181)(8 182)(9 183)(10 184)(11 185)(12 186)(13 99)(14 100)(15 101)(16 102)(17 97)(18 98)(19 108)(20 103)(21 104)(22 105)(23 106)(24 107)(25 224)(26 225)(27 226)(28 227)(29 228)(30 223)(31 229)(32 230)(33 231)(34 232)(35 233)(36 234)(37 162)(38 157)(39 158)(40 159)(41 160)(42 161)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 74)(56 75)(57 76)(58 77)(59 78)(60 73)(61 140)(62 141)(63 142)(64 143)(65 144)(66 139)(67 138)(68 133)(69 134)(70 135)(71 136)(72 137)(79 166)(80 167)(81 168)(82 163)(83 164)(84 165)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)(91 110)(92 111)(93 112)(94 113)(95 114)(96 109)(145 276)(146 271)(147 272)(148 273)(149 274)(150 275)(169 249)(170 250)(171 251)(172 252)(173 247)(174 248)(175 245)(176 246)(177 241)(178 242)(179 243)(180 244)(187 266)(188 267)(189 268)(190 269)(191 270)(192 265)(193 264)(194 259)(195 260)(196 261)(197 262)(198 263)(199 218)(200 219)(201 220)(202 221)(203 222)(204 217)(205 285)(206 286)(207 287)(208 288)(209 283)(210 284)(211 281)(212 282)(213 277)(214 278)(215 279)(216 280)(235 254)(236 255)(237 256)(238 257)(239 258)(240 253)
(1 94 55 41)(2 95 56 42)(3 96 57 37)(4 91 58 38)(5 92 59 39)(6 93 60 40)(7 273 253 220)(8 274 254 221)(9 275 255 222)(10 276 256 217)(11 271 257 218)(12 272 258 219)(13 66 23 67)(14 61 24 68)(15 62 19 69)(16 63 20 70)(17 64 21 71)(18 65 22 72)(25 270 33 263)(26 265 34 264)(27 266 35 259)(28 267 36 260)(29 268 31 261)(30 269 32 262)(43 88 49 82)(44 89 50 83)(45 90 51 84)(46 85 52 79)(47 86 53 80)(48 87 54 81)(73 159 132 112)(74 160 127 113)(75 161 128 114)(76 162 129 109)(77 157 130 110)(78 158 131 111)(97 143 104 136)(98 144 105 137)(99 139 106 138)(100 140 107 133)(101 141 108 134)(102 142 103 135)(115 163 121 154)(116 164 122 155)(117 165 123 156)(118 166 124 151)(119 167 125 152)(120 168 126 153)(145 237 204 184)(146 238 199 185)(147 239 200 186)(148 240 201 181)(149 235 202 182)(150 236 203 183)(169 216 177 208)(170 211 178 209)(171 212 179 210)(172 213 180 205)(173 214 175 206)(174 215 176 207)(187 233 194 226)(188 234 195 227)(189 229 196 228)(190 230 197 223)(191 231 198 224)(192 232 193 225)(241 288 249 280)(242 283 250 281)(243 284 251 282)(244 285 252 277)(245 286 247 278)(246 287 248 279)
(1 44 23)(2 45 24)(3 46 19)(4 47 20)(5 48 21)(6 43 22)(7 281 32)(8 282 33)(9 277 34)(10 278 35)(11 279 36)(12 280 31)(13 55 50)(14 56 51)(15 57 52)(16 58 53)(17 59 54)(18 60 49)(25 254 284)(26 255 285)(27 256 286)(28 257 287)(29 258 288)(30 253 283)(37 79 62)(38 80 63)(39 81 64)(40 82 65)(41 83 66)(42 84 61)(67 94 89)(68 95 90)(69 96 85)(70 91 86)(71 92 87)(72 93 88)(73 115 98)(74 116 99)(75 117 100)(76 118 101)(77 119 102)(78 120 97)(103 130 125)(104 131 126)(105 132 121)(106 127 122)(107 128 123)(108 129 124)(109 151 134)(110 152 135)(111 153 136)(112 154 137)(113 155 138)(114 156 133)(139 160 164)(140 161 165)(141 162 166)(142 157 167)(143 158 168)(144 159 163)(145 175 194)(146 176 195)(147 177 196)(148 178 197)(149 179 198)(150 180 193)(169 189 200)(170 190 201)(171 191 202)(172 192 203)(173 187 204)(174 188 199)(181 211 230)(182 212 231)(183 213 232)(184 214 233)(185 215 234)(186 216 229)(205 225 236)(206 226 237)(207 227 238)(208 228 239)(209 223 240)(210 224 235)(217 247 266)(218 248 267)(219 249 268)(220 250 269)(221 251 270)(222 252 265)(241 261 272)(242 262 273)(243 263 274)(244 264 275)(245 259 276)(246 260 271)
(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)
(1 253 4 256)(2 258 5 255)(3 257 6 254)(7 58 10 55)(8 57 11 60)(9 56 12 59)(13 281 16 278)(14 280 17 277)(15 279 18 282)(19 287 22 284)(20 286 23 283)(21 285 24 288)(25 46 28 43)(26 45 29 48)(27 44 30 47)(31 54 34 51)(32 53 35 50)(33 52 36 49)(37 271 40 274)(38 276 41 273)(39 275 42 272)(61 241 64 244)(62 246 65 243)(63 245 66 242)(67 250 70 247)(68 249 71 252)(69 248 72 251)(73 182 76 185)(74 181 77 184)(75 186 78 183)(79 260 82 263)(80 259 83 262)(81 264 84 261)(85 267 88 270)(86 266 89 269)(87 265 90 268)(91 217 94 220)(92 222 95 219)(93 221 96 218)(97 213 100 216)(98 212 101 215)(99 211 102 214)(103 206 106 209)(104 205 107 208)(105 210 108 207)(109 199 112 202)(110 204 113 201)(111 203 114 200)(115 231 118 234)(116 230 119 233)(117 229 120 232)(121 224 124 227)(122 223 125 226)(123 228 126 225)(127 240 130 237)(128 239 131 236)(129 238 132 235)(133 169 136 172)(134 174 137 171)(135 173 138 170)(139 178 142 175)(140 177 143 180)(141 176 144 179)(145 160 148 157)(146 159 149 162)(147 158 150 161)(151 188 154 191)(152 187 155 190)(153 192 156 189)(163 198 166 195)(164 197 167 194)(165 196 168 193)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 3D 4A ··· 4H 4I ··· 4X 6A ··· 6AB 12A ··· 12AF order 1 2 ··· 2 3 3 3 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 1 ··· 1 9 ··· 9 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + + - + + image C1 C2 C2 C2 C4 C4 S3 Dic3 D6 D6 C4×S3 kernel C2×C4×C3⋊Dic3 C4×C3⋊Dic3 C22×C3⋊Dic3 C2×C6×C12 C2×C3⋊Dic3 C6×C12 C22×C12 C2×C12 C2×C12 C22×C6 C2×C6 # reps 1 4 2 1 16 8 4 16 8 4 32

Matrix representation of C2×C4×C3⋊Dic3 in GL6(𝔽13)

 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 12 0 0 0 0 0 0 12
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 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 9 0 0 0 0 0 0 3
,
 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 10 0 0 0 0 0 0 4
,
 7 3 0 0 0 0 10 6 0 0 0 0 0 0 3 3 0 0 0 0 6 10 0 0 0 0 0 0 0 12 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,10,0,0,0,0,0,0,4],[7,10,0,0,0,0,3,6,0,0,0,0,0,0,3,6,0,0,0,0,3,10,0,0,0,0,0,0,0,1,0,0,0,0,12,0] >;

C2×C4×C3⋊Dic3 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("C2xC4xC3:Dic3");
// GroupNames label

G:=SmallGroup(288,779);
// by ID

G=gap.SmallGroup(288,779);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,100,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^3=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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