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G = C23×C3⋊Dic3  order 288 = 25·32

Direct product of C23 and C3⋊Dic3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C23×C3⋊Dic3, C62.283C23, (C2×C62)⋊11C4, C62⋊26(C2×C4), C32⋊8(C23×C4), C24.3(C3⋊S3), (C23×C6).14S3, C6.67(S3×C23), (C3×C6).66C24, (C22×C6)⋊7Dic3, C6⋊2(C22×Dic3), C3⋊2(C23×Dic3), (C22×C6).168D6, (C22×C62).5C2, (C2×C62).124C22, (C3×C6)⋊8(C22×C4), C2.2(C23×C3⋊S3), (C2×C6)⋊12(C2×Dic3), C23.41(C2×C3⋊S3), (C2×C6).291(C22×S3), C22.33(C22×C3⋊S3), SmallGroup(288,1016)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C23×C3⋊Dic3
Chief series
C1 — C3 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C23×C3⋊Dic3
Lower central
C32 — C23×C3⋊Dic3

Subgroups: 1524 in 708 conjugacy classes, 453 normal (7 characteristic)
C1, C2, C2 [×14], C3 [×4], C4 [×8], C22 [×35], C6 [×60], C2×C4 [×28], C23 [×15], C32, Dic3 [×32], C2×C6 [×140], C22×C4 [×14], C24, C3×C6, C3×C6 [×14], C2×Dic3 [×112], C22×C6 [×60], C23×C4, C3⋊Dic3 [×8], C62 [×35], C22×Dic3 [×56], C23×C6 [×4], C2×C3⋊Dic3 [×28], C2×C62 [×15], C23×Dic3 [×4], C22×C3⋊Dic3 [×14], C22×C62, C23×C3⋊Dic3

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3 [×4], C2×C4 [×28], C23 [×15], Dic3 [×32], D6 [×28], C22×C4 [×14], C24, C3⋊S3, C2×Dic3 [×112], C22×S3 [×28], C23×C4, C3⋊Dic3 [×8], C2×C3⋊S3 [×7], C22×Dic3 [×56], S3×C23 [×4], C2×C3⋊Dic3 [×28], C22×C3⋊S3 [×7], C23×Dic3 [×4], C22×C3⋊Dic3 [×14], C23×C3⋊S3, C23×C3⋊Dic3

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d3=e6=1, f2=e3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >

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

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

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

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

(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)

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

Matrix representation ►G ⊆ GL7(𝔽13)

12000000
0100000
00120000
0001000
0000100
00000120
00000012
,
12000000
0100000
0010000
0001000
0000100
0000010
0000001
,
1000000
01200000
00120000
0001000
0000100
00000120
00000012
,
1000000
0100000
0010000
0000100
000121200
0000001
000001212
,
1000000
0100000
0010000
000121200
0001000
00000012
0000011
,
1000000
0100000
0010000
0001000
000121200
00000119
00000112

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

96 conjugacy classes

class 1 2A···2O3A3B3C3D4A···4P6A···6BH
order12···233334···46···6
size11···122229···92···2

96 irreducible representations

dim1111222
type++++-+
imageC1C2C2C4S3Dic3D6
kernelC23×C3⋊Dic3C22×C3⋊Dic3C22×C62C2×C62C23×C6C22×C6C22×C6
# reps11411643228

In GAP, Magma, Sage, TeX 

C_2^3\times C_3\rtimes Dic_3
 
% in TeX
 
G:=Group("C2^3xC3:Dic3");
 
// GroupNames label
 
G:=SmallGroup(288,1016);
 
// by ID
 
G=gap.SmallGroup(288,1016);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,2693,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^3=e^6=1,f^2=e^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
 
// generators/relations
 

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