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## G = C23×C3⋊Dic3order 288 = 25·32

### Direct product of C23 and C3⋊Dic3

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
 Upper central C1 — C24

Generators and relations for C23×C3⋊Dic3
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 >

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

Smallest permutation representation of C23×C3⋊Dic3
Regular action on 288 points
Generators in S288
(1 101)(2 102)(3 97)(4 98)(5 99)(6 100)(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 89)(26 90)(27 85)(28 86)(29 87)(30 88)(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)(79 154)(80 155)(81 156)(82 151)(83 152)(84 153)(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 80)(2 81)(3 82)(4 83)(5 84)(6 79)(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 68)(26 69)(27 70)(28 71)(29 72)(30 67)(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)(97 151)(98 152)(99 153)(100 154)(101 155)(102 156)(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 47)(2 48)(3 43)(4 44)(5 45)(6 46)(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 35)(26 36)(27 31)(28 32)(29 33)(30 34)(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)(133 154)(134 155)(135 156)(136 151)(137 152)(138 153)(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 26)(2 24 27)(3 19 28)(4 20 29)(5 21 30)(6 22 25)(7 13 287)(8 14 288)(9 15 283)(10 16 284)(11 17 285)(12 18 286)(31 48 39)(32 43 40)(33 44 41)(34 45 42)(35 46 37)(36 47 38)(49 66 57)(50 61 58)(51 62 59)(52 63 60)(53 64 55)(54 65 56)(67 84 75)(68 79 76)(69 80 77)(70 81 78)(71 82 73)(72 83 74)(85 102 93)(86 97 94)(87 98 95)(88 99 96)(89 100 91)(90 101 92)(103 120 111)(104 115 112)(105 116 113)(106 117 114)(107 118 109)(108 119 110)(121 138 129)(122 133 130)(123 134 131)(124 135 132)(125 136 127)(126 137 128)(139 156 150)(140 151 145)(141 152 146)(142 153 147)(143 154 148)(144 155 149)(157 170 164)(158 171 165)(159 172 166)(160 173 167)(161 174 168)(162 169 163)(175 188 182)(176 189 183)(177 190 184)(178 191 185)(179 192 186)(180 187 181)(193 206 200)(194 207 201)(195 208 202)(196 209 203)(197 210 204)(198 205 199)(211 224 218)(212 225 219)(213 226 220)(214 227 221)(215 228 222)(216 223 217)(229 242 236)(230 243 237)(231 244 238)(232 245 239)(233 246 240)(234 241 235)(247 260 254)(248 261 255)(249 262 256)(250 263 257)(251 264 258)(252 259 253)(265 278 272)(266 279 273)(267 280 274)(268 281 275)(269 282 276)(270 277 271)
(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)
(1 180 4 177)(2 179 5 176)(3 178 6 175)(7 121 10 124)(8 126 11 123)(9 125 12 122)(13 129 16 132)(14 128 17 131)(15 127 18 130)(19 185 22 182)(20 184 23 181)(21 183 24 186)(25 188 28 191)(26 187 29 190)(27 192 30 189)(31 171 34 174)(32 170 35 173)(33 169 36 172)(37 167 40 164)(38 166 41 163)(39 165 42 168)(43 157 46 160)(44 162 47 159)(45 161 48 158)(49 225 52 228)(50 224 53 227)(51 223 54 226)(55 221 58 218)(56 220 59 217)(57 219 60 222)(61 211 64 214)(62 216 65 213)(63 215 66 212)(67 207 70 210)(68 206 71 209)(69 205 72 208)(73 203 76 200)(74 202 77 199)(75 201 78 204)(79 193 82 196)(80 198 83 195)(81 197 84 194)(85 261 88 264)(86 260 89 263)(87 259 90 262)(91 257 94 254)(92 256 95 253)(93 255 96 258)(97 247 100 250)(98 252 101 249)(99 251 102 248)(103 243 106 246)(104 242 107 245)(105 241 108 244)(109 239 112 236)(110 238 113 235)(111 237 114 240)(115 229 118 232)(116 234 119 231)(117 233 120 230)(133 283 136 286)(134 288 137 285)(135 287 138 284)(139 279 142 282)(140 278 143 281)(141 277 144 280)(145 272 148 275)(146 271 149 274)(147 276 150 273)(151 265 154 268)(152 270 155 267)(153 269 156 266)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2O 3A 3B 3C 3D 4A ··· 4P 6A ··· 6BH order 1 2 ··· 2 3 3 3 3 4 ··· 4 6 ··· 6 size 1 1 ··· 1 2 2 2 2 9 ··· 9 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 S3 Dic3 D6 kernel C23×C3⋊Dic3 C22×C3⋊Dic3 C22×C62 C2×C62 C23×C6 C22×C6 C22×C6 # reps 1 14 1 16 4 32 28

Matrix representation of C23×C3⋊Dic3 in GL7(𝔽13)

 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 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 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 12 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 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 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 11 9 0 0 0 0 0 11 2

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] >;

C23×C3⋊Dic3 in GAP, Magma, Sage, TeX

C_2^3\times C_3\rtimes {\rm 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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