C2^3xC6^2 - GroupNames

G = C₂³×C₆² order 288 = 2⁵·3²

Abelian group of type [2,2,2,6,6]

direct product, abelian, monomial

Aliases: C₂³×C₆², SmallGroup(288,1045)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³×C₆²

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₆² — C₂×C₆² — C₂²×C₆² — C₂³×C₆²

Lower central

C₁ — C₂³×C₆²

Upper central

C₁ — C₂³×C₆²

Generators and relations for C₂³×C₆²
G = < a,b,c,d,e | a²=b²=c²=d⁶=e⁶=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 2244, all normal (4 characteristic)
C₁, C₂, C₃, C₂², C₆, C₂³, C₃², C₂×C₆, C₂⁴, C₃×C₆, C₂²×C₆, C₂⁵, C₆², C₂³×C₆, C₂×C₆², C₂⁴×C₆, C₂²×C₆², C₂³×C₆²
Quotients: C₁, C₂, C₃, C₂², C₆, C₂³, C₃², C₂×C₆, C₂⁴, C₃×C₆, C₂²×C₆, C₂⁵, C₆², C₂³×C₆, C₂×C₆², C₂⁴×C₆, C₂²×C₆², C₂³×C₆²

Smallest permutation representation of C₂³×C₆²
►Regular action on 288 points

Generators in S₂₈₈

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

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

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

(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 18 39 61)(2 33 59 13 40 62)(3 34 60 14 41 63)(4 35 55 15 42 64)(5 36 56 16 37 65)(6 31 57 17 38 66)(7 22 261 283 29 265)(8 23 262 284 30 266)(9 24 263 285 25 267)(10 19 264 286 26 268)(11 20 259 287 27 269)(12 21 260 288 28 270)(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)(115 143 163 125 150 172)(116 144 164 126 145 173)(117 139 165 121 146 174)(118 140 166 122 147 169)(119 141 167 123 148 170)(120 142 168 124 149 171)(151 179 199 161 186 208)(152 180 200 162 181 209)(153 175 201 157 182 210)(154 176 202 158 183 205)(155 177 203 159 184 206)(156 178 204 160 185 207)(187 215 235 197 222 244)(188 216 236 198 217 245)(189 211 237 193 218 246)(190 212 238 194 219 241)(191 213 239 195 220 242)(192 214 240 196 221 243)(223 251 271 233 258 280)(224 252 272 234 253 281)(225 247 273 229 254 282)(226 248 274 230 255 277)(227 249 275 231 256 278)(228 250 276 232 257 279)

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

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

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

288 conjugacy classes

class	1	2A	···	2AE	3A	···	3H	6A	···	6IN
order	1	2	···	2	3	···	3	6	···	6
size	1	1	···	1	1	···	1	1	···	1

288 irreducible representations

dim	1	1	1	1
type	+	+
image	C₁	C₂	C₃	C₆
kernel	C₂³×C₆²	C₂²×C₆²	C₂⁴×C₆	C₂³×C₆
# reps	1	31	8	248

Matrix representation of C₂³×C₆² ►in GL₅(𝔽₇)

6	0	0	0	0
0	6	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	6

6	0	0	0	0
0	6	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

6	0	0	0	0
0	1	0	0	0
0	0	6	0	0
0	0	0	1	0
0	0	0	0	6

5	0	0	0	0
0	1	0	0	0
0	0	4	0	0
0	0	0	6	0
0	0	0	0	6

5	0	0	0	0
0	1	0	0	0
0	0	3	0	0
0	0	0	3	0
0	0	0	0	2

G:=sub<GL(5,GF(7))| [6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6],[6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[6,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,6],[5,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,6,0,0,0,0,0,6],[5,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,2] >;

C₂³×C₆² in GAP, Magma, Sage, TeX

C_2^3\times C_6^2

% in TeX

G:=Group("C2^3xC6^2");

// GroupNames label

G:=SmallGroup(288,1045);

// by ID

G=gap.SmallGroup(288,1045);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^6=e^6=1,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,c*e=e*c,d*e=e*d>;

// generators/relations

G = C23×C62 order 288 = 25·32

Abelian group of type [2,2,2,6,6]

G = C₂³×C₆² order 288 = 2⁵·3²