direct product, metabelian, nilpotent (class 2), monomial
Aliases: C32×C42.C2, C122.16C2, C62.294C23, C6.27(C6×Q8), (C4×C12).24C6, C42.4(C3×C6), (C3×C12).31Q8, C12.15(C3×Q8), C4.3(Q8×C32), (C2×C4).12C62, (C6×C12).272C22, C22.15(C2×C62), C2.4(Q8×C3×C6), C4⋊C4.4(C3×C6), (C3×C4⋊C4).21C6, C6.55(C3×C4○D4), (C3×C6).81(C2×Q8), (C2×C12).75(C2×C6), C2.8(C32×C4○D4), (C32×C4⋊C4).18C2, (C3×C6).172(C4○D4), (C2×C6).100(C22×C6), SmallGroup(288,822)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C42.C2
G = < a,b,c,d,e | a3=b3=c4=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd2, ede-1=c2d >
Subgroups: 204 in 168 conjugacy classes, 132 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C32, C12, C12, C2×C6, C42, C4⋊C4, C3×C6, C3×C6, C2×C12, C42.C2, C3×C12, C3×C12, C62, C4×C12, C3×C4⋊C4, C6×C12, C6×C12, C3×C42.C2, C122, C32×C4⋊C4, C32×C42.C2
Quotients: C1, C2, C3, C22, C6, Q8, C23, C32, C2×C6, C2×Q8, C4○D4, C3×C6, C3×Q8, C22×C6, C42.C2, C62, C6×Q8, C3×C4○D4, Q8×C32, C2×C62, C3×C42.C2, Q8×C3×C6, C32×C4○D4, C32×C42.C2
►Regular action on 288 points
Generators in S288
(1 93 85)(2 94 86)(3 95 87)(4 96 88)(5 104 32)(6 101 29)(7 102 30)(8 103 31)(9 89 17)(10 90 18)(11 91 19)(12 92 20)(13 81 21)(14 82 22)(15 83 23)(16 84 24)(25 79 33)(26 80 34)(27 77 35)(28 78 36)(37 105 97)(38 106 98)(39 107 99)(40 108 100)(41 121 49)(42 122 50)(43 123 51)(44 124 52)(45 113 53)(46 114 54)(47 115 55)(48 116 56)(57 147 65)(58 148 66)(59 145 67)(60 146 68)(61 141 133)(62 142 134)(63 143 135)(64 144 136)(69 137 129)(70 138 130)(71 139 131)(72 140 132)(73 209 201)(74 210 202)(75 211 203)(76 212 204)(109 125 117)(110 126 118)(111 127 119)(112 128 120)(149 231 157)(150 232 158)(151 229 159)(152 230 160)(153 223 161)(154 224 162)(155 221 163)(156 222 164)(165 181 173)(166 182 174)(167 183 175)(168 184 176)(169 251 243)(170 252 244)(171 249 241)(172 250 242)(177 247 239)(178 248 240)(179 245 237)(180 246 238)(185 267 193)(186 268 194)(187 265 195)(188 266 196)(189 259 197)(190 260 198)(191 257 199)(192 258 200)(205 287 279)(206 288 280)(207 285 277)(208 286 278)(213 283 275)(214 284 276)(215 281 273)(216 282 274)(217 233 225)(218 234 226)(219 235 227)(220 236 228)(253 269 261)(254 270 262)(255 271 263)(256 272 264)
(1 81 9)(2 82 10)(3 83 11)(4 84 12)(5 40 78)(6 37 79)(7 38 80)(8 39 77)(13 17 85)(14 18 86)(15 19 87)(16 20 88)(21 89 93)(22 90 94)(23 91 95)(24 92 96)(25 29 97)(26 30 98)(27 31 99)(28 32 100)(33 101 105)(34 102 106)(35 103 107)(36 104 108)(41 109 113)(42 110 114)(43 111 115)(44 112 116)(45 49 117)(46 50 118)(47 51 119)(48 52 120)(53 121 125)(54 122 126)(55 123 127)(56 124 128)(57 61 129)(58 62 130)(59 63 131)(60 64 132)(65 133 137)(66 134 138)(67 135 139)(68 136 140)(69 147 141)(70 148 142)(71 145 143)(72 146 144)(73 287 213)(74 288 214)(75 285 215)(76 286 216)(149 219 223)(150 220 224)(151 217 221)(152 218 222)(153 157 227)(154 158 228)(155 159 225)(156 160 226)(161 231 235)(162 232 236)(163 229 233)(164 230 234)(165 169 239)(166 170 240)(167 171 237)(168 172 238)(173 243 247)(174 244 248)(175 241 245)(176 242 246)(177 181 251)(178 182 252)(179 183 249)(180 184 250)(185 255 259)(186 256 260)(187 253 257)(188 254 258)(189 193 263)(190 194 264)(191 195 261)(192 196 262)(197 267 271)(198 268 272)(199 265 269)(200 266 270)(201 205 275)(202 206 276)(203 207 273)(204 208 274)(209 279 283)(210 280 284)(211 277 281)(212 278 282)
(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 59 111 25)(2 60 112 26)(3 57 109 27)(4 58 110 28)(5 24 142 54)(6 21 143 55)(7 22 144 56)(8 23 141 53)(9 131 43 97)(10 132 44 98)(11 129 41 99)(12 130 42 100)(13 135 47 101)(14 136 48 102)(15 133 45 103)(16 134 46 104)(17 139 51 105)(18 140 52 106)(19 137 49 107)(20 138 50 108)(29 81 63 115)(30 82 64 116)(31 83 61 113)(32 84 62 114)(33 85 67 119)(34 86 68 120)(35 87 65 117)(36 88 66 118)(37 89 71 123)(38 90 72 124)(39 91 69 121)(40 92 70 122)(73 269 184 236)(74 270 181 233)(75 271 182 234)(76 272 183 235)(77 95 147 125)(78 96 148 126)(79 93 145 127)(80 94 146 128)(149 274 186 237)(150 275 187 238)(151 276 188 239)(152 273 185 240)(153 278 190 241)(154 279 191 242)(155 280 192 243)(156 277 189 244)(157 282 194 245)(158 283 195 246)(159 284 196 247)(160 281 193 248)(161 286 198 249)(162 287 199 250)(163 288 200 251)(164 285 197 252)(165 217 202 254)(166 218 203 255)(167 219 204 256)(168 220 201 253)(169 221 206 258)(170 222 207 259)(171 223 208 260)(172 224 205 257)(173 225 210 262)(174 226 211 263)(175 227 212 264)(176 228 209 261)(177 229 214 266)(178 230 215 267)(179 231 216 268)(180 232 213 265)
(1 218 111 255)(2 256 112 219)(3 220 109 253)(4 254 110 217)(5 286 142 249)(6 250 143 287)(7 288 144 251)(8 252 141 285)(9 152 43 185)(10 186 44 149)(11 150 41 187)(12 188 42 151)(13 156 47 189)(14 190 48 153)(15 154 45 191)(16 192 46 155)(17 160 51 193)(18 194 52 157)(19 158 49 195)(20 196 50 159)(21 164 55 197)(22 198 56 161)(23 162 53 199)(24 200 54 163)(25 168 59 201)(26 202 60 165)(27 166 57 203)(28 204 58 167)(29 172 63 205)(30 206 64 169)(31 170 61 207)(32 208 62 171)(33 176 67 209)(34 210 68 173)(35 174 65 211)(36 212 66 175)(37 180 71 213)(38 214 72 177)(39 178 69 215)(40 216 70 179)(73 79 184 145)(74 146 181 80)(75 77 182 147)(76 148 183 78)(81 222 115 259)(82 260 116 223)(83 224 113 257)(84 258 114 221)(85 226 119 263)(86 264 120 227)(87 228 117 261)(88 262 118 225)(89 230 123 267)(90 268 124 231)(91 232 121 265)(92 266 122 229)(93 234 127 271)(94 272 128 235)(95 236 125 269)(96 270 126 233)(97 238 131 275)(98 276 132 239)(99 240 129 273)(100 274 130 237)(101 242 135 279)(102 280 136 243)(103 244 133 277)(104 278 134 241)(105 246 139 283)(106 284 140 247)(107 248 137 281)(108 282 138 245)
G:=sub<Sym(288)| (1,93,85)(2,94,86)(3,95,87)(4,96,88)(5,104,32)(6,101,29)(7,102,30)(8,103,31)(9,89,17)(10,90,18)(11,91,19)(12,92,20)(13,81,21)(14,82,22)(15,83,23)(16,84,24)(25,79,33)(26,80,34)(27,77,35)(28,78,36)(37,105,97)(38,106,98)(39,107,99)(40,108,100)(41,121,49)(42,122,50)(43,123,51)(44,124,52)(45,113,53)(46,114,54)(47,115,55)(48,116,56)(57,147,65)(58,148,66)(59,145,67)(60,146,68)(61,141,133)(62,142,134)(63,143,135)(64,144,136)(69,137,129)(70,138,130)(71,139,131)(72,140,132)(73,209,201)(74,210,202)(75,211,203)(76,212,204)(109,125,117)(110,126,118)(111,127,119)(112,128,120)(149,231,157)(150,232,158)(151,229,159)(152,230,160)(153,223,161)(154,224,162)(155,221,163)(156,222,164)(165,181,173)(166,182,174)(167,183,175)(168,184,176)(169,251,243)(170,252,244)(171,249,241)(172,250,242)(177,247,239)(178,248,240)(179,245,237)(180,246,238)(185,267,193)(186,268,194)(187,265,195)(188,266,196)(189,259,197)(190,260,198)(191,257,199)(192,258,200)(205,287,279)(206,288,280)(207,285,277)(208,286,278)(213,283,275)(214,284,276)(215,281,273)(216,282,274)(217,233,225)(218,234,226)(219,235,227)(220,236,228)(253,269,261)(254,270,262)(255,271,263)(256,272,264), (1,81,9)(2,82,10)(3,83,11)(4,84,12)(5,40,78)(6,37,79)(7,38,80)(8,39,77)(13,17,85)(14,18,86)(15,19,87)(16,20,88)(21,89,93)(22,90,94)(23,91,95)(24,92,96)(25,29,97)(26,30,98)(27,31,99)(28,32,100)(33,101,105)(34,102,106)(35,103,107)(36,104,108)(41,109,113)(42,110,114)(43,111,115)(44,112,116)(45,49,117)(46,50,118)(47,51,119)(48,52,120)(53,121,125)(54,122,126)(55,123,127)(56,124,128)(57,61,129)(58,62,130)(59,63,131)(60,64,132)(65,133,137)(66,134,138)(67,135,139)(68,136,140)(69,147,141)(70,148,142)(71,145,143)(72,146,144)(73,287,213)(74,288,214)(75,285,215)(76,286,216)(149,219,223)(150,220,224)(151,217,221)(152,218,222)(153,157,227)(154,158,228)(155,159,225)(156,160,226)(161,231,235)(162,232,236)(163,229,233)(164,230,234)(165,169,239)(166,170,240)(167,171,237)(168,172,238)(173,243,247)(174,244,248)(175,241,245)(176,242,246)(177,181,251)(178,182,252)(179,183,249)(180,184,250)(185,255,259)(186,256,260)(187,253,257)(188,254,258)(189,193,263)(190,194,264)(191,195,261)(192,196,262)(197,267,271)(198,268,272)(199,265,269)(200,266,270)(201,205,275)(202,206,276)(203,207,273)(204,208,274)(209,279,283)(210,280,284)(211,277,281)(212,278,282), (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), (1,59,111,25)(2,60,112,26)(3,57,109,27)(4,58,110,28)(5,24,142,54)(6,21,143,55)(7,22,144,56)(8,23,141,53)(9,131,43,97)(10,132,44,98)(11,129,41,99)(12,130,42,100)(13,135,47,101)(14,136,48,102)(15,133,45,103)(16,134,46,104)(17,139,51,105)(18,140,52,106)(19,137,49,107)(20,138,50,108)(29,81,63,115)(30,82,64,116)(31,83,61,113)(32,84,62,114)(33,85,67,119)(34,86,68,120)(35,87,65,117)(36,88,66,118)(37,89,71,123)(38,90,72,124)(39,91,69,121)(40,92,70,122)(73,269,184,236)(74,270,181,233)(75,271,182,234)(76,272,183,235)(77,95,147,125)(78,96,148,126)(79,93,145,127)(80,94,146,128)(149,274,186,237)(150,275,187,238)(151,276,188,239)(152,273,185,240)(153,278,190,241)(154,279,191,242)(155,280,192,243)(156,277,189,244)(157,282,194,245)(158,283,195,246)(159,284,196,247)(160,281,193,248)(161,286,198,249)(162,287,199,250)(163,288,200,251)(164,285,197,252)(165,217,202,254)(166,218,203,255)(167,219,204,256)(168,220,201,253)(169,221,206,258)(170,222,207,259)(171,223,208,260)(172,224,205,257)(173,225,210,262)(174,226,211,263)(175,227,212,264)(176,228,209,261)(177,229,214,266)(178,230,215,267)(179,231,216,268)(180,232,213,265), (1,218,111,255)(2,256,112,219)(3,220,109,253)(4,254,110,217)(5,286,142,249)(6,250,143,287)(7,288,144,251)(8,252,141,285)(9,152,43,185)(10,186,44,149)(11,150,41,187)(12,188,42,151)(13,156,47,189)(14,190,48,153)(15,154,45,191)(16,192,46,155)(17,160,51,193)(18,194,52,157)(19,158,49,195)(20,196,50,159)(21,164,55,197)(22,198,56,161)(23,162,53,199)(24,200,54,163)(25,168,59,201)(26,202,60,165)(27,166,57,203)(28,204,58,167)(29,172,63,205)(30,206,64,169)(31,170,61,207)(32,208,62,171)(33,176,67,209)(34,210,68,173)(35,174,65,211)(36,212,66,175)(37,180,71,213)(38,214,72,177)(39,178,69,215)(40,216,70,179)(73,79,184,145)(74,146,181,80)(75,77,182,147)(76,148,183,78)(81,222,115,259)(82,260,116,223)(83,224,113,257)(84,258,114,221)(85,226,119,263)(86,264,120,227)(87,228,117,261)(88,262,118,225)(89,230,123,267)(90,268,124,231)(91,232,121,265)(92,266,122,229)(93,234,127,271)(94,272,128,235)(95,236,125,269)(96,270,126,233)(97,238,131,275)(98,276,132,239)(99,240,129,273)(100,274,130,237)(101,242,135,279)(102,280,136,243)(103,244,133,277)(104,278,134,241)(105,246,139,283)(106,284,140,247)(107,248,137,281)(108,282,138,245)>;
G:=Group( (1,93,85)(2,94,86)(3,95,87)(4,96,88)(5,104,32)(6,101,29)(7,102,30)(8,103,31)(9,89,17)(10,90,18)(11,91,19)(12,92,20)(13,81,21)(14,82,22)(15,83,23)(16,84,24)(25,79,33)(26,80,34)(27,77,35)(28,78,36)(37,105,97)(38,106,98)(39,107,99)(40,108,100)(41,121,49)(42,122,50)(43,123,51)(44,124,52)(45,113,53)(46,114,54)(47,115,55)(48,116,56)(57,147,65)(58,148,66)(59,145,67)(60,146,68)(61,141,133)(62,142,134)(63,143,135)(64,144,136)(69,137,129)(70,138,130)(71,139,131)(72,140,132)(73,209,201)(74,210,202)(75,211,203)(76,212,204)(109,125,117)(110,126,118)(111,127,119)(112,128,120)(149,231,157)(150,232,158)(151,229,159)(152,230,160)(153,223,161)(154,224,162)(155,221,163)(156,222,164)(165,181,173)(166,182,174)(167,183,175)(168,184,176)(169,251,243)(170,252,244)(171,249,241)(172,250,242)(177,247,239)(178,248,240)(179,245,237)(180,246,238)(185,267,193)(186,268,194)(187,265,195)(188,266,196)(189,259,197)(190,260,198)(191,257,199)(192,258,200)(205,287,279)(206,288,280)(207,285,277)(208,286,278)(213,283,275)(214,284,276)(215,281,273)(216,282,274)(217,233,225)(218,234,226)(219,235,227)(220,236,228)(253,269,261)(254,270,262)(255,271,263)(256,272,264), (1,81,9)(2,82,10)(3,83,11)(4,84,12)(5,40,78)(6,37,79)(7,38,80)(8,39,77)(13,17,85)(14,18,86)(15,19,87)(16,20,88)(21,89,93)(22,90,94)(23,91,95)(24,92,96)(25,29,97)(26,30,98)(27,31,99)(28,32,100)(33,101,105)(34,102,106)(35,103,107)(36,104,108)(41,109,113)(42,110,114)(43,111,115)(44,112,116)(45,49,117)(46,50,118)(47,51,119)(48,52,120)(53,121,125)(54,122,126)(55,123,127)(56,124,128)(57,61,129)(58,62,130)(59,63,131)(60,64,132)(65,133,137)(66,134,138)(67,135,139)(68,136,140)(69,147,141)(70,148,142)(71,145,143)(72,146,144)(73,287,213)(74,288,214)(75,285,215)(76,286,216)(149,219,223)(150,220,224)(151,217,221)(152,218,222)(153,157,227)(154,158,228)(155,159,225)(156,160,226)(161,231,235)(162,232,236)(163,229,233)(164,230,234)(165,169,239)(166,170,240)(167,171,237)(168,172,238)(173,243,247)(174,244,248)(175,241,245)(176,242,246)(177,181,251)(178,182,252)(179,183,249)(180,184,250)(185,255,259)(186,256,260)(187,253,257)(188,254,258)(189,193,263)(190,194,264)(191,195,261)(192,196,262)(197,267,271)(198,268,272)(199,265,269)(200,266,270)(201,205,275)(202,206,276)(203,207,273)(204,208,274)(209,279,283)(210,280,284)(211,277,281)(212,278,282), (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), (1,59,111,25)(2,60,112,26)(3,57,109,27)(4,58,110,28)(5,24,142,54)(6,21,143,55)(7,22,144,56)(8,23,141,53)(9,131,43,97)(10,132,44,98)(11,129,41,99)(12,130,42,100)(13,135,47,101)(14,136,48,102)(15,133,45,103)(16,134,46,104)(17,139,51,105)(18,140,52,106)(19,137,49,107)(20,138,50,108)(29,81,63,115)(30,82,64,116)(31,83,61,113)(32,84,62,114)(33,85,67,119)(34,86,68,120)(35,87,65,117)(36,88,66,118)(37,89,71,123)(38,90,72,124)(39,91,69,121)(40,92,70,122)(73,269,184,236)(74,270,181,233)(75,271,182,234)(76,272,183,235)(77,95,147,125)(78,96,148,126)(79,93,145,127)(80,94,146,128)(149,274,186,237)(150,275,187,238)(151,276,188,239)(152,273,185,240)(153,278,190,241)(154,279,191,242)(155,280,192,243)(156,277,189,244)(157,282,194,245)(158,283,195,246)(159,284,196,247)(160,281,193,248)(161,286,198,249)(162,287,199,250)(163,288,200,251)(164,285,197,252)(165,217,202,254)(166,218,203,255)(167,219,204,256)(168,220,201,253)(169,221,206,258)(170,222,207,259)(171,223,208,260)(172,224,205,257)(173,225,210,262)(174,226,211,263)(175,227,212,264)(176,228,209,261)(177,229,214,266)(178,230,215,267)(179,231,216,268)(180,232,213,265), (1,218,111,255)(2,256,112,219)(3,220,109,253)(4,254,110,217)(5,286,142,249)(6,250,143,287)(7,288,144,251)(8,252,141,285)(9,152,43,185)(10,186,44,149)(11,150,41,187)(12,188,42,151)(13,156,47,189)(14,190,48,153)(15,154,45,191)(16,192,46,155)(17,160,51,193)(18,194,52,157)(19,158,49,195)(20,196,50,159)(21,164,55,197)(22,198,56,161)(23,162,53,199)(24,200,54,163)(25,168,59,201)(26,202,60,165)(27,166,57,203)(28,204,58,167)(29,172,63,205)(30,206,64,169)(31,170,61,207)(32,208,62,171)(33,176,67,209)(34,210,68,173)(35,174,65,211)(36,212,66,175)(37,180,71,213)(38,214,72,177)(39,178,69,215)(40,216,70,179)(73,79,184,145)(74,146,181,80)(75,77,182,147)(76,148,183,78)(81,222,115,259)(82,260,116,223)(83,224,113,257)(84,258,114,221)(85,226,119,263)(86,264,120,227)(87,228,117,261)(88,262,118,225)(89,230,123,267)(90,268,124,231)(91,232,121,265)(92,266,122,229)(93,234,127,271)(94,272,128,235)(95,236,125,269)(96,270,126,233)(97,238,131,275)(98,276,132,239)(99,240,129,273)(100,274,130,237)(101,242,135,279)(102,280,136,243)(103,244,133,277)(104,278,134,241)(105,246,139,283)(106,284,140,247)(107,248,137,281)(108,282,138,245) );
G=PermutationGroup([[(1,93,85),(2,94,86),(3,95,87),(4,96,88),(5,104,32),(6,101,29),(7,102,30),(8,103,31),(9,89,17),(10,90,18),(11,91,19),(12,92,20),(13,81,21),(14,82,22),(15,83,23),(16,84,24),(25,79,33),(26,80,34),(27,77,35),(28,78,36),(37,105,97),(38,106,98),(39,107,99),(40,108,100),(41,121,49),(42,122,50),(43,123,51),(44,124,52),(45,113,53),(46,114,54),(47,115,55),(48,116,56),(57,147,65),(58,148,66),(59,145,67),(60,146,68),(61,141,133),(62,142,134),(63,143,135),(64,144,136),(69,137,129),(70,138,130),(71,139,131),(72,140,132),(73,209,201),(74,210,202),(75,211,203),(76,212,204),(109,125,117),(110,126,118),(111,127,119),(112,128,120),(149,231,157),(150,232,158),(151,229,159),(152,230,160),(153,223,161),(154,224,162),(155,221,163),(156,222,164),(165,181,173),(166,182,174),(167,183,175),(168,184,176),(169,251,243),(170,252,244),(171,249,241),(172,250,242),(177,247,239),(178,248,240),(179,245,237),(180,246,238),(185,267,193),(186,268,194),(187,265,195),(188,266,196),(189,259,197),(190,260,198),(191,257,199),(192,258,200),(205,287,279),(206,288,280),(207,285,277),(208,286,278),(213,283,275),(214,284,276),(215,281,273),(216,282,274),(217,233,225),(218,234,226),(219,235,227),(220,236,228),(253,269,261),(254,270,262),(255,271,263),(256,272,264)], [(1,81,9),(2,82,10),(3,83,11),(4,84,12),(5,40,78),(6,37,79),(7,38,80),(8,39,77),(13,17,85),(14,18,86),(15,19,87),(16,20,88),(21,89,93),(22,90,94),(23,91,95),(24,92,96),(25,29,97),(26,30,98),(27,31,99),(28,32,100),(33,101,105),(34,102,106),(35,103,107),(36,104,108),(41,109,113),(42,110,114),(43,111,115),(44,112,116),(45,49,117),(46,50,118),(47,51,119),(48,52,120),(53,121,125),(54,122,126),(55,123,127),(56,124,128),(57,61,129),(58,62,130),(59,63,131),(60,64,132),(65,133,137),(66,134,138),(67,135,139),(68,136,140),(69,147,141),(70,148,142),(71,145,143),(72,146,144),(73,287,213),(74,288,214),(75,285,215),(76,286,216),(149,219,223),(150,220,224),(151,217,221),(152,218,222),(153,157,227),(154,158,228),(155,159,225),(156,160,226),(161,231,235),(162,232,236),(163,229,233),(164,230,234),(165,169,239),(166,170,240),(167,171,237),(168,172,238),(173,243,247),(174,244,248),(175,241,245),(176,242,246),(177,181,251),(178,182,252),(179,183,249),(180,184,250),(185,255,259),(186,256,260),(187,253,257),(188,254,258),(189,193,263),(190,194,264),(191,195,261),(192,196,262),(197,267,271),(198,268,272),(199,265,269),(200,266,270),(201,205,275),(202,206,276),(203,207,273),(204,208,274),(209,279,283),(210,280,284),(211,277,281),(212,278,282)], [(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)], [(1,59,111,25),(2,60,112,26),(3,57,109,27),(4,58,110,28),(5,24,142,54),(6,21,143,55),(7,22,144,56),(8,23,141,53),(9,131,43,97),(10,132,44,98),(11,129,41,99),(12,130,42,100),(13,135,47,101),(14,136,48,102),(15,133,45,103),(16,134,46,104),(17,139,51,105),(18,140,52,106),(19,137,49,107),(20,138,50,108),(29,81,63,115),(30,82,64,116),(31,83,61,113),(32,84,62,114),(33,85,67,119),(34,86,68,120),(35,87,65,117),(36,88,66,118),(37,89,71,123),(38,90,72,124),(39,91,69,121),(40,92,70,122),(73,269,184,236),(74,270,181,233),(75,271,182,234),(76,272,183,235),(77,95,147,125),(78,96,148,126),(79,93,145,127),(80,94,146,128),(149,274,186,237),(150,275,187,238),(151,276,188,239),(152,273,185,240),(153,278,190,241),(154,279,191,242),(155,280,192,243),(156,277,189,244),(157,282,194,245),(158,283,195,246),(159,284,196,247),(160,281,193,248),(161,286,198,249),(162,287,199,250),(163,288,200,251),(164,285,197,252),(165,217,202,254),(166,218,203,255),(167,219,204,256),(168,220,201,253),(169,221,206,258),(170,222,207,259),(171,223,208,260),(172,224,205,257),(173,225,210,262),(174,226,211,263),(175,227,212,264),(176,228,209,261),(177,229,214,266),(178,230,215,267),(179,231,216,268),(180,232,213,265)], [(1,218,111,255),(2,256,112,219),(3,220,109,253),(4,254,110,217),(5,286,142,249),(6,250,143,287),(7,288,144,251),(8,252,141,285),(9,152,43,185),(10,186,44,149),(11,150,41,187),(12,188,42,151),(13,156,47,189),(14,190,48,153),(15,154,45,191),(16,192,46,155),(17,160,51,193),(18,194,52,157),(19,158,49,195),(20,196,50,159),(21,164,55,197),(22,198,56,161),(23,162,53,199),(24,200,54,163),(25,168,59,201),(26,202,60,165),(27,166,57,203),(28,204,58,167),(29,172,63,205),(30,206,64,169),(31,170,61,207),(32,208,62,171),(33,176,67,209),(34,210,68,173),(35,174,65,211),(36,212,66,175),(37,180,71,213),(38,214,72,177),(39,178,69,215),(40,216,70,179),(73,79,184,145),(74,146,181,80),(75,77,182,147),(76,148,183,78),(81,222,115,259),(82,260,116,223),(83,224,113,257),(84,258,114,221),(85,226,119,263),(86,264,120,227),(87,228,117,261),(88,262,118,225),(89,230,123,267),(90,268,124,231),(91,232,121,265),(92,266,122,229),(93,234,127,271),(94,272,128,235),(95,236,125,269),(96,270,126,233),(97,238,131,275),(98,276,132,239),(99,240,129,273),(100,274,130,237),(101,242,135,279),(102,280,136,243),(103,244,133,277),(104,278,134,241),(105,246,139,283),(106,284,140,247),(107,248,137,281),(108,282,138,245)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6X | 12A | ··· | 12AV | 12AW | ··· | 12CB |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | Q8 | C4○D4 | C3×Q8 | C3×C4○D4 |
| kernel | C32×C42.C2 | C122 | C32×C4⋊C4 | C3×C42.C2 | C4×C12 | C3×C4⋊C4 | C3×C12 | C3×C6 | C12 | C6 |
| # reps | 1 | 1 | 6 | 8 | 8 | 48 | 2 | 4 | 16 | 32 |
Matrix representation of C32×C42.C2 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,0,12,0,0,1,0],[5,0,0,0,0,5,0,0,0,0,0,12,0,0,1,0],[5,0,0,0,0,8,0,0,0,0,5,0,0,0,0,8] >;
C32×C42.C2 in GAP, Magma, Sage, TeX
C_3^2\times C_4^2.C_2
% in TeX
G:=Group("C3^2xC4^2.C2"); // GroupNames label
G:=SmallGroup(288,822);
// by ID
G=gap.SmallGroup(288,822);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,504,1037,1016,3110,394]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^4=1,e^2=d^2,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*d^2,e*d*e^-1=c^2*d>;
// generators/relations