direct product, metacyclic, nilpotent (class 2), monomial
Aliases: C32×C4⋊C8, C12⋊3C24, C122.14C2, C4⋊(C3×C24), (C3×C12)⋊9C8, (C2×C24).8C6, C2.2(C6×C24), (C6×C24).6C2, (C4×C12).22C6, C6.15(C2×C24), (C6×C12).23C4, C12.90(C3×D4), C42.2(C3×C6), C12.18(C3×Q8), (C3×C12).32Q8, (C2×C12).23C12, (C3×C12).185D4, C4.4(Q8×C32), (C2×C4).32C62, C22.9(C6×C12), C4.18(D4×C32), C62.118(C2×C4), (C3×C6).26M4(2), C6.13(C3×M4(2)), (C6×C12).381C22, C2.3(C32×M4(2)), C6.17(C3×C4⋊C4), (C2×C8).2(C3×C6), (C2×C4).4(C3×C12), (C3×C6).48(C2×C8), C2.2(C32×C4⋊C4), (C3×C6).46(C4⋊C4), (C2×C6).53(C2×C12), (C2×C12).172(C2×C6), SmallGroup(288,323)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C4⋊C8
G = < a,b,c,d | a3=b3=c4=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 132 in 114 conjugacy classes, 96 normal (24 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C32, C12, C12, C2×C6, C42, C2×C8, C3×C6, C24, C2×C12, C4⋊C8, C3×C12, C3×C12, C3×C12, C62, C4×C12, C2×C24, C3×C24, C6×C12, C3×C4⋊C8, C122, C6×C24, C32×C4⋊C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, Q8, C32, C12, C2×C6, C4⋊C4, C2×C8, M4(2), C3×C6, C24, C2×C12, C3×D4, C3×Q8, C4⋊C8, C3×C12, C62, C3×C4⋊C4, C2×C24, C3×M4(2), C3×C24, C6×C12, D4×C32, Q8×C32, C3×C4⋊C8, C32×C4⋊C4, C6×C24, C32×M4(2), C32×C4⋊C8
►Regular action on 288 points
Generators in S288
(1 185 169)(2 186 170)(3 187 171)(4 188 172)(5 189 173)(6 190 174)(7 191 175)(8 192 176)(9 208 64)(10 201 57)(11 202 58)(12 203 59)(13 204 60)(14 205 61)(15 206 62)(16 207 63)(17 177 33)(18 178 34)(19 179 35)(20 180 36)(21 181 37)(22 182 38)(23 183 39)(24 184 40)(25 161 41)(26 162 42)(27 163 43)(28 164 44)(29 165 45)(30 166 46)(31 167 47)(32 168 48)(49 158 65)(50 159 66)(51 160 67)(52 153 68)(53 154 69)(54 155 70)(55 156 71)(56 157 72)(73 209 193)(74 210 194)(75 211 195)(76 212 196)(77 213 197)(78 214 198)(79 215 199)(80 216 200)(81 241 97)(82 242 98)(83 243 99)(84 244 100)(85 245 101)(86 246 102)(87 247 103)(88 248 104)(89 225 105)(90 226 106)(91 227 107)(92 228 108)(93 229 109)(94 230 110)(95 231 111)(96 232 112)(113 149 129)(114 150 130)(115 151 131)(116 152 132)(117 145 133)(118 146 134)(119 147 135)(120 148 136)(121 281 265)(122 282 266)(123 283 267)(124 284 268)(125 285 269)(126 286 270)(127 287 271)(128 288 272)(137 273 257)(138 274 258)(139 275 259)(140 276 260)(141 277 261)(142 278 262)(143 279 263)(144 280 264)(217 249 233)(218 250 234)(219 251 235)(220 252 236)(221 253 237)(222 254 238)(223 255 239)(224 256 240)
(1 161 17)(2 162 18)(3 163 19)(4 164 20)(5 165 21)(6 166 22)(7 167 23)(8 168 24)(9 80 157)(10 73 158)(11 74 159)(12 75 160)(13 76 153)(14 77 154)(15 78 155)(16 79 156)(25 33 169)(26 34 170)(27 35 171)(28 36 172)(29 37 173)(30 38 174)(31 39 175)(32 40 176)(41 177 185)(42 178 186)(43 179 187)(44 180 188)(45 181 189)(46 182 190)(47 183 191)(48 184 192)(49 57 193)(50 58 194)(51 59 195)(52 60 196)(53 61 197)(54 62 198)(55 63 199)(56 64 200)(65 201 209)(66 202 210)(67 203 211)(68 204 212)(69 205 213)(70 206 214)(71 207 215)(72 208 216)(81 217 225)(82 218 226)(83 219 227)(84 220 228)(85 221 229)(86 222 230)(87 223 231)(88 224 232)(89 97 233)(90 98 234)(91 99 235)(92 100 236)(93 101 237)(94 102 238)(95 103 239)(96 104 240)(105 241 249)(106 242 250)(107 243 251)(108 244 252)(109 245 253)(110 246 254)(111 247 255)(112 248 256)(113 121 257)(114 122 258)(115 123 259)(116 124 260)(117 125 261)(118 126 262)(119 127 263)(120 128 264)(129 265 273)(130 266 274)(131 267 275)(132 268 276)(133 269 277)(134 270 278)(135 271 279)(136 272 280)(137 149 281)(138 150 282)(139 151 283)(140 152 284)(141 145 285)(142 146 286)(143 147 287)(144 148 288)
(1 115 223 53)(2 54 224 116)(3 117 217 55)(4 56 218 118)(5 119 219 49)(6 50 220 120)(7 113 221 51)(8 52 222 114)(9 106 286 44)(10 45 287 107)(11 108 288 46)(12 47 281 109)(13 110 282 48)(14 41 283 111)(15 112 284 42)(16 43 285 105)(17 259 87 197)(18 198 88 260)(19 261 81 199)(20 200 82 262)(21 263 83 193)(22 194 84 264)(23 257 85 195)(24 196 86 258)(25 267 95 205)(26 206 96 268)(27 269 89 207)(28 208 90 270)(29 271 91 201)(30 202 92 272)(31 265 93 203)(32 204 94 266)(33 275 103 213)(34 214 104 276)(35 277 97 215)(36 216 98 278)(37 279 99 209)(38 210 100 280)(39 273 101 211)(40 212 102 274)(57 165 127 227)(58 228 128 166)(59 167 121 229)(60 230 122 168)(61 161 123 231)(62 232 124 162)(63 163 125 225)(64 226 126 164)(65 173 135 235)(66 236 136 174)(67 175 129 237)(68 238 130 176)(69 169 131 239)(70 240 132 170)(71 171 133 233)(72 234 134 172)(73 181 143 243)(74 244 144 182)(75 183 137 245)(76 246 138 184)(77 177 139 247)(78 248 140 178)(79 179 141 241)(80 242 142 180)(145 249 156 187)(146 188 157 250)(147 251 158 189)(148 190 159 252)(149 253 160 191)(150 192 153 254)(151 255 154 185)(152 186 155 256)
(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)
G:=sub<Sym(288)| (1,185,169)(2,186,170)(3,187,171)(4,188,172)(5,189,173)(6,190,174)(7,191,175)(8,192,176)(9,208,64)(10,201,57)(11,202,58)(12,203,59)(13,204,60)(14,205,61)(15,206,62)(16,207,63)(17,177,33)(18,178,34)(19,179,35)(20,180,36)(21,181,37)(22,182,38)(23,183,39)(24,184,40)(25,161,41)(26,162,42)(27,163,43)(28,164,44)(29,165,45)(30,166,46)(31,167,47)(32,168,48)(49,158,65)(50,159,66)(51,160,67)(52,153,68)(53,154,69)(54,155,70)(55,156,71)(56,157,72)(73,209,193)(74,210,194)(75,211,195)(76,212,196)(77,213,197)(78,214,198)(79,215,199)(80,216,200)(81,241,97)(82,242,98)(83,243,99)(84,244,100)(85,245,101)(86,246,102)(87,247,103)(88,248,104)(89,225,105)(90,226,106)(91,227,107)(92,228,108)(93,229,109)(94,230,110)(95,231,111)(96,232,112)(113,149,129)(114,150,130)(115,151,131)(116,152,132)(117,145,133)(118,146,134)(119,147,135)(120,148,136)(121,281,265)(122,282,266)(123,283,267)(124,284,268)(125,285,269)(126,286,270)(127,287,271)(128,288,272)(137,273,257)(138,274,258)(139,275,259)(140,276,260)(141,277,261)(142,278,262)(143,279,263)(144,280,264)(217,249,233)(218,250,234)(219,251,235)(220,252,236)(221,253,237)(222,254,238)(223,255,239)(224,256,240), (1,161,17)(2,162,18)(3,163,19)(4,164,20)(5,165,21)(6,166,22)(7,167,23)(8,168,24)(9,80,157)(10,73,158)(11,74,159)(12,75,160)(13,76,153)(14,77,154)(15,78,155)(16,79,156)(25,33,169)(26,34,170)(27,35,171)(28,36,172)(29,37,173)(30,38,174)(31,39,175)(32,40,176)(41,177,185)(42,178,186)(43,179,187)(44,180,188)(45,181,189)(46,182,190)(47,183,191)(48,184,192)(49,57,193)(50,58,194)(51,59,195)(52,60,196)(53,61,197)(54,62,198)(55,63,199)(56,64,200)(65,201,209)(66,202,210)(67,203,211)(68,204,212)(69,205,213)(70,206,214)(71,207,215)(72,208,216)(81,217,225)(82,218,226)(83,219,227)(84,220,228)(85,221,229)(86,222,230)(87,223,231)(88,224,232)(89,97,233)(90,98,234)(91,99,235)(92,100,236)(93,101,237)(94,102,238)(95,103,239)(96,104,240)(105,241,249)(106,242,250)(107,243,251)(108,244,252)(109,245,253)(110,246,254)(111,247,255)(112,248,256)(113,121,257)(114,122,258)(115,123,259)(116,124,260)(117,125,261)(118,126,262)(119,127,263)(120,128,264)(129,265,273)(130,266,274)(131,267,275)(132,268,276)(133,269,277)(134,270,278)(135,271,279)(136,272,280)(137,149,281)(138,150,282)(139,151,283)(140,152,284)(141,145,285)(142,146,286)(143,147,287)(144,148,288), (1,115,223,53)(2,54,224,116)(3,117,217,55)(4,56,218,118)(5,119,219,49)(6,50,220,120)(7,113,221,51)(8,52,222,114)(9,106,286,44)(10,45,287,107)(11,108,288,46)(12,47,281,109)(13,110,282,48)(14,41,283,111)(15,112,284,42)(16,43,285,105)(17,259,87,197)(18,198,88,260)(19,261,81,199)(20,200,82,262)(21,263,83,193)(22,194,84,264)(23,257,85,195)(24,196,86,258)(25,267,95,205)(26,206,96,268)(27,269,89,207)(28,208,90,270)(29,271,91,201)(30,202,92,272)(31,265,93,203)(32,204,94,266)(33,275,103,213)(34,214,104,276)(35,277,97,215)(36,216,98,278)(37,279,99,209)(38,210,100,280)(39,273,101,211)(40,212,102,274)(57,165,127,227)(58,228,128,166)(59,167,121,229)(60,230,122,168)(61,161,123,231)(62,232,124,162)(63,163,125,225)(64,226,126,164)(65,173,135,235)(66,236,136,174)(67,175,129,237)(68,238,130,176)(69,169,131,239)(70,240,132,170)(71,171,133,233)(72,234,134,172)(73,181,143,243)(74,244,144,182)(75,183,137,245)(76,246,138,184)(77,177,139,247)(78,248,140,178)(79,179,141,241)(80,242,142,180)(145,249,156,187)(146,188,157,250)(147,251,158,189)(148,190,159,252)(149,253,160,191)(150,192,153,254)(151,255,154,185)(152,186,155,256), (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)>;
G:=Group( (1,185,169)(2,186,170)(3,187,171)(4,188,172)(5,189,173)(6,190,174)(7,191,175)(8,192,176)(9,208,64)(10,201,57)(11,202,58)(12,203,59)(13,204,60)(14,205,61)(15,206,62)(16,207,63)(17,177,33)(18,178,34)(19,179,35)(20,180,36)(21,181,37)(22,182,38)(23,183,39)(24,184,40)(25,161,41)(26,162,42)(27,163,43)(28,164,44)(29,165,45)(30,166,46)(31,167,47)(32,168,48)(49,158,65)(50,159,66)(51,160,67)(52,153,68)(53,154,69)(54,155,70)(55,156,71)(56,157,72)(73,209,193)(74,210,194)(75,211,195)(76,212,196)(77,213,197)(78,214,198)(79,215,199)(80,216,200)(81,241,97)(82,242,98)(83,243,99)(84,244,100)(85,245,101)(86,246,102)(87,247,103)(88,248,104)(89,225,105)(90,226,106)(91,227,107)(92,228,108)(93,229,109)(94,230,110)(95,231,111)(96,232,112)(113,149,129)(114,150,130)(115,151,131)(116,152,132)(117,145,133)(118,146,134)(119,147,135)(120,148,136)(121,281,265)(122,282,266)(123,283,267)(124,284,268)(125,285,269)(126,286,270)(127,287,271)(128,288,272)(137,273,257)(138,274,258)(139,275,259)(140,276,260)(141,277,261)(142,278,262)(143,279,263)(144,280,264)(217,249,233)(218,250,234)(219,251,235)(220,252,236)(221,253,237)(222,254,238)(223,255,239)(224,256,240), (1,161,17)(2,162,18)(3,163,19)(4,164,20)(5,165,21)(6,166,22)(7,167,23)(8,168,24)(9,80,157)(10,73,158)(11,74,159)(12,75,160)(13,76,153)(14,77,154)(15,78,155)(16,79,156)(25,33,169)(26,34,170)(27,35,171)(28,36,172)(29,37,173)(30,38,174)(31,39,175)(32,40,176)(41,177,185)(42,178,186)(43,179,187)(44,180,188)(45,181,189)(46,182,190)(47,183,191)(48,184,192)(49,57,193)(50,58,194)(51,59,195)(52,60,196)(53,61,197)(54,62,198)(55,63,199)(56,64,200)(65,201,209)(66,202,210)(67,203,211)(68,204,212)(69,205,213)(70,206,214)(71,207,215)(72,208,216)(81,217,225)(82,218,226)(83,219,227)(84,220,228)(85,221,229)(86,222,230)(87,223,231)(88,224,232)(89,97,233)(90,98,234)(91,99,235)(92,100,236)(93,101,237)(94,102,238)(95,103,239)(96,104,240)(105,241,249)(106,242,250)(107,243,251)(108,244,252)(109,245,253)(110,246,254)(111,247,255)(112,248,256)(113,121,257)(114,122,258)(115,123,259)(116,124,260)(117,125,261)(118,126,262)(119,127,263)(120,128,264)(129,265,273)(130,266,274)(131,267,275)(132,268,276)(133,269,277)(134,270,278)(135,271,279)(136,272,280)(137,149,281)(138,150,282)(139,151,283)(140,152,284)(141,145,285)(142,146,286)(143,147,287)(144,148,288), (1,115,223,53)(2,54,224,116)(3,117,217,55)(4,56,218,118)(5,119,219,49)(6,50,220,120)(7,113,221,51)(8,52,222,114)(9,106,286,44)(10,45,287,107)(11,108,288,46)(12,47,281,109)(13,110,282,48)(14,41,283,111)(15,112,284,42)(16,43,285,105)(17,259,87,197)(18,198,88,260)(19,261,81,199)(20,200,82,262)(21,263,83,193)(22,194,84,264)(23,257,85,195)(24,196,86,258)(25,267,95,205)(26,206,96,268)(27,269,89,207)(28,208,90,270)(29,271,91,201)(30,202,92,272)(31,265,93,203)(32,204,94,266)(33,275,103,213)(34,214,104,276)(35,277,97,215)(36,216,98,278)(37,279,99,209)(38,210,100,280)(39,273,101,211)(40,212,102,274)(57,165,127,227)(58,228,128,166)(59,167,121,229)(60,230,122,168)(61,161,123,231)(62,232,124,162)(63,163,125,225)(64,226,126,164)(65,173,135,235)(66,236,136,174)(67,175,129,237)(68,238,130,176)(69,169,131,239)(70,240,132,170)(71,171,133,233)(72,234,134,172)(73,181,143,243)(74,244,144,182)(75,183,137,245)(76,246,138,184)(77,177,139,247)(78,248,140,178)(79,179,141,241)(80,242,142,180)(145,249,156,187)(146,188,157,250)(147,251,158,189)(148,190,159,252)(149,253,160,191)(150,192,153,254)(151,255,154,185)(152,186,155,256), (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) );
G=PermutationGroup([[(1,185,169),(2,186,170),(3,187,171),(4,188,172),(5,189,173),(6,190,174),(7,191,175),(8,192,176),(9,208,64),(10,201,57),(11,202,58),(12,203,59),(13,204,60),(14,205,61),(15,206,62),(16,207,63),(17,177,33),(18,178,34),(19,179,35),(20,180,36),(21,181,37),(22,182,38),(23,183,39),(24,184,40),(25,161,41),(26,162,42),(27,163,43),(28,164,44),(29,165,45),(30,166,46),(31,167,47),(32,168,48),(49,158,65),(50,159,66),(51,160,67),(52,153,68),(53,154,69),(54,155,70),(55,156,71),(56,157,72),(73,209,193),(74,210,194),(75,211,195),(76,212,196),(77,213,197),(78,214,198),(79,215,199),(80,216,200),(81,241,97),(82,242,98),(83,243,99),(84,244,100),(85,245,101),(86,246,102),(87,247,103),(88,248,104),(89,225,105),(90,226,106),(91,227,107),(92,228,108),(93,229,109),(94,230,110),(95,231,111),(96,232,112),(113,149,129),(114,150,130),(115,151,131),(116,152,132),(117,145,133),(118,146,134),(119,147,135),(120,148,136),(121,281,265),(122,282,266),(123,283,267),(124,284,268),(125,285,269),(126,286,270),(127,287,271),(128,288,272),(137,273,257),(138,274,258),(139,275,259),(140,276,260),(141,277,261),(142,278,262),(143,279,263),(144,280,264),(217,249,233),(218,250,234),(219,251,235),(220,252,236),(221,253,237),(222,254,238),(223,255,239),(224,256,240)], [(1,161,17),(2,162,18),(3,163,19),(4,164,20),(5,165,21),(6,166,22),(7,167,23),(8,168,24),(9,80,157),(10,73,158),(11,74,159),(12,75,160),(13,76,153),(14,77,154),(15,78,155),(16,79,156),(25,33,169),(26,34,170),(27,35,171),(28,36,172),(29,37,173),(30,38,174),(31,39,175),(32,40,176),(41,177,185),(42,178,186),(43,179,187),(44,180,188),(45,181,189),(46,182,190),(47,183,191),(48,184,192),(49,57,193),(50,58,194),(51,59,195),(52,60,196),(53,61,197),(54,62,198),(55,63,199),(56,64,200),(65,201,209),(66,202,210),(67,203,211),(68,204,212),(69,205,213),(70,206,214),(71,207,215),(72,208,216),(81,217,225),(82,218,226),(83,219,227),(84,220,228),(85,221,229),(86,222,230),(87,223,231),(88,224,232),(89,97,233),(90,98,234),(91,99,235),(92,100,236),(93,101,237),(94,102,238),(95,103,239),(96,104,240),(105,241,249),(106,242,250),(107,243,251),(108,244,252),(109,245,253),(110,246,254),(111,247,255),(112,248,256),(113,121,257),(114,122,258),(115,123,259),(116,124,260),(117,125,261),(118,126,262),(119,127,263),(120,128,264),(129,265,273),(130,266,274),(131,267,275),(132,268,276),(133,269,277),(134,270,278),(135,271,279),(136,272,280),(137,149,281),(138,150,282),(139,151,283),(140,152,284),(141,145,285),(142,146,286),(143,147,287),(144,148,288)], [(1,115,223,53),(2,54,224,116),(3,117,217,55),(4,56,218,118),(5,119,219,49),(6,50,220,120),(7,113,221,51),(8,52,222,114),(9,106,286,44),(10,45,287,107),(11,108,288,46),(12,47,281,109),(13,110,282,48),(14,41,283,111),(15,112,284,42),(16,43,285,105),(17,259,87,197),(18,198,88,260),(19,261,81,199),(20,200,82,262),(21,263,83,193),(22,194,84,264),(23,257,85,195),(24,196,86,258),(25,267,95,205),(26,206,96,268),(27,269,89,207),(28,208,90,270),(29,271,91,201),(30,202,92,272),(31,265,93,203),(32,204,94,266),(33,275,103,213),(34,214,104,276),(35,277,97,215),(36,216,98,278),(37,279,99,209),(38,210,100,280),(39,273,101,211),(40,212,102,274),(57,165,127,227),(58,228,128,166),(59,167,121,229),(60,230,122,168),(61,161,123,231),(62,232,124,162),(63,163,125,225),(64,226,126,164),(65,173,135,235),(66,236,136,174),(67,175,129,237),(68,238,130,176),(69,169,131,239),(70,240,132,170),(71,171,133,233),(72,234,134,172),(73,181,143,243),(74,244,144,182),(75,183,137,245),(76,246,138,184),(77,177,139,247),(78,248,140,178),(79,179,141,241),(80,242,142,180),(145,249,156,187),(146,188,157,250),(147,251,158,189),(148,190,159,252),(149,253,160,191),(150,192,153,254),(151,255,154,185),(152,186,155,256)], [(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)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6X | 8A | ··· | 8H | 12A | ··· | 12AF | 12AG | ··· | 12BL | 24A | ··· | 24BL |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C8 | C12 | C24 | D4 | Q8 | M4(2) | C3×D4 | C3×Q8 | C3×M4(2) |
| kernel | C32×C4⋊C8 | C122 | C6×C24 | C3×C4⋊C8 | C6×C12 | C4×C12 | C2×C24 | C3×C12 | C2×C12 | C12 | C3×C12 | C3×C12 | C3×C6 | C12 | C12 | C6 |
| # reps | 1 | 1 | 2 | 8 | 4 | 8 | 16 | 8 | 32 | 64 | 1 | 1 | 2 | 8 | 8 | 16 |
Matrix representation of C32×C4⋊C8 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 27 |
| 22 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,46,0,0,0,0,27],[22,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
C32×C4⋊C8 in GAP, Magma, Sage, TeX
C_3^2\times C_4\rtimes C_8
% in TeX
G:=Group("C3^2xC4:C8"); // GroupNames label
G:=SmallGroup(288,323);
// by ID
G=gap.SmallGroup(288,323);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,260,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^4=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations