direct product, metacyclic, nilpotent (class 2), monomial
Aliases: C32×C8⋊C4, C24⋊7C12, C122.1C2, C2.2C122, C8⋊3(C3×C12), (C3×C24)⋊15C4, C4.9(C6×C12), C6.9(C4×C12), (C4×C12).7C6, (C6×C24).27C2, (C6×C12).21C4, (C2×C24).32C6, C42.1(C3×C6), (C2×C12).11C12, C12.61(C2×C12), (C3×C6).22C42, C22.7(C6×C12), (C2×C4).30C62, C62.116(C2×C4), (C3×C6).24M4(2), C6.11(C3×M4(2)), (C6×C12).379C22, C2.1(C32×M4(2)), (C2×C8).7(C3×C6), (C2×C4).2(C3×C12), (C2×C6).51(C2×C12), (C2×C12).170(C2×C6), (C3×C12).145(C2×C4), SmallGroup(288,315)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C8⋊C4
G = < a,b,c,d | a3=b3=c8=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 132 in 120 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, C32, C12, C12, C2×C6, C42, C2×C8, C3×C6, C3×C6, C24, C2×C12, C8⋊C4, C3×C12, C3×C12, C62, C4×C12, C2×C24, C3×C24, C6×C12, C6×C12, C3×C8⋊C4, C122, C6×C24, C32×C8⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C42, M4(2), C3×C6, C2×C12, C8⋊C4, C3×C12, C62, C4×C12, C3×M4(2), C6×C12, C3×C8⋊C4, C122, C32×M4(2), C32×C8⋊C4
►Regular action on 288 points
Generators in S288
(1 170 154)(2 171 155)(3 172 156)(4 173 157)(5 174 158)(6 175 159)(7 176 160)(8 169 153)(9 243 25)(10 244 26)(11 245 27)(12 246 28)(13 247 29)(14 248 30)(15 241 31)(16 242 32)(17 227 33)(18 228 34)(19 229 35)(20 230 36)(21 231 37)(22 232 38)(23 225 39)(24 226 40)(41 224 57)(42 217 58)(43 218 59)(44 219 60)(45 220 61)(46 221 62)(47 222 63)(48 223 64)(49 283 267)(50 284 268)(51 285 269)(52 286 270)(53 287 271)(54 288 272)(55 281 265)(56 282 266)(65 275 259)(66 276 260)(67 277 261)(68 278 262)(69 279 263)(70 280 264)(71 273 257)(72 274 258)(73 161 89)(74 162 90)(75 163 91)(76 164 92)(77 165 93)(78 166 94)(79 167 95)(80 168 96)(81 145 97)(82 146 98)(83 147 99)(84 148 100)(85 149 101)(86 150 102)(87 151 103)(88 152 104)(105 199 127)(106 200 128)(107 193 121)(108 194 122)(109 195 123)(110 196 124)(111 197 125)(112 198 126)(113 184 129)(114 177 130)(115 178 131)(116 179 132)(117 180 133)(118 181 134)(119 182 135)(120 183 136)(137 201 185)(138 202 186)(139 203 187)(140 204 188)(141 205 189)(142 206 190)(143 207 191)(144 208 192)(209 249 233)(210 250 234)(211 251 235)(212 252 236)(213 253 237)(214 254 238)(215 255 239)(216 256 240)
(1 146 74)(2 147 75)(3 148 76)(4 149 77)(5 150 78)(6 151 79)(7 152 80)(8 145 73)(9 211 227)(10 212 228)(11 213 229)(12 214 230)(13 215 231)(14 216 232)(15 209 225)(16 210 226)(17 25 235)(18 26 236)(19 27 237)(20 28 238)(21 29 239)(22 30 240)(23 31 233)(24 32 234)(33 243 251)(34 244 252)(35 245 253)(36 246 254)(37 247 255)(38 248 256)(39 241 249)(40 242 250)(41 49 259)(42 50 260)(43 51 261)(44 52 262)(45 53 263)(46 54 264)(47 55 257)(48 56 258)(57 267 275)(58 268 276)(59 269 277)(60 270 278)(61 271 279)(62 272 280)(63 265 273)(64 266 274)(65 224 283)(66 217 284)(67 218 285)(68 219 286)(69 220 287)(70 221 288)(71 222 281)(72 223 282)(81 89 153)(82 90 154)(83 91 155)(84 92 156)(85 93 157)(86 94 158)(87 95 159)(88 96 160)(97 161 169)(98 162 170)(99 163 171)(100 164 172)(101 165 173)(102 166 174)(103 167 175)(104 168 176)(105 143 182)(106 144 183)(107 137 184)(108 138 177)(109 139 178)(110 140 179)(111 141 180)(112 142 181)(113 121 185)(114 122 186)(115 123 187)(116 124 188)(117 125 189)(118 126 190)(119 127 191)(120 128 192)(129 193 201)(130 194 202)(131 195 203)(132 196 204)(133 197 205)(134 198 206)(135 199 207)(136 200 208)
(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)
(1 115 41 211)(2 120 42 216)(3 117 43 213)(4 114 44 210)(5 119 45 215)(6 116 46 212)(7 113 47 209)(8 118 48 214)(9 74 187 259)(10 79 188 264)(11 76 189 261)(12 73 190 258)(13 78 191 263)(14 75 192 260)(15 80 185 257)(16 77 186 262)(17 82 195 267)(18 87 196 272)(19 84 197 269)(20 81 198 266)(21 86 199 271)(22 83 200 268)(23 88 193 265)(24 85 194 270)(25 90 203 275)(26 95 204 280)(27 92 205 277)(28 89 206 274)(29 94 207 279)(30 91 208 276)(31 96 201 273)(32 93 202 278)(33 98 109 283)(34 103 110 288)(35 100 111 285)(36 97 112 282)(37 102 105 287)(38 99 106 284)(39 104 107 281)(40 101 108 286)(49 227 146 123)(50 232 147 128)(51 229 148 125)(52 226 149 122)(53 231 150 127)(54 228 151 124)(55 225 152 121)(56 230 145 126)(57 235 154 131)(58 240 155 136)(59 237 156 133)(60 234 157 130)(61 239 158 135)(62 236 159 132)(63 233 160 129)(64 238 153 134)(65 243 162 139)(66 248 163 144)(67 245 164 141)(68 242 165 138)(69 247 166 143)(70 244 167 140)(71 241 168 137)(72 246 161 142)(169 181 223 254)(170 178 224 251)(171 183 217 256)(172 180 218 253)(173 177 219 250)(174 182 220 255)(175 179 221 252)(176 184 222 249)
G:=sub<Sym(288)| (1,170,154)(2,171,155)(3,172,156)(4,173,157)(5,174,158)(6,175,159)(7,176,160)(8,169,153)(9,243,25)(10,244,26)(11,245,27)(12,246,28)(13,247,29)(14,248,30)(15,241,31)(16,242,32)(17,227,33)(18,228,34)(19,229,35)(20,230,36)(21,231,37)(22,232,38)(23,225,39)(24,226,40)(41,224,57)(42,217,58)(43,218,59)(44,219,60)(45,220,61)(46,221,62)(47,222,63)(48,223,64)(49,283,267)(50,284,268)(51,285,269)(52,286,270)(53,287,271)(54,288,272)(55,281,265)(56,282,266)(65,275,259)(66,276,260)(67,277,261)(68,278,262)(69,279,263)(70,280,264)(71,273,257)(72,274,258)(73,161,89)(74,162,90)(75,163,91)(76,164,92)(77,165,93)(78,166,94)(79,167,95)(80,168,96)(81,145,97)(82,146,98)(83,147,99)(84,148,100)(85,149,101)(86,150,102)(87,151,103)(88,152,104)(105,199,127)(106,200,128)(107,193,121)(108,194,122)(109,195,123)(110,196,124)(111,197,125)(112,198,126)(113,184,129)(114,177,130)(115,178,131)(116,179,132)(117,180,133)(118,181,134)(119,182,135)(120,183,136)(137,201,185)(138,202,186)(139,203,187)(140,204,188)(141,205,189)(142,206,190)(143,207,191)(144,208,192)(209,249,233)(210,250,234)(211,251,235)(212,252,236)(213,253,237)(214,254,238)(215,255,239)(216,256,240), (1,146,74)(2,147,75)(3,148,76)(4,149,77)(5,150,78)(6,151,79)(7,152,80)(8,145,73)(9,211,227)(10,212,228)(11,213,229)(12,214,230)(13,215,231)(14,216,232)(15,209,225)(16,210,226)(17,25,235)(18,26,236)(19,27,237)(20,28,238)(21,29,239)(22,30,240)(23,31,233)(24,32,234)(33,243,251)(34,244,252)(35,245,253)(36,246,254)(37,247,255)(38,248,256)(39,241,249)(40,242,250)(41,49,259)(42,50,260)(43,51,261)(44,52,262)(45,53,263)(46,54,264)(47,55,257)(48,56,258)(57,267,275)(58,268,276)(59,269,277)(60,270,278)(61,271,279)(62,272,280)(63,265,273)(64,266,274)(65,224,283)(66,217,284)(67,218,285)(68,219,286)(69,220,287)(70,221,288)(71,222,281)(72,223,282)(81,89,153)(82,90,154)(83,91,155)(84,92,156)(85,93,157)(86,94,158)(87,95,159)(88,96,160)(97,161,169)(98,162,170)(99,163,171)(100,164,172)(101,165,173)(102,166,174)(103,167,175)(104,168,176)(105,143,182)(106,144,183)(107,137,184)(108,138,177)(109,139,178)(110,140,179)(111,141,180)(112,142,181)(113,121,185)(114,122,186)(115,123,187)(116,124,188)(117,125,189)(118,126,190)(119,127,191)(120,128,192)(129,193,201)(130,194,202)(131,195,203)(132,196,204)(133,197,205)(134,198,206)(135,199,207)(136,200,208), (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), (1,115,41,211)(2,120,42,216)(3,117,43,213)(4,114,44,210)(5,119,45,215)(6,116,46,212)(7,113,47,209)(8,118,48,214)(9,74,187,259)(10,79,188,264)(11,76,189,261)(12,73,190,258)(13,78,191,263)(14,75,192,260)(15,80,185,257)(16,77,186,262)(17,82,195,267)(18,87,196,272)(19,84,197,269)(20,81,198,266)(21,86,199,271)(22,83,200,268)(23,88,193,265)(24,85,194,270)(25,90,203,275)(26,95,204,280)(27,92,205,277)(28,89,206,274)(29,94,207,279)(30,91,208,276)(31,96,201,273)(32,93,202,278)(33,98,109,283)(34,103,110,288)(35,100,111,285)(36,97,112,282)(37,102,105,287)(38,99,106,284)(39,104,107,281)(40,101,108,286)(49,227,146,123)(50,232,147,128)(51,229,148,125)(52,226,149,122)(53,231,150,127)(54,228,151,124)(55,225,152,121)(56,230,145,126)(57,235,154,131)(58,240,155,136)(59,237,156,133)(60,234,157,130)(61,239,158,135)(62,236,159,132)(63,233,160,129)(64,238,153,134)(65,243,162,139)(66,248,163,144)(67,245,164,141)(68,242,165,138)(69,247,166,143)(70,244,167,140)(71,241,168,137)(72,246,161,142)(169,181,223,254)(170,178,224,251)(171,183,217,256)(172,180,218,253)(173,177,219,250)(174,182,220,255)(175,179,221,252)(176,184,222,249)>;
G:=Group( (1,170,154)(2,171,155)(3,172,156)(4,173,157)(5,174,158)(6,175,159)(7,176,160)(8,169,153)(9,243,25)(10,244,26)(11,245,27)(12,246,28)(13,247,29)(14,248,30)(15,241,31)(16,242,32)(17,227,33)(18,228,34)(19,229,35)(20,230,36)(21,231,37)(22,232,38)(23,225,39)(24,226,40)(41,224,57)(42,217,58)(43,218,59)(44,219,60)(45,220,61)(46,221,62)(47,222,63)(48,223,64)(49,283,267)(50,284,268)(51,285,269)(52,286,270)(53,287,271)(54,288,272)(55,281,265)(56,282,266)(65,275,259)(66,276,260)(67,277,261)(68,278,262)(69,279,263)(70,280,264)(71,273,257)(72,274,258)(73,161,89)(74,162,90)(75,163,91)(76,164,92)(77,165,93)(78,166,94)(79,167,95)(80,168,96)(81,145,97)(82,146,98)(83,147,99)(84,148,100)(85,149,101)(86,150,102)(87,151,103)(88,152,104)(105,199,127)(106,200,128)(107,193,121)(108,194,122)(109,195,123)(110,196,124)(111,197,125)(112,198,126)(113,184,129)(114,177,130)(115,178,131)(116,179,132)(117,180,133)(118,181,134)(119,182,135)(120,183,136)(137,201,185)(138,202,186)(139,203,187)(140,204,188)(141,205,189)(142,206,190)(143,207,191)(144,208,192)(209,249,233)(210,250,234)(211,251,235)(212,252,236)(213,253,237)(214,254,238)(215,255,239)(216,256,240), (1,146,74)(2,147,75)(3,148,76)(4,149,77)(5,150,78)(6,151,79)(7,152,80)(8,145,73)(9,211,227)(10,212,228)(11,213,229)(12,214,230)(13,215,231)(14,216,232)(15,209,225)(16,210,226)(17,25,235)(18,26,236)(19,27,237)(20,28,238)(21,29,239)(22,30,240)(23,31,233)(24,32,234)(33,243,251)(34,244,252)(35,245,253)(36,246,254)(37,247,255)(38,248,256)(39,241,249)(40,242,250)(41,49,259)(42,50,260)(43,51,261)(44,52,262)(45,53,263)(46,54,264)(47,55,257)(48,56,258)(57,267,275)(58,268,276)(59,269,277)(60,270,278)(61,271,279)(62,272,280)(63,265,273)(64,266,274)(65,224,283)(66,217,284)(67,218,285)(68,219,286)(69,220,287)(70,221,288)(71,222,281)(72,223,282)(81,89,153)(82,90,154)(83,91,155)(84,92,156)(85,93,157)(86,94,158)(87,95,159)(88,96,160)(97,161,169)(98,162,170)(99,163,171)(100,164,172)(101,165,173)(102,166,174)(103,167,175)(104,168,176)(105,143,182)(106,144,183)(107,137,184)(108,138,177)(109,139,178)(110,140,179)(111,141,180)(112,142,181)(113,121,185)(114,122,186)(115,123,187)(116,124,188)(117,125,189)(118,126,190)(119,127,191)(120,128,192)(129,193,201)(130,194,202)(131,195,203)(132,196,204)(133,197,205)(134,198,206)(135,199,207)(136,200,208), (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), (1,115,41,211)(2,120,42,216)(3,117,43,213)(4,114,44,210)(5,119,45,215)(6,116,46,212)(7,113,47,209)(8,118,48,214)(9,74,187,259)(10,79,188,264)(11,76,189,261)(12,73,190,258)(13,78,191,263)(14,75,192,260)(15,80,185,257)(16,77,186,262)(17,82,195,267)(18,87,196,272)(19,84,197,269)(20,81,198,266)(21,86,199,271)(22,83,200,268)(23,88,193,265)(24,85,194,270)(25,90,203,275)(26,95,204,280)(27,92,205,277)(28,89,206,274)(29,94,207,279)(30,91,208,276)(31,96,201,273)(32,93,202,278)(33,98,109,283)(34,103,110,288)(35,100,111,285)(36,97,112,282)(37,102,105,287)(38,99,106,284)(39,104,107,281)(40,101,108,286)(49,227,146,123)(50,232,147,128)(51,229,148,125)(52,226,149,122)(53,231,150,127)(54,228,151,124)(55,225,152,121)(56,230,145,126)(57,235,154,131)(58,240,155,136)(59,237,156,133)(60,234,157,130)(61,239,158,135)(62,236,159,132)(63,233,160,129)(64,238,153,134)(65,243,162,139)(66,248,163,144)(67,245,164,141)(68,242,165,138)(69,247,166,143)(70,244,167,140)(71,241,168,137)(72,246,161,142)(169,181,223,254)(170,178,224,251)(171,183,217,256)(172,180,218,253)(173,177,219,250)(174,182,220,255)(175,179,221,252)(176,184,222,249) );
G=PermutationGroup([[(1,170,154),(2,171,155),(3,172,156),(4,173,157),(5,174,158),(6,175,159),(7,176,160),(8,169,153),(9,243,25),(10,244,26),(11,245,27),(12,246,28),(13,247,29),(14,248,30),(15,241,31),(16,242,32),(17,227,33),(18,228,34),(19,229,35),(20,230,36),(21,231,37),(22,232,38),(23,225,39),(24,226,40),(41,224,57),(42,217,58),(43,218,59),(44,219,60),(45,220,61),(46,221,62),(47,222,63),(48,223,64),(49,283,267),(50,284,268),(51,285,269),(52,286,270),(53,287,271),(54,288,272),(55,281,265),(56,282,266),(65,275,259),(66,276,260),(67,277,261),(68,278,262),(69,279,263),(70,280,264),(71,273,257),(72,274,258),(73,161,89),(74,162,90),(75,163,91),(76,164,92),(77,165,93),(78,166,94),(79,167,95),(80,168,96),(81,145,97),(82,146,98),(83,147,99),(84,148,100),(85,149,101),(86,150,102),(87,151,103),(88,152,104),(105,199,127),(106,200,128),(107,193,121),(108,194,122),(109,195,123),(110,196,124),(111,197,125),(112,198,126),(113,184,129),(114,177,130),(115,178,131),(116,179,132),(117,180,133),(118,181,134),(119,182,135),(120,183,136),(137,201,185),(138,202,186),(139,203,187),(140,204,188),(141,205,189),(142,206,190),(143,207,191),(144,208,192),(209,249,233),(210,250,234),(211,251,235),(212,252,236),(213,253,237),(214,254,238),(215,255,239),(216,256,240)], [(1,146,74),(2,147,75),(3,148,76),(4,149,77),(5,150,78),(6,151,79),(7,152,80),(8,145,73),(9,211,227),(10,212,228),(11,213,229),(12,214,230),(13,215,231),(14,216,232),(15,209,225),(16,210,226),(17,25,235),(18,26,236),(19,27,237),(20,28,238),(21,29,239),(22,30,240),(23,31,233),(24,32,234),(33,243,251),(34,244,252),(35,245,253),(36,246,254),(37,247,255),(38,248,256),(39,241,249),(40,242,250),(41,49,259),(42,50,260),(43,51,261),(44,52,262),(45,53,263),(46,54,264),(47,55,257),(48,56,258),(57,267,275),(58,268,276),(59,269,277),(60,270,278),(61,271,279),(62,272,280),(63,265,273),(64,266,274),(65,224,283),(66,217,284),(67,218,285),(68,219,286),(69,220,287),(70,221,288),(71,222,281),(72,223,282),(81,89,153),(82,90,154),(83,91,155),(84,92,156),(85,93,157),(86,94,158),(87,95,159),(88,96,160),(97,161,169),(98,162,170),(99,163,171),(100,164,172),(101,165,173),(102,166,174),(103,167,175),(104,168,176),(105,143,182),(106,144,183),(107,137,184),(108,138,177),(109,139,178),(110,140,179),(111,141,180),(112,142,181),(113,121,185),(114,122,186),(115,123,187),(116,124,188),(117,125,189),(118,126,190),(119,127,191),(120,128,192),(129,193,201),(130,194,202),(131,195,203),(132,196,204),(133,197,205),(134,198,206),(135,199,207),(136,200,208)], [(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)], [(1,115,41,211),(2,120,42,216),(3,117,43,213),(4,114,44,210),(5,119,45,215),(6,116,46,212),(7,113,47,209),(8,118,48,214),(9,74,187,259),(10,79,188,264),(11,76,189,261),(12,73,190,258),(13,78,191,263),(14,75,192,260),(15,80,185,257),(16,77,186,262),(17,82,195,267),(18,87,196,272),(19,84,197,269),(20,81,198,266),(21,86,199,271),(22,83,200,268),(23,88,193,265),(24,85,194,270),(25,90,203,275),(26,95,204,280),(27,92,205,277),(28,89,206,274),(29,94,207,279),(30,91,208,276),(31,96,201,273),(32,93,202,278),(33,98,109,283),(34,103,110,288),(35,100,111,285),(36,97,112,282),(37,102,105,287),(38,99,106,284),(39,104,107,281),(40,101,108,286),(49,227,146,123),(50,232,147,128),(51,229,148,125),(52,226,149,122),(53,231,150,127),(54,228,151,124),(55,225,152,121),(56,230,145,126),(57,235,154,131),(58,240,155,136),(59,237,156,133),(60,234,157,130),(61,239,158,135),(62,236,159,132),(63,233,160,129),(64,238,153,134),(65,243,162,139),(66,248,163,144),(67,245,164,141),(68,242,165,138),(69,247,166,143),(70,244,167,140),(71,241,168,137),(72,246,161,142),(169,181,223,254),(170,178,224,251),(171,183,217,256),(172,180,218,253),(173,177,219,250),(174,182,220,255),(175,179,221,252),(176,184,222,249)]])
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 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | M4(2) | C3×M4(2) |
| kernel | C32×C8⋊C4 | C122 | C6×C24 | C3×C8⋊C4 | C3×C24 | C6×C12 | C4×C12 | C2×C24 | C24 | C2×C12 | C3×C6 | C6 |
| # reps | 1 | 1 | 2 | 8 | 8 | 4 | 8 | 16 | 64 | 32 | 4 | 32 |
Matrix representation of C32×C8⋊C4 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 61 | 62 |
| 0 | 0 | 62 | 12 |
| 72 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,72,0,0,0,0,61,62,0,0,62,12],[72,0,0,0,0,27,0,0,0,0,0,72,0,0,1,0] >;
C32×C8⋊C4 in GAP, Magma, Sage, TeX
C_3^2\times C_8\rtimes C_4
% in TeX
G:=Group("C3^2xC8:C4"); // GroupNames label
G:=SmallGroup(288,315);
// by ID
G=gap.SmallGroup(288,315);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,252,2045,512,172]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=d^4=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^5>;
// generators/relations